Content-Type: multipart/mixed; boundary="-------------9811161127406" This is a multi-part message in MIME format. ---------------9811161127406 Content-Type: text/plain; name="98-717.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-717.keywords" countable Markov maps, decay of correlations, transfer operator, birth and death processes ---------------9811161127406 Content-Type: application/postscript; name="artdenum.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="artdenum.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: artdenum.dvi %%Pages: 33 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips artdenum.dvi -o artdenum.ps %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1998.11.16:1807 %%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 /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{ rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 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put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet 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b(will)i(denote)f(b)o(y)f Fs(K)s Fu(\()p Fs(f)5 b Fu(\))19 b(the)g(smallest)h(p)q(ositiv)o(e)h(n)o(um)o(b)q(er)f(satisfying)f (this)h(prop)q(ert)o(y)l(,)g(it)g(will)h(b)q(e)f(called)-12 2271 y(Lipsc)o(hitz)c(constan)o(t)e(of)f Fs(f)5 b Fu(.)20 b(Let)14 b Fs(L)h Fu(b)q(e)f(the)h(space)f(of)g(u.l.L.)g(functions)h (that)f(are)g(b)q(ounded)h(on)f(\006.)20 b(F)l(or)13 b Fs(f)18 b Fn(2)13 b Fs(L)p Fu(,)-12 2328 y(let)j Fn(k)p Fs(f)5 b Fn(k)12 b Fu(=)h(max\()p Fs(K)s Fu(\()p Fs(f)5 b Fu(\))p Fs(;)j Fn(k)p Fs(f)d Fn(k)489 2335 y Fq(1)524 2328 y Fu(\),)14 b(this)i(de\014nes)g(a)f(norm)g(on)g Fs(L)g Fu(whic)o(h)h(turns)f Fs(L)g Fu(in)o(to)g(a)g(Banac)o(h)h (space.)-12 2384 y(Let)g Fn(F)k Fu(b)q(e)c(the)g(b)q(orelian)h (sigma-algebra)f(on)f(\006,)h Fs(m)f Fu(b)q(e)i(a)e(b)q(orelian)i (probabilit)o(y)g(on)f(\006)f(whose)h(supp)q(ort)f(is)i(\006)-12 2441 y(and)h(\010)f(a)g(u.l.L.)g(function.)27 b(According)18 b(to)f(thermo)q(dynamic)h(formalism)f(w)o(e)g(will)i(call)g(\010)e(a)g Ft(p)n(otential)p Fu(.)25 b(W)l(e)-12 2497 y(will)17 b(alw)o(a)o(ys)e(assume)g(that)f(\010)h(satis\014es)g(the)h(follo)o (wing)g(assumptions.)-12 2578 y Fo(Standing)j(assumptions)f(on)f(the)h (p)q(oten)o(tial)i(\(SA\).)44 2672 y Fu(1.)i(\010)15 b(is)h(a)e(u.l.L.)i(function,)939 2796 y(2)p eop %%Page: 3 3 3 2 bop 44 3 a Fu(2.)22 b(sup)102 42 y Fr(x)p Fq(2)p Fp(\006)189 -40 y Fi(X)178 55 y Fr(\033)q(y)q Fp(=)p Fr(x)273 3 y Fs(e)294 -16 y Fp(\010\()p Fr(y)q Fp(\))381 3 y Fs(<)14 b Fn(1)p Fu(,)i(so)f(the)h(transfer)e(op)q(erator)h Fn(L)1022 10 y Fp(\010)1066 3 y Fu(asso)q(ciated)g(to)g(\010)h(is)g(w)o (ell)h(de\014ned)g(and)e(acts)g(on)102 105 y Fs(C)135 112 y Fr(u)157 105 y Fu(\(\006\):)k(for)14 b Fs(f)k Fn(2)13 b Fs(C)443 112 y Fr(u)465 105 y Fu(\(\006\),)h Fs(x)f Fn(2)g Fu(\006)759 213 y Fn(L)790 220 y Fp(\010)818 213 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))12 b(=)978 170 y Fi(X)967 266 y Fr(\033)q(y)q Fp(=)p Fr(x)1062 213 y Fs(e)1083 195 y Fp(\010\()p Fr(y)q Fp(\))1157 213 y Fs(f)5 b Fu(\()p Fs(y)r Fu(\))p 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b(the)g(fact)f(that)f(\010)h(is)h(u.l.L.,) g(an)g(easy)f(computation)g(leads)h(to)f(the)h(follo)o(wing)g Ft(Bounde)n(d)g(Distortion)-12 1032 y(pr)n(op)n(erty)e(for)g Fn(L)270 1039 y Fp(\010)297 1032 y Fu(:)-12 1088 y(there)e(exists)h Fs(C)f(>)e Fu(0)i(suc)o(h)h(that)e(for)h(an)o(y)g Fs(x)g Fu(and)g Fs(y)i Fu(in)f(the)g(same)e(1-cylinder,)j(for)e Fs(n)e Fn(2)f Fj(N)p Fu(,)668 1157 y Fn(L)699 1138 y Fr(n)699 1168 y Fp(\010)727 1157 y Fo(1)p Fu(\()p Fs(x)p Fu(\))g Fn(\024)h(L)906 1138 y Fr(n)906 1168 y Fp(\010)934 1157 y Fo(1)p Fu(\()p Fs(y)r Fu(\))p Fs(e)1041 1138 y Fr(C)r(d)p Fp(\()p Fr(x;y)q Fp(\))1164 1157 y Fs(:)609 b Fu(\()p Fo(BD)p Fu(\))-12 1226 y(The)15 b(fact)g(that)f Fs(m)h Fu(is)h(a)e(conformal)h(measure)g(for)f Fn(L)895 1233 y Fp(\010)938 1226 y Fu(implies)j(that)e(if)g Fs(\026)e Fu(=)g Fs(hm)i Fu(with)h Fs(h)c Fn(2)h Fs(C)1619 1233 y Fr(u)1641 1226 y Fu(\(\006\))h(then,)h Fs(\026)h Fu(is)-12 1283 y Fs(\033)r Fu(-in)o(v)m(arian)o(t)e(if)g(and)g(only)g(if)g Fs(h)g 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Fs(c)539 1820 y Fr(n)562 1813 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))j(=)726 1747 y Fi(\014)726 1774 y(\014)726 1802 y(\014)726 1829 y(\014)741 1751 y(Z)795 1813 y Fs(f)5 b Fu(\()p Fs(g)11 b Fn(\016)f Fs(\033)934 1794 y Fr(n)957 1813 y Fu(\))p Fs(d\026)f Fn(\000)1081 1751 y Fi(Z)1134 1813 y Fs(f)c(d\026)1220 1751 y Fi(Z)1273 1813 y Fs(g)r(d\026)1348 1747 y Fi(\014)1348 1774 y(\014)1348 1802 y(\014)1348 1829 y(\014)1370 1813 y Fs(:)-12 1921 y Fu(The)16 b(measure)g Fs(\026)g Fu(is)h Ft(mixing)e Fu(if)i(and)f(only)g(if)h(the)f(co)q(e\016cien)o(ts)g Fs(c)1102 1928 y Fr(n)1126 1921 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))14 b(go)h(to)g(zero)h(for)f(an)o(y)h Fs(f)j Fn(2)14 b Fs(C)1734 1928 y Fr(u)1756 1921 y Fu(\(\006\))h(and)-12 1978 y Fs(g)g Fn(2)e Fs(L)99 1961 y Fp(1)119 1978 y Fu(\()p Fs(m)p Fu(\).)19 b(In)d(this)g(case,)g(estimates)f(on)g(the)h(sp)q(eed) g(of)f(con)o(v)o(ergence)h(to)f(zero)g(of)g Fs(c)1478 1985 y Fr(n)1501 1978 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))14 b(or)h(equiv)m(alen)o(tly)-12 2034 y(estimates)h(on)g(the)g Ft(de)n(c)n(ay)h(of)g(c)n(orr)n(elations)e Fu(ma)o(y)h(lead)g(to)g(the) g(Cen)o(tral)g(Limit)h(Theorem)f(\(see)g([Li3]\))g(and)g(to)-12 2091 y(the)k(determination)g(of)g(asymptotic)f(la)o(ws)g(for)g(en)o (trance)h(times)g(\(see)f([G,)14 b(S)q(])19 b(and)h([Sau)o(]\).)33 b(The)20 b(follo)o(wing)-12 2147 y(trivial)c(computation)f(relates)g (the)g(deca)o(y)g(of)g(correlations)g(to)f(the)h(asymptotic)g(b)q(eha)o (vior)g(of)g(the)g(iterates)g(of)-12 2204 y Fn(L)19 2211 y Fp(\010)47 2204 y Fu(:)574 2281 y Fs(c)594 2288 y Fr(n)617 2281 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))k(=)781 2215 y Fi(\014)781 2242 y(\014)781 2269 y(\014)781 2297 y(\014)797 2219 y(Z)842 2281 y Fu([)p Fn(L)886 2262 y Fr(n)886 2292 y Fp(\010)914 2281 y Fu(\()p Fs(f)5 b(h)p Fu(\))k Fn(\000)i Fs(hm)p Fu(\()p Fs(f)5 b(h)p Fu(\)])p Fs(g)r(dm)1314 2215 y Fi(\014)1314 2242 y(\014)1314 2269 y(\014)1314 2297 y(\014)1820 2281 y Fu(\(1.1\))-12 2389 y(so)16 b(that)f(if)h Fn(L)217 2373 y Fr(n)217 2403 y Fp(\010)245 2389 y Fs(f)j Fn(!)14 b Fs(hm)p Fu(\()p Fs(f)5 b Fu(\))15 b(in)i(some)f(reasonable)g (w)o(a)o(y)f(then)h Fs(\026)g Fu(is)h(mixing)f(and)g(estimates)g(on)g (the)g(sp)q(eed)h(of)-12 2446 y(this)f(con)o(v)o(ergence)f(w)o(ould)h (precise)g(the)f(deca)o(y)h(of)e(correlations.)-12 2502 y(Let)i(us)f(state)f(our)h(main)h(results.)939 2796 y(3)p eop %%Page: 4 4 4 3 bop -12 1 a Fh(1.1)56 b(Results.)-12 87 y Fu(Under)16 b(the)f(follo)o(wing)h(additional)h(h)o(yp)q(othesis:)530 156 y Fn(9)p Fs(M)h(>)13 b Fu(0)i(suc)o(h)h(that)e Fn(8)p Fs(n)f Fn(2)g Fj(N)h Fn(kL)1114 137 y Fr(n)1114 167 y Fp(\010)1142 156 y Fo(1)p Fn(k)1191 163 y Fq(1)1241 156 y Fn(\024)f Fs(M)5 b(:)471 b Fu(\()p Fo(K)p Fu(\))-12 225 y(w)o(e)15 b(ha)o(v)o(e)g(the)g(follo)o(wing)h(result.)-12 331 y Fo(Theorem)h(1.1)23 b Fg(Let)c Fu(\006)g Fg(b)q(e)g(ap)q(erio)q (dic,)j Fu(\010)d Fg(satisfying)g(\()p Fo(SA)p Fg(\))f(and)h(\()p Fo(K)p Fg(\),)f(then)h(there)g(exists)h(a)e(\014xed)i(p)q(oin)o(t)-12 387 y Fs(h)13 b Fn(2)g Fs(L)g Fg(for)h Fn(L)214 394 y Fp(\010)241 387 y Fg(,)g Fs(h)f(>)g Fu(0)g Fg(on)h Fu(\006)p Fg(,)f Fs(m)p Fu(\()p Fs(h)p Fu(\))f(=)h(1)p Fg(,)g(this)i(\014xed)f(p) q(oin)o(t)g(is)g(unique)h(up)f(to)g(m)o(ultiplication)i(b)o(y)d(a)h (constan)o(t.)-12 444 y(Moreo)o(v)o(er,)g(w)o(e)g(ha)o(v)o(e)h(the)h (follo)o(wing)f(con)o(v)o(ergence)h(for)e Fs(f)k Fn(2)13 b Fs(C)1079 451 y Fr(u)1101 444 y Fu(\(\006\))p Fg(:)779 546 y Fn(L)810 527 y Fr(n)810 557 y Fp(\010)837 546 y Fs(f)877 520 y Fr(n)p Fq(!1)887 546 y Fn(\000)-8 b(!)23 b Fs(hm)p Fu(\()p Fs(f)5 b Fu(\))p Fs(;)696 b Fu(\(1.2\))-12 648 y Fg(this)16 b(con)o(v)o(ergence)f(is)h(uniform)g(on)f(the)g (compact)g(subsets)g(of)g Fu(\006)g Fg(and)g(tak)o(es)g(place)h(in)g Fs(L)1530 632 y Fp(1)1549 648 y Fu(\()p Fs(m)p Fu(\))p Fg(.)-12 738 y Fu(The)i(pro)q(of)f(of)g(this)h(theorem)f(follo)o(ws)g (Sarig's)g(pro)q(of)g(\(theorem)g(4)g(in)i([Sa)o(]\))e(excepted)h(for)f (some)g(details)i(in)-12 794 y(the)c(construction)h(of)f Fs(h)p Fu(.)20 b(Let)15 b(us)h(giv)o(e)f(the)g(outline)i(of)e(the)g (argumen)o(ts.)-12 851 y(F)l(rom)h(\()p Fo(K)p Fu(\))g(and)g(\()p Fo(BD)p Fu(\),)g(w)o(e)g(deduce)i(that)e(the)g(sequence)i(\()p Fn(L)1072 834 y Fr(n)1072 864 y Fp(\010)1100 851 y Fo(1)p Fu(\))1144 858 y Fr(n)p Fq(2)p Fm(N)1231 851 y Fu(is)f(an)f(equicon)o (tin)o(uous)i(and)f(b)q(ounded)-12 907 y(sequence)g(of)d(elemen)o(ts)i (of)f Fs(L)p Fu(.)20 b(Let)782 1003 y Fs(Q)818 1010 y Fr(n)855 1003 y Fu(=)910 972 y(1)p 907 992 28 2 v 907 1034 a Fs(n)947 946 y Fr(n)p Fq(\000)p Fp(1)948 960 y Fi(X)949 1057 y Fr(p)p Fp(=0)1021 1003 y Fn(L)1052 981 y Fr(p)1052 1017 y Fp(\010)1080 1003 y Fo(1)p Fs(:)-12 1133 y Fu(Ascoli's)h(theorem)e(on)h(separable)g(sets)f(implies)j(that)d (the)h(sequence)h(\()p Fs(Q)1297 1140 y Fr(n)1320 1133 y Fu(\))1338 1140 y Fr(n)p Fq(2)p Fm(N)1428 1133 y Fu(admits)f(an)f (accum)o(ulation)-12 1189 y(p)q(oin)o(t)g(for)f(the)h(top)q(ology)f(of) h(uniform)g(con)o(v)o(ergence)g(on)f(compact)h(sets)f(and)h(for)f(the)h Fs(L)1562 1173 y Fp(1)1581 1189 y Fu(\()p Fs(m)p Fu(\))f(top)q(ology)g (b)o(y)-12 1246 y(Leb)q(esgue's)g(dominated)f(con)o(v)o(ergence)f (theorem.)24 b(Let)17 b Fs(h)f Fu(b)q(e)i(suc)o(h)e(an)h(accum)o (ulation)g(p)q(oin)o(t.)25 b(Using)17 b(\()p Fo(BD)p Fu(\))-12 1302 y(and)g(\()p Fo(K)p Fu(\),)e(w)o(e)h(get)f(that)h Fs(h)g Fu(b)q(elongs)h(to)f Fs(L)p Fu(.)23 b(Using)16 b(that)g Fs(m)g Fu(is)h(a)f(conformal)g(measure)g(whose)g(supp)q(ort)g (is)h(\006,)-12 1359 y(w)o(e)c(get)g(that)g Fs(h)h Fu(is)g(a)f(\014xed) h(p)q(oin)o(t)g(for)f Fn(L)668 1366 y Fp(\010)709 1359 y Fu(whic)o(h)h(is)g(non)g(zero)f(b)q(ecause)i(Leb)q(esgue's)f (dominated)g(con)o(v)o(ergence)-12 1415 y(theorem)h(implies)i(that)e Fs(m)p Fu(\()p Fs(h)p Fu(\))d(=)h(1.)20 b(No)o(w,)14 b(if)h Fs(h)p Fu(\()p Fs(x)p Fu(\))e(=)g(0)h(for)h(some)g Fs(x)e Fn(2)g Fu(\006,)h(since)i Fs(h)d Fn(\025)g Fu(0,)i(for)f(an)o(y) h Fs(n)e Fn(2)g Fj(N)h Fu(and)-12 1472 y(an)o(y)i Fs(z)f Fn(2)f(f)p Fs(y)k(=)d(\033)284 1455 y Fr(n)307 1472 y Fs(y)h Fu(=)e Fs(x)p Fn(g)p Fu(,)h Fs(h)p Fu(\()p Fs(z)r Fu(\))e(=)h(0.)21 b(Since)d(\006)d(is)h(irreducible)j(\(indeed)e(it)f (is)g(ap)q(erio)q(dic\),)i(this)e(set)f(is)i(dense)-12 1528 y(in)f(\006,)e(so)h(that)f Fs(h)f Fn(\021)g Fu(0)h(since)i(it)f (is)g(con)o(tin)o(uous.)21 b(This)15 b(con)o(tradicts)g(the)f(fact)h (that)f Fs(h)h Fu(is)g(non)g(zero.)20 b(So,)14 b Fs(h)f(>)g Fu(0.)-12 1585 y(The)i(rest)f(of)g(the)g(pro)q(of)g(follo)o(ws)h (Sarig's)f(pro)q(of)g(and)g(uses)h(some)f(general)h(argumen)o(ts)e(on)i (Mark)o(o)o(v)e(dynamics)-12 1641 y(from)i([A,D,U)n(].)1596 b Fn(4)-12 1754 y Fu(In)16 b(fact,)f(w)o(e)g(ha)o(v)o(e)g(a)g(more)g (precise)i(description)g(of)e(the)h(sp)q(ectrum)f(of)g Fn(L)1267 1761 y Fp(\010)1295 1754 y Fu(.)21 b(Suc)o(h)16 b(a)f(description)i(is)f(usual)h(in)-12 1810 y(quasi-compactness)i (setting.)28 b(Our)19 b(purp)q(ose)g(is)f(to)g(giv)o(e)g(estimates)g (on)g(the)g(deca)o(y)g(of)g(correlations)g(so)g(w)o(e)-12 1867 y(will)g(omit)f(the)f(pro)q(of)g(of)g(the)h(follo)o(wing)g(result) g(whic)o(h)g(ma)o(y)f(b)q(e)h(found)g(in)g([Ma)o(].)24 b(Let)16 b(us)h(just)f(remark)g(that)-12 1923 y(\()p Fo(K)p Fu(\))f(implies)i(that)d(the)i(sp)q(ectral)g(radius)f(of)g Fn(L)801 1930 y Fp(\010)844 1923 y Fu(on)g(the)h(space)f Fs(C)1140 1930 y Fr(u)1162 1923 y Fu(\(\006\))f(is)i(less)g(or)f(equal) h(than)f(1.)-12 2030 y Fo(Theorem)i(1.2)23 b Fg(Let)12 b Fu(\006)h Fg(b)q(e)g(irreducible,)j Fu(\010)c Fg(satis\014es)h(\()p Fo(SA)p Fg(\))e(and)i Fu(\()p Fo(K)p Fu(\))f Fg(then)g Fu(1)h Fg(is)g(a)f(simple)i(eigen)o(v)m(alue)h(for)c Fn(L)1885 2037 y Fp(\010)-12 2086 y Fg(acting)j(on)h Fs(L)p Fg(,)e(if)i Fs(h)f Fg(is)h(the)f(normalized)h(eigenfunction)h (then)f Fs(h)f Fg(is)h(strictly)f(p)q(ositiv)o(e)h(on)f Fu(\006)g Fg(and)h(the)f(in)o(v)m(arian)o(t)-12 2143 y(measure)h Fs(\026)e Fu(=)g Fs(hm)h Fg(is)h(ergo)q(dic.)21 b(Moreo)o(v)o(er,)13 b Fn(L)797 2150 y Fp(\010)839 2143 y Fg(has)i(only)g(\014nitely)h(man)o(y)e(eigen)o(v)m(alues)i(of)f(mo)q (dulus)g Fu(1)p Fg(,)f(there)-12 2199 y(are)e(all)h(simple.)20 b(If)12 b Fu(\006)g Fg(is)h(ap)q(erio)q(dic)h(then)e Fu(1)g Fg(is)g(the)h(only)f(eigen)o(v)m(alue)i(of)e(maximal)g(mo)q (dulus)h(and)g(the)f(in)o(v)m(arian)o(t)-12 2255 y(measure)j Fs(\026)e Fu(=)g Fs(hm)j Fg(is)f(mixing.)21 b(W)l(e)15 b(ha)o(v)o(e)g(the)h(con)o(v)o(ergence)f(for)g Fs(f)i Fn(2)c Fs(C)1246 2262 y Fr(u)1268 2255 y Fu(\(\006\))798 2358 y Fn(L)829 2339 y Fr(n)829 2369 y Fp(\010)857 2358 y Fs(f)896 2332 y Fr(n)p Fq(!1)906 2358 y Fn(\000)-7 b(!)22 b Fs(\031)r Fu(\()p Fs(f)5 b Fu(\))p Fs(;)-12 2460 y Fg(uniformly)16 b(on)f(compact)f(subsets)h(of)f Fu(\006)h Fg(and)g(in)h Fs(L)867 2443 y Fp(1)886 2460 y Fu(\()p Fs(m)p Fu(\))p Fg(,)e(where)h Fs(\031)h Fg(is)g(the)f(sp)q (ectral)g(pro)s(jection)g(on)f(the)h(\014nite)-12 2516 y(dimensional)h(space)e(asso)q(ciated)g(to)e(the)i(eigen)o(v)m(alues)i (of)d(mo)q(dulus)i Fu(1)p Fg(.)k(In)14 b(particular,)g(if)g Fu(\006)f Fg(is)i(ap)q(erio)q(dic)g(then)779 2618 y Fn(L)810 2600 y Fr(n)810 2630 y Fp(\010)837 2618 y Fs(f)877 2593 y Fr(n)p Fq(!1)887 2618 y Fn(\000)-8 b(!)23 b Fs(hm)p Fu(\()p Fs(f)5 b Fu(\))p Fs(:)939 2796 y Fu(4)p eop %%Page: 5 5 5 4 bop -12 1 a Fu(F)l(rom)14 b(no)o(w)g(on,)g(w)o(e)g(assume)g(that)g (\006)g(is)h(ap)q(erio)q(dic)h(and)f(\010)f(satis\014es)g(\()p Fo(K)p Fu(\))g(and)h(\()p Fo(SA)p Fu(\).)d(The)j(normalized)h(\014xed) -12 57 y(p)q(oin)o(t)21 b(\(giv)o(en)g(b)o(y)f(theorem)g(1.1\))f(for)h Fn(L)706 64 y Fp(\010)754 57 y Fu(will)i(alw)o(a)o(ys)e(b)q(e)h (denoted)g(b)o(y)f Fs(h)h Fu(and)f Fs(\026)h Fu(will)h(b)q(e)f(the)g (in)o(v)m(arian)o(t)-12 114 y(measure)e Fs(\026)g Fu(=)g Fs(hm)p Fu(.)31 b(W)l(e)19 b(will)i(giv)o(e)e(additional)h(conditions)h (under)e(whic)o(h)h(the)f(sp)q(eed)h(of)e(con)o(v)o(ergence)i(in)-12 170 y(\(1.2\))14 b(can)h(b)q(e)h(estimated.)-12 227 y(W)l(e)f(will)i (sa)o(y)e(that)f(\010)h(satis\014es)h(\()p Fo(Exp1)p Fu(\))e(if)166 296 y Fn(9)p Fs(k)215 303 y Fp(1)234 296 y Fs(;)23 b Fn(9)p Fs(n)322 303 y Fp(1)357 296 y Fu(suc)o(h)16 b(that)e Fn(8)p Fs(k)g(>)f(k)693 303 y Fp(1)712 296 y Fs(;)8 b Fn(9)p Fs(\032)782 303 y Fr(k)815 296 y Fs(<)13 b Fu(1)i(suc)o(h)h(that)e Fn(8)p Fs(n)f(>)g(n)1242 303 y Fp(1)1262 296 y Fs(;)i Fu(sup)1283 337 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])1374 296 y Fn(L)1405 277 y Fr(k)1405 307 y Fp(\010)1433 296 y Fo(1)p Fu(\()p Fs(x)p Fu(\))d Fn(\024)h Fs(\032)1605 303 y Fr(k)1626 296 y Fs(:)107 b Fu(\()p Fo(Exp1)p Fu(\))7 401 y(\()p Fo(Exp1)p Fu(\))19 b(means)g(that)g(the)g(cylinders)i(close)f(to)e(in\014nit)o(y)j(do)e (not)g(con)o(tribute)g(to)q(o)g(m)o(uc)o(h)g(to)g(the)g(transfer)-12 457 y(op)q(erator,)g(in)g(fact)f(their)i(con)o(tribution)f(is)g (assumed)g(to)f(b)q(e)i(uniformly)g(strictly)f(smaller)g(than)g(one.)31 b(This)-12 513 y(condition)24 b(is)e(su\016cien)o(t)h(to)f(guaran)o(t)o (y)f(exp)q(onen)o(tial)j(deca)o(y)e(of)g(correlations)g(for)g(observ)m (ables)h(in)g Fs(L)f Fu(\(see)-12 570 y(theorem)15 b(1.3)f(b)q(elo)o (w\).)-12 626 y(The)j(system)f(\(\006)p Fs(;)8 b(\033)r Fu(\))14 b(is)j Ft(without)i(big)e(br)n(anches)g(at)h(in\014nity)d Fu(if)i(it)f(exists)h Fs(K)h Fn(2)d Fj(N)g Fu(suc)o(h)i(that)f(for)g (an)o(y)g Fs(n)f Fn(2)g Fj(N)p Fu(,)-12 683 y(for)f Fs(x)e Fn(2)h Fu([)p Fs(n)p Fu(],)h Fs(\033)r(x)g Fu(b)q(elongs)h(to)f([)p Fs(p)p Fu(])f(with)i Fs(p)d Fn(\025)h Fs(n)c Fn(\000)g Fs(K)s Fu(.)19 b(In)c(other)f(w)o(ords,)f(the)i(matrix)e(whic)o(h)j (de\014nes)f(\006)f(has)g(the)-12 739 y(follo)o(wing)i(form:)686 792 y(0)692 838 y(.)692 855 y(.)692 872 y(.)677 928 y Fs(K)723 756 y Fi(0)723 836 y(B)723 863 y(B)723 891 y(B)723 918 y(B)723 945 y(B)723 973 y(B)723 1000 y(B)723 1027 y(B)723 1054 y(B)723 1082 y(B)723 1109 y(B)723 1138 y(@)784 793 y Fn(\003)41 b(\001)8 b(\001)g(\001)40 b(\001)8 b(\001)g(\001)39 b(\001)8 b(\001)g(\001)40 b(\001)8 b(\001)g(\001)789 840 y Fu(.)789 856 y(.)789 873 y(.)784 929 y Fn(\003)784 986 y Fu(0)56 b Fn(\003)789 1032 y Fu(.)789 1049 y(.)789 1066 y(.)851 1037 y(.)868 1049 y(.)886 1062 y(.)945 1037 y(.)963 1049 y(.)981 1062 y(.)784 1122 y(0)151 b(0)71 b Fn(\003)789 1169 y Fu(.)789 1185 y(.)789 1202 y(.)1040 1173 y(.)1058 1185 y(.)1075 1198 y(.)1134 1173 y(.)1152 1185 y(.)1170 1198 y(.)1206 756 y Fi(1)1206 836 y(C)1206 863 y(C)1206 891 y(C)1206 918 y(C)1206 945 y(C)1206 973 y(C)1206 1000 y(C)1206 1027 y(C)1206 1054 y(C)1206 1082 y(C)1206 1109 y(C)1206 1138 y(A)-12 1285 y Fu(with)16 b Fn(\003)c(2)h(f)p Fu(0)p Fs(;)8 b Fu(1)p Fn(g)p Fu(.)18 b(If)d Fs(n)386 1292 y Fp(1)422 1285 y Fu(and)g Fs(N)i Fn(\025)c Fs(n)639 1292 y Fp(1)674 1285 y Fu(are)i(\014xed)h(in)o (tegers,)f(let)g(us)h(note)301 1387 y Fs(\016)323 1369 y Fq(0)321 1399 y Fr(k)q(;j)381 1387 y Fu(:=)c(sup)q Fn(fL)564 1369 y Fr(k)564 1399 y Fp(\010)592 1387 y Fo(1)p Fu(\()p Fs(x)p Fu(\))i Fs(=)h(x)e Fn(2)g Fu([)p Fs(n)p Fu(])p Fs(;)21 b(N)d Fn(\024)13 b Fs(n)f Fn(\024)h Fs(N)i Fu(+)10 b Fs(k)q(K)s(j)s Fn(g)p Fs(;)22 b(j)15 b Fn(\025)e Fu(0)p Fs(;)22 b(k)14 b Fn(2)e Fj(N)p Fs(:)-12 1489 y Fu(W)l(e)17 b(will)i(sa)o(y)e(that)f(\010)h(satis\014es)h(\()p Fo(S-Exp1)p Fu(\))f(if)h(there)f(exists)h Fs(n)1090 1496 y Fp(1)1126 1489 y Fn(2)e Fj(N)g Fu(suc)o(h)h(that)g(for)g Fs(N)j Fn(\025)c Fs(n)1634 1496 y Fp(1)1654 1489 y Fu(,)i(there)f (exists)-12 1546 y Fs(k)12 1553 y Fp(1)32 1546 y Fu(\()p Fs(N)5 b Fu(\))14 b(suc)o(h)h(that)g(for)f Fs(k)g Fn(\025)f Fs(k)504 1553 y Fp(1)523 1546 y Fu(,)i(there)g(exists)h Fs(R)p Fu(\()p Fs(k)q Fu(\),)e(0)e Fs(<)h(R)p Fu(\()p Fs(k)q Fu(\))f Fs(<)h(N)i Fu(+)10 b Fs(k)q(K)18 b Fu(and)473 1641 y Fs(\016)495 1622 y Fq(0)493 1653 y Fr(k)q(;j)553 1641 y Fn(\024)601 1577 y Fi(\022)635 1641 y Fu(1)10 b Fn(\000)763 1610 y Fs(R)p Fu(\()p Fs(k)q Fu(\))p 718 1631 186 2 v 718 1672 a Fs(N)15 b Fu(+)10 b Fs(K)s(k)q(j)908 1577 y Fi(\023)942 1588 y Fr(\013)974 1641 y Fs(;)e(\013)k(>)h Fu(0)p Fs(;)22 b Fn(8)p Fs(j)15 b Fn(\025)e Fu(0)p Fs(:)414 b Fu(\()p Fo(S-Exp1)p Fu(\))-12 1735 y(\()p Fo(S-Exp1)p Fu(\))17 b(means)g(that)f(the)h(con)o(tribution)h(to)e(the)h(transfer)f (op)q(erator)g(of)g(the)h(cylinders)i(close)f(to)e(in\014nit)o(y)-12 1791 y(is)i(strictly)f(smaller)g(than)g(one)g(but)g(not)g(uniformly)l (.)26 b(Under)17 b(this)g(condition)h(and)f(the)g(assumption)g(that)g (\006)-12 1848 y(has)e(no)g(big)g(branc)o(hes)h(at)e(in\014nit)o(y)l(,) i(w)o(e)f(can)g(estimate)g(the)g(deca)o(y)g(of)f(correlation)h(for)g (observ)m(ables)h(in)g Fs(L)e Fu(\(see)-12 1904 y(theorem)h(1.4)f(b)q (elo)o(w\).)-12 1961 y(Before)h(stating)g(our)g(main)h(results,)f(let)h (us)f(remark)f(that:)655 2054 y(\()p Fo(Exp1)p Fu(\))g Fn(\))i Fu(\()p Fo(S-Exp1)p Fu(\))f Fn(\))h Fu(\()p Fo(K)p Fu(\).)-12 2148 y(The)g(\014rst)e(implication)k(is)e(trivial:)k(tak)o (e)15 b Fs(\016)732 2120 y Ff(0)730 2162 y Fr(k)q(;j)790 2148 y Fu(=)e Fs(\032)862 2155 y Fr(k)898 2148 y Fu(for)i(all)h Fs(j)f Fn(2)e Fj(N)p Fu(.)18 b(Let)e(us)f(pro)o(v)o(e)g(the)g(second)h (implication.)-12 2205 y(Let)g Fs(N)h Fn(\025)c Fs(n)199 2212 y Fp(1)234 2205 y Fu(b)q(e)j(\014xed)g(and)f(let)h Fs(k)584 2212 y Fp(1)616 2205 y Fu(=)d Fs(k)688 2212 y Fp(1)707 2205 y Fu(\()p Fs(N)5 b Fu(\),)14 b(\()p Fo(S-Exp1)p Fu(\))h(implies)i(that)d(for)h Fs(n)e Fn(\025)g Fs(N)5 b Fu(,)14 b Fs(k)g Fn(\025)f Fs(k)1615 2212 y Fp(1)1634 2205 y Fu(,)792 2307 y(sup)784 2348 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])876 2307 y Fn(L)907 2288 y Fr(k)907 2318 y Fp(\010)935 2307 y Fo(1)p Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fu(1)p Fs(;)-12 2445 y Fu(moreo)o(v)o(er,)h(since)i(w)o(e)f(alw)o(a)o (ys)g(assume)g(that)f Fn(kL)830 2452 y Fp(\010)858 2445 y Fo(1)p Fn(k)907 2452 y Fq(1)957 2445 y Fs(<)f Fn(1)i Fu(\()p Fo(SA)p Fu(\),)f(w)o(e)h(ha)o(v)o(e)770 2547 y(sup)763 2587 y Fr(k)q()14 b Fu(0)i(suc)o(h)g(that)f(for)h(an)o(y)f Fs(n)g Fn(2)f Fj(N)p Fu(,)g Fn(kL)1543 519 y Fr(k)1543 549 y Fp(\010)1571 535 y Fo(1)p Fn(k)1620 542 y Fq(1)1671 535 y Fn(\024)h Fs(M)5 b Fu(,)15 b(whic)o(h)-12 592 y(is)h(\()p Fo(K)p Fu(\).)1749 b Fn(4)-12 705 y Fu(Finally)l(,)23 b(let)d(us)g(recall)h(that)f(t)o(w)o(o)e(p)q(oten)o(tial)j(\011)843 712 y Fp(1)883 705 y Fu(and)f(\011)1011 712 y Fp(2)1051 705 y Fu(are)g Ft(c)n(ohomolo)n(gous)g Fu(if)g(there)g(exists)h(a)e(p)q (ositiv)o(e)-12 761 y(function)d Fs(v)f Fn(2)d Fs(C)278 768 y Fr(u)300 761 y Fu(\(\006\))j(suc)o(h)g(that)g(\011)620 768 y Fp(1)652 761 y Fu(=)e(\011)735 768 y Fp(2)766 761 y Fu(+)d Fs(v)i Fn(\000)e Fs(v)i Fn(\016)e Fs(\033)r Fu(.)19 b(The)d(function)g Fs(v)h Fu(is)e(called)i Ft(change)f(of)h(p)n (otential)p Fu(.)-12 866 y Fo(Theorem)g(1.3)23 b Fg(Let)15 b Fu(\006)g Fg(b)q(e)h(ap)q(erio)q(dic.)22 b(If)16 b Fu(\010)f Fg(satis\014es)g(\()p Fo(SA)p Fg(\))f(and)i(\()p Fo(Exp1)p Fg(\))e(then)i Fn(L)1477 873 y Fp(\010)1520 866 y Fg(is)g(quasi-compact)f(on)-12 922 y Fs(L)j Fg(so)f(w)o(e)g(ha)o (v)o(e)g(the)h(follo)o(wing)g(exp)q(onen)o(tial)h(deca)o(y)e(of)g (correlations:)25 b(there)18 b(exist)g Fu(0)e Fs(<)h(\015)h(<)f Fu(1)g Fg(and)h Fs(C)h(>)e Fu(0)-12 978 y Fg(suc)o(h)f(that)e(for)h Fs(f)j Fn(2)12 b Fs(L)p Fg(,)j Fs(g)f Fn(2)f Fs(L)510 962 y Fp(1)530 978 y Fu(\()p Fs(m)p Fu(\))p Fg(,)691 1077 y Fs(c)711 1084 y Fr(n)734 1077 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))j Fn(\024)i Fs(C)18 b(\015)976 1058 y Fr(n)998 1077 y Fn(k)p Fs(f)5 b Fn(k)15 b(k)p Fs(g)r Fn(k)1156 1087 y Fr(L)1180 1077 y Fk(1)s Fs(:)609 b Fu(\(1.3\))-12 1176 y Fg(Moreo)o(v)o(er,)14 b(\()p Fo(Exp1)p Fg(\))g(is)i(a)f (necessary)g(condition)i(for)d(quasi-compactness)i(in)g(the)f(follo)o (wing)h(sense:)-12 1232 y(let)11 b Fu(\010)g Fg(v)o(erify)g(\()p Fo(K)p Fg(\))f(and)h(\()p Fo(SA)p Fg(\))e(then)i Fn(L)630 1239 y Fp(\010)669 1232 y Fg(is)g(quasi-compact)g(on)g Fs(L)f Fg(if)i(and)f(only)g(if)g(there)g(exists)g Fu(\011)g Fg(cohomologous)-12 1289 y(to)j Fu(\010)g Fg(with)h(a)f(c)o(hange)h(of) f(p)q(oten)o(tial)h(in)g Fs(L)f Fg(and)h(b)q(ounded)h(a)o(w)o(a)o(y)d (from)h(zero)g(suc)o(h)h(that)e Fn(L)1537 1296 y Fp(\011)1582 1289 y Fg(satis\014es)h(\()p Fo(Exp1)p Fg(\).)-12 1377 y Fu(Let)23 b(us)g(recall)g(that)f(a)g(linear)i(op)q(erator)e Fs(P)29 b Fu(on)22 b(a)h(Banac)o(h)f(space)h Fs(B)i Fu(is)e Ft(quasi-c)n(omp)n(act)g Fu(if)g(there)g(exists)-12 1434 y(0)13 b Fs(<)g Fu(\002)g Fs(<)f Fu(1)i(suc)o(h)h(that)e(if)i Fs(\025)e Fu(b)q(elong)i(to)f(the)g(sp)q(ectrum)h(of)e Fs(P)21 b Fu(and)14 b Fn(j)p Fs(\025)p Fn(j)e Fs(>)h(s)p Fu(\()p Fs(P)6 b Fu(\)\002)14 b(where)h Fs(s)p Fu(\()p Fs(P)6 b Fu(\))14 b(is)g(the)h(sp)q(ectral)-12 1490 y(radius)h(of)f Fs(P)21 b Fu(then)16 b Fs(\025)f Fu(is)g(an)g(eigen)o(v)m(alue)j(with)d (\014nite)h(m)o(ultiplicit)o(y)l(.)-12 1547 y(The)g(fact)f(that)g(\()p Fo(Exp1)p Fu(\))g(is)h(a)g(necessary)f(condition)i(for)e (quasi-compactness)i(in)f(the)g(sense)g(of)f(theorem)g(1.3)-12 1603 y(is)i(easy)f(to)f(see.)23 b(Indeed,)17 b(let)f(us)h(assume)f (that)f Fn(L)861 1610 y Fp(\010)905 1603 y Fu(is)h(quasi-compact)h(on)f Fs(L)p Fu(,)f(b)q(ecause)i(of)f(theorem)g(1.1,)f(w)o(e)-12 1660 y(ha)o(v)o(e)g(for)g Fs(f)i Fn(2)c Fs(L)p Fu(,)i Fs(k)f Fn(2)f Fj(N)p Fu(,)671 1716 y Fn(kL)725 1697 y Fr(k)725 1727 y Fp(\010)753 1716 y Fs(f)i Fn(\000)10 b Fs(hm)p Fu(\()p Fs(f)5 b Fu(\))p Fn(k)12 b(\024)h Fs(C)18 b(\015)1125 1697 y Fr(k)1145 1716 y Fn(k)p Fs(f)5 b Fn(k)p Fs(;)589 b Fu(\(1.4\))-12 1798 y(with)18 b Fs(C)i(>)e Fu(0)f(and)h(0)f Fs(<)g(\015)i(<)f Fu(1.)27 b(Let)18 b(\011)h(b)q(e)f(cohomologous)f(to)h(\010)f(with)h(c)o(hange)g(of)g(p)q (oten)o(tial)g Fs(v)i Fu(in)e Fs(L)g Fu(and)-12 1854 y(b)q(ounded)f(a)o(w)o(a)o(y)c(from)i(zero.)20 b(W)l(e)15 b(ha)o(v)o(e:)801 1927 y Fn(L)832 1908 y Fr(k)832 1938 y Fp(\011)862 1927 y Fo(1)e Fu(=)954 1896 y(1)p 954 1917 24 2 v 954 1958 a Fs(v)982 1927 y Fn(L)1013 1908 y Fr(k)1013 1938 y Fp(\010)1041 1927 y Fu(\()p Fs(v)r Fu(\))-12 2022 y(so,)i(using)h(\(1.4\))d(w)o(e)i(get:)605 2142 y Fn(L)636 2123 y Fr(k)636 2153 y Fp(\011)666 2142 y Fo(1)41 b Fu(=)816 2111 y(1)p 815 2131 V 815 2173 a Fs(v)844 2142 y Fu([)p Fn(L)888 2123 y Fr(k)888 2153 y Fp(\010)915 2142 y Fu(\()p Fs(v)r Fu(\))9 b Fn(\000)i Fs(hm)p Fu(\()p Fs(v)r Fu(\)])e(+)1228 2111 y Fs(h)p 1228 2131 27 2 v 1229 2173 a(v)1259 2142 y(m)p Fu(\()p Fs(v)r Fu(\))543 2254 y Fn(L)574 2236 y Fr(k)574 2266 y Fp(\011)604 2254 y Fo(1)p Fu(\()p Fs(x)p Fu(\))41 b Fn(\024)h Fs(C)860 2224 y Fn(k)p Fs(v)r Fn(k)p 851 2244 87 2 v 851 2286 a Fu(inf)12 b Fs(v)942 2254 y(\015)969 2236 y Fr(k)1000 2254 y Fu(+)1050 2224 y Fs(h)p 1050 2244 27 2 v 1051 2286 a(v)1081 2254 y Fu(\()p Fs(x)p Fu(\))p Fs(m)p Fu(\()p Fs(v)r Fu(\))p Fs(:)-12 2366 y Fu(It)17 b(is)h(alw)o(a)o(ys)e(p)q(ossible)j(to)e(\014nd)h Fs(v)f Fn(2)g Fs(L)g Fu(\(indeed,)h Fs(v)h Fu(can)e(b)q(e)h(c)o(hosen)g (to)e(b)q(e)i(constan)o(t)e(on)h(1-cylinders\))i(suc)o(h)-12 2422 y(that)c Fs(m)p Fu(\()p Fs(v)r Fu(\))c(=)i(1,)i(inf)d Fs(v)i(>)f Fu(0)i(and)776 2500 y(sup)770 2539 y Fr(n)p Fq(\025)p Fr(N)866 2500 y Fu(sup)858 2542 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])955 2470 y Fs(h)p 955 2490 V 956 2531 a(v)987 2500 y Fu(\()p Fs(x)p Fu(\))c Fs(<)i Fu(1)-12 2615 y(pro)o(vided)j Fs(N)k Fu(is)c(big)f(enough.)21 b(This)15 b(conclude)i(the)e(pro)q(of)g(of)g(the)g(necessit)o(y)h(part) f(of)g(theorem)f(1.3.)124 b Fn(4)-12 2672 y Fu(The)22 b(rest)e(of)h(theorem)g(1.3)f(will)j(b)q(e)f(pro)o(v)o(en)f(in)h (section)g(2.)37 b(This)22 b(w)o(on't)e(b)q(e)i(done)g(using)f(the)h (standard)939 2796 y(6)p eop %%Page: 7 7 7 6 bop -12 1 a Fu(approac)o(h)15 b(consisting)i(in)f(the)g (application)h(of)e(the)h(Ionescu-Marinescu)h(T)l(ulcea)g(theorem)e (\([IT,M)o(]\))g(but)h(w)o(e)-12 57 y(will)j(use)e(Birkho\013)g(cones)h (and)f(pro)s(jectiv)o(e)g(metrics.)26 b(The)17 b(main)g(adv)m(an)o (tages)g(of)f(this)i(tec)o(hnic)g(are:)23 b(\014rst)17 b(it)-12 114 y(pro)o(vides)e(a)g(constructiv)o(e)g(b)q(ound)h(for)e (the)h(rate)f(of)g(con)o(v)o(ergence,)h(second)g(the)g(cones)g(are)g (adaptable)g(to)f(non)-12 170 y(quasi-compact)i(cases.)k(Indeed)c(in)g (section)g(3)f(w)o(e)g(will)i(pro)o(v)o(e)d(the)i(follo)o(wing)f (result.)-12 276 y Fo(Theorem)i(1.4)23 b Fg(Let)e Fu(\006)f 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y(There)j(exists)g(a)f(sequence)i Fs(u)507 655 y Fr(n)548 648 y Fg(whic)o(h)f(go)q(es)f(to)g(zero)h(when)g Fs(n)g Fg(go)q(es)f(to)g(in\014nit)o(y)i(suc)o(h)f(that)e(for)h Fs(f)22 b Fn(2)17 b Fs(L)g Fg(and)-12 704 y Fs(g)d Fn(2)f Fs(L)98 688 y Fq(1)135 704 y Fu(\()p Fs(m)p Fu(\))p Fg(,)713 761 y Fs(C)746 768 y Fr(n)769 761 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))j Fn(\024)i Fs(u)959 768 y Fr(n)982 761 y Fn(k)p Fs(f)5 b Fn(k)15 b(k)p Fs(g)r Fn(k)1140 768 y Fq(1)1176 761 y Fs(:)-12 844 y Fg(Moreo)o(v)o(er,)f(for)g(\014xed)i Fs(Q)p Fg(,)797 876 y Fs(u)823 883 y Fr(n)p 761 897 121 2 v 761 938 a Fs(u)787 945 y Fr(n)811 938 y Fu(\()p Fs(Q)p Fu(\))900 907 y Fs(<)d(K)18 b Fn(8)p Fs(n)13 b Fn(2)g Fj(N)-12 1022 y Fg(and)g(the)f(sequences)h Fs(u)378 1029 y Fr(n)402 1022 y Fu(\()p Fs(Q)p Fu(\))e Fg(and)i Fs(u)597 1029 y Fr(n)632 1022 y Fg(dep)q(end)h(on)e(the)g(pro)q(duct)h(of)f(the) g Fs(\016)1236 994 y Ff(0)1234 1036 y Fr(k)q(;j)1294 1022 y Fg(and)g(on)g(the)h(measure)f(of)g(cylinders)-12 1079 y(close)k(to)f(in\014nit)o(y)l(.)-12 1168 y Fu(In)d(section)g(4,)g (w)o(e)f(giv)o(e)h(large)f(classes)h(of)f(examples)h(satisfying)g(\()p Fo(Exp1)p Fu(\))f(or)g(\()p Fo(S-Exp1)p Fu(\))g(and)h(compute)f (explicit)-12 1225 y(b)q(ounds)16 b(for)f(some)g(birth-and-death)h(lik) o(e)g(dynamics)g(and)g(non)f(uniformly)h(expanding)h(maps.)-12 1281 y(W)l(e)e(will)i(no)o(w)e(recall)h(de\014nitions)h(and)f(results)f (on)g(cones)h(and)f(pro)s(jectiv)o(e)g(metrics.)-12 1403 y Fh(1.2)56 b(Cones)19 b(and)g(pro)s(jectiv)n(e)e(metrics.)-12 1489 y Fu(The)f(theory)f(of)g(cones)h(and)g(pro)s(jectiv)o(e)f(metrics) h(of)f(G.)f(Birkho\013)i([Bi1])f(is)h(a)f(p)q(o)o(w)o(erful)h(to)q(ol)g (to)f(study)g(linear)-12 1545 y(op)q(erators.)35 b(P)l(.)20 b(F)l(errero)f(and)i(B.)f(Sc)o(hmitt)h([F,S)14 b(1])20 b(applied)i(it)f(to)e(estimate)h(the)h(correlations)g(deca)o(y)f(for) -12 1601 y(random)14 b(dynamical)i(systems.)j(Then,)14 b(this)h(strategy)e(had)i(b)q(een)g(used)g(b)o(y)f(man)o(y)g(authors.) 19 b(Let)c(us)f(men)o(tion)-12 1658 y Fn(\017)i Fu(C.)f(Liv)o(erani)i ([Li1],)e(C.)g(Liv)o(erani,)i(B.)f(Saussol)g(and)g(S.)f(V)l(aien)o(ti)i ([L,)e(S,)g(V1])g(for)g(one)h(dimensional)i(Lasota-)-12 1714 y(Y)l(ork)o(e)d(t)o(yp)q(e)g(dynamics)h(with)g(\014nite)g(or)f (coun)o(table)h(partition,)-12 1771 y Fn(\017)f Fu(C.)g(Liv)o(erani)h ([Li2)q(])e(and)i(M.)e(Viana)i([V)o(])f(for)g(Anoso)o(v)f(and)i(Axiom)f (A)h(di\013eomorphisms,)-12 1827 y Fn(\017)k Fu(V.)f(Baladi,)j(A.)d (Kondah,)i(B.)e(Sc)o(hmitt)h([B,K,S],)g(T.)f(Bogensc)o(h)q(\177)-24 b(utz)20 b(\([Bog)o(]\))f(and)h(J.)g(Buzzi)h(\([Buz]\))e(for)-12 1884 y(random)c(dynamical)h(systems.)-12 1940 y(They)f(all)h(used)g (Birkho\013)f(cones)g(to)g(obtain)g(exp)q(onen)o(tial)h(deca)o(y)f(of)g (correlations.)20 b(In)15 b([K,M,S])f(the)h(Birkho\013)-12 1997 y(cones)e(tec)o(hniques)h(w)o(ere)e(used)i(in)f(a)g(di\013eren)o (t)g(w)o(a)o(y)e(to)h(obtain)h(sub-exp)q(onen)o(tial)i(deca)o(y)e(of)f (correlations.)19 b(The)-12 2053 y(w)o(a)o(y)11 b(w)o(e)h(use)h(cone's) f(tec)o(hniques)i(in)f(section)f(3)g(follo)o(ws)h(some)f(ideas)g(of)g (P)l(.)g(F)l(errero)g(and)g(B.)g(Sc)o(hmitt)h(\([F,S)h(2)o(]\).)-12 2166 y(Let)20 b(us)h(recall)g(de\014nitions)h(and)e(prop)q(erties)g(of) g(cones)g(and)h(pro)s(jectiv)o(e)e(metrics)i(\(see)f([Li2])g(or)f([L,)c (S,)g(V1])-12 2222 y(for)g(a)g(more)f(complete)i(presen)o(tation\).)-12 2335 y(Let)g Fs(B)i Fu(b)q(e)d(a)g(v)o(ector)g(space)g(and)h Fs(C)f Fn(\032)e Fs(B)18 b Fu(a)d(cone)g(with)h(the)f(follo)o(wing)h (prop)q(erties.)56 2428 y Fn(\017)23 b Fs(C)18 b Fu(is)d(con)o(v)o(ex.) 56 2522 y Fn(\017)23 b Fs(C)13 b Fn(\\)d(\000)p Fs(C)16 b Fu(=)d Fn(;)p Fu(.)56 2615 y Fn(\017)23 b Fu(if)14 b Fs(\013)171 2622 y Fr(n)209 2615 y Fu(is)h(a)e(sequence)j(of)d(real)i (n)o(um)o(b)q(ers)f(suc)o(h)h(that)e Fs(\013)1026 2622 y Fr(n)1062 2615 y Fn(!)g Fs(\013)i Fu(and)f Fs(x)8 b Fn(\000)g Fs(\013)1357 2622 y Fr(n)1381 2615 y Fs(y)14 b Fn(2)f Fs(C)k Fn(8)p Fs(n)e Fu(then)f Fs(x)8 b Fn(\000)g Fs(\013y)15 b Fn(2)e Fs(C)s Fu(.)102 2672 y(This)j(prop)q(ert)o(y)e(is) i(called)h(\\in)o(tegral)e(closure".)939 2796 y(7)p eop %%Page: 8 8 8 7 bop -12 1 a Fu(F)l(or)15 b(suc)o(h)g(a)g(cone,)g(the)h (pseudo-metric)g Fs(\022)719 8 y Fr(C)764 1 y Fu(on)f Fs(C)j Fu(is)e(de\014ned)g(in)g(the)g(follo)o(wing)f(w)o(a)o(y)l(.)20 b(Let)15 b Fs(x;)8 b(y)14 b Fn(2)f Fs(C)s Fu(,)521 86 y Fs(\026)p Fu(\()p Fs(x;)8 b(y)r Fu(\))k(=)h(inf)t Fn(f)p Fs(\014)h(>)f Fu(0)i(suc)o(h)h(that)e Fs(\014)r(x)c Fn(\000)h Fs(y)j Fn(2)f Fs(C)s Fn(g)p Fs(;)514 172 y(\025)p Fu(\()p Fs(x;)8 b(y)r Fu(\))i(=)j(sup)q Fn(f)p Fs(\013)f(>)h Fu(0)i(suc)o(h)h(that)e Fs(y)e Fn(\000)f Fs(\013x)i Fn(2)g Fs(C)s Fn(g)p Fs(;)-12 247 y Fu(with)22 b(the)f(con)o(v)o(en)o(tion:)32 b Fs(\026)p Fu(\()p Fs(x;)8 b(y)r Fu(\))21 b(=)i Fn(1)f Fu(and)f Fs(\025)p Fu(\()p Fs(x;)8 b(y)r Fu(\))21 b(=)i(0)e(if)g(the)g (corresp)q(onding)i(sets)e(are)f(empt)o(y)l(.)38 b(Let)-12 303 y Fs(\022)9 310 y Fr(C)39 303 y Fu(\()p Fs(x;)8 b(y)r Fu(\))k(=)j(log)279 282 y Fr(\026)p 279 293 22 2 v 279 319 a(\025)305 303 y Fu(.)22 b Fs(\022)361 310 y Fr(C)407 303 y Fu(is)17 b(called)g(pseudo-metric)h(b)q(ecause)f(it)f(is)g(not)g (necessarily)h(\014nite.)24 b(Moreo)o(v)o(er,)14 b(it)i(is)h(a)-12 359 y(pro)s(jectiv)o(e)e(pseudo-metric:)21 b(if)16 b Fs(x)f Fu(and)g Fs(x)707 366 y Fp(1)742 359 y Fu(are)g(prop)q(ortional) h(then)f(for)g(an)o(y)g Fs(y)f Fn(2)f Fs(C)s Fu(,)i Fs(\022)1503 366 y Fr(C)1533 359 y Fu(\()p Fs(x;)8 b(y)r Fu(\))j(=)i Fs(\022)1720 366 y Fr(C)1750 359 y Fu(\()p Fs(x)1794 366 y Fp(1)1813 359 y Fs(;)8 b(y)r Fu(\).)-12 472 y(The)16 b(follo)o(wing)g(t)o(w)o(o)d(results)j(rev)o(eal)f(the)h(usefulness)g (of)f(pro)s(jectiv)o(e)g(metrics.)-12 529 y(Let)h Fs(C)i Fu(and)e Fs(C)246 512 y Fq(0)273 529 y Fu(b)q(e)h(t)o(w)o(o)d(cones,)i Fs(P)22 b Fu(a)15 b(linear)i(op)q(erator)e Fs(P)k Fu(:)13 b Fs(C)k Fn(!)c Fs(C)1175 512 y Fq(0)1187 529 y Fu(.)21 b(Let)16 b(\001)f(denotes)h(the)g(diameter)g(of)f Fs(P)6 b(C)-12 585 y Fu(in)16 b Fs(C)77 569 y Fq(0)89 585 y Fu(:)724 642 y(\001)d(=)27 b(sup)823 681 y Fr(f)r(;g)q Fq(2)p Fr(C)928 642 y Fs(\022)949 649 y Fr(C)976 640 y Ff(0)990 642 y Fu(\()p Fs(P)6 b(f)r(;)i(P)e(g)r Fu(\))p Fs(:)-12 772 y Fo(Theorem)17 b(1.5)23 b Fu([Bi1])15 b Fg(F)l(or)f(an)o(y)h Fs(f)r(;)8 b(g)16 b Fg(in)g Fs(C)s Fg(,)f(w)o(e)g(ha)o(v)o(e:)608 882 y Fs(\022)629 890 y Fr(C)656 880 y Ff(0)670 882 y Fu(\()p Fs(P)6 b(f)r(;)i(P)e(g)r Fu(\))12 b Fn(\024)h Fu(tanh)1004 818 y Fi(\022)1042 851 y Fu(\001)p 1042 872 38 2 v 1050 913 a(4)1085 818 y Fi(\023)1126 882 y Fs(\022)1147 889 y Fr(C)1177 882 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))p Fs(:)-12 993 y Fu(This)17 b(theorem)f(implies)j(that)d Fs(P)21 b Fu(:)15 b Fs(C)i Fn(!)e Fs(C)751 977 y Fq(0)779 993 y Fu(is)i(alw)o(a)o(ys)f(a)g(con)o (traction)g(\(in)h(wide)g(sense\))g(for)f(the)h(pro)s(jectiv)o(e)-12 1050 y(metrics.)j(If)c(\001)c Fs(<)h Fn(1)j Fu(then)f(it)h(is)g(a)e (strict)i(con)o(traction.)-12 1106 y(The)k(follo)o(wing)f(result)h (relies)g(the)g(metric)f Fs(\022)781 1113 y Fr(C)830 1106 y Fu(to)g(certain)g(norms)g(on)g Fs(B)r Fu(.)32 b(A)20 b(norm)e Fn(k)h(k)g Fu(on)g Fs(B)j Fu(is)d(a)g Ft(norm)-12 1163 y(adapte)n(d)j(to)e Fs(C)j Fu(if)d(for)f Fs(f)25 b Fu(and)20 b Fs(g)i Fu(in)e Fs(B)j Fu(suc)o(h)d(that)f Fs(f)f Fu(+)c Fs(g)21 b Fu(b)q(elongs)g(to)e Fs(C)k Fu(and)d Fs(f)e Fn(\000)13 b Fs(g)22 b Fu(b)q(elongs)e(to)f Fs(C)k Fu(then)-12 1219 y Fn(k)p Fs(g)r Fn(k)13 b(\024)h(k)p Fs(f)5 b Fn(k)p Fu(.)21 b Fs(\032)15 b Fu(is)h(a)g Ft(homo)n(gene)n (ous)g(form)i(adapte)n(d)f(to)g Fs(C)i Fu(if)d Fs(\032)f Fu(maps)h Fs(C)i Fu(to)d Fj(R)1352 1203 y Fp(+)1379 1219 y Fu(,)g(for)g(an)o(y)h Fs(\025)d(>)h Fu(0)h(and)h Fs(f)j Fn(2)14 b Fs(C)s Fu(,)-12 1275 y Fs(\032)p Fu(\()p Fs(\025f)5 b Fu(\))11 b(=)i Fs(\025\032)p Fu(\()p Fs(f)5 b Fu(\))14 b(and)h(if)h Fs(f)f Fn(\000)c Fs(g)j Fn(2)f Fs(C)k Fu(implies)h Fs(\032)p Fu(\()p Fs(g)r Fu(\))11 b Fn(\024)i Fs(\032)p Fu(\()p Fs(f)5 b Fu(\).)-12 1373 y Fo(Theorem)17 b(1.6)23 b Fu([Bi1],)14 b([L,)h(S,)g(V1)o(])g Fg(Let)f Fs(C)k Fg(b)q(e)d(a)f(cone,)h(let)g Fn(k)f(k)g Fg(and)h Fs(\032)f Fg(b)q(e)i(adapted)e(to)g Fs(C)s Fg(.)19 b(F)l(or)14 b(an)o(y)h Fs(f)k Fg(and)c Fs(g)-12 1430 y Fg(in)h Fs(C)i Fg(suc)o(h)e(that)e Fs(\032)p Fu(\()p Fs(f)5 b Fu(\))12 b(=)h Fs(\032)p Fu(\()p Fs(g)r Fu(\))e Fn(6)p Fu(=)h(0)j Fg(w)o(e)g(ha)o(v)o(e:)584 1515 y Fn(k)p Fs(f)g Fn(\000)c Fs(g)r Fn(k)g(\024)i Fu(\()p Fs(e)835 1496 y Fr(\022)q Fp(\()p Fr(f)r(;g)q Fp(\))939 1515 y Fn(\000)d Fu(1\))e(min\()p Fn(k)p Fs(f)d Fn(k)p Fs(;)j Fn(k)p Fs(g)r Fn(k)p Fu(\))p Fs(:)-12 1656 y Fv(2)67 b(Quasi-compact)22 b(case)f(\(pro)r(of)i(of)e (theorem)i(1.3\))-12 1757 y Fu(In)14 b(this)f(section,)h(w)o(e)e(pro)o (v)o(e)h(the)g(\014rst)f(part)h(of)f(theorem)h(1.3,)f(so)h(w)o(e)f (assume)h(that)f(\006)h(is)g(ap)q(erio)q(dic,)i(\010)e(satis\014es)-12 1814 y(\()p Fo(SA)p Fu(\))i(and)h(\()p Fo(Exp1)p Fu(\).)21 b(As)15 b(w)o(e)h(already)g(men)o(tioned,)g(\()p Fo(Exp1)p Fu(\))f(implies)j(\()p Fo(K)p Fu(\).)c(Let)i(us)g(note)g Fs(\026)e Fu(=)f Fs(hm)j Fu(where)g Fs(h)-12 1870 y Fu(is)f(the)g (normalized)h(\014xed)f(p)q(oin)o(t)g(for)f Fn(L)670 1877 y Fp(\010)712 1870 y Fu(giv)o(en)h(b)o(y)f(theorem)h(1.1.)j(Since) e(\010)f(satis\014es)f(\()p Fo(K)p Fu(\),)g Fs(h)g Fu(is)h(b)q(ounded)h (b)o(y)-12 1933 y Fs(M)i Fu(=)13 b(sup)166 1944 y Fr(k)195 1933 y Fn(kL)249 1916 y Fr(k)249 1946 y Fp(\010)277 1933 y Fo(1)p Fn(k)326 1940 y Fq(1)363 1933 y Fu(,)i(so)g(for)f(an)o(y)h Fs(A)e Fn(2)g(F)5 b Fu(,)767 1910 y Fr(\026)p Fp(\()p Fr(A)p Fp(\))p 762 1922 86 2 v 762 1949 a Fr(m)p Fp(\()p Fr(A)p Fp(\))864 1933 y Fn(\024)13 b Fs(M)5 b Fu(.)-12 1999 y(F)l(or)17 b Fs(f)23 b Fn(2)17 b Fs(C)197 2006 y Fr(u)219 1999 y Fu(\(\006\),)h(the)g(iterates)f(of)h Fn(L)650 2006 y Fp(\010)678 1999 y Fu(\()p Fs(f)5 b Fu(\))17 b(are)h(con)o(v)o(erging)g(to)f Fs(hm)p Fu(\()p Fs(f)5 b Fu(\))17 b(in)i Fs(L)1357 1983 y Fp(1)1377 1999 y Fu(\()p Fs(m)p Fu(\))e(\(b)o(y)g(theorem)h(1.1\),)f(this)-12 2056 y(implies)i(the)f(follo)o(wing)g(mixing)g(prop)q(ert)o(y)f(whic)o (h)h(will)h(b)q(e)f(used)f(to)g(obtain)g(deca)o(y)g(of)g(correlations)h (for)e(ob-)-12 2112 y(serv)m(ables)g(in)h Fs(L)p Fu(.)316 2198 y Fn(8)p Fs(A)c Fn(2)g(F)5 b Fs(;)22 b(g)13 b Fu(=)g Fo(1)613 2205 y Fr(A)642 2198 y Fs(;)22 b Fn(8)p Fs(f)c Fn(2)13 b Fs(C)818 2205 y Fr(u)840 2198 y Fu(\(\006\))p Fs(;)22 b Fn(j)p Fs(m)p Fu(\()p Fs(g)11 b Fn(\016)e Fs(\033)1108 2179 y Fr(n)1131 2198 y Fs(f)c Fu(\))10 b Fn(\000)h Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fs(m)p Fu(\()p Fs(f)5 b Fu(\))p Fn(j)1446 2172 y Fr(n)p Fq(!1)1455 2198 y Fn(\000)-7 b(!)22 b Fu(0)p Fs(:)234 b Fu(\(2.1\))-12 2283 y(Indeed,)17 b(w)o(e)d(ha)o(v)o(e,)370 2391 y Fn(j)o Fs(m)p Fu(\()p Fs(g)d Fn(\016)f Fs(\033)534 2372 y Fr(n)567 2391 y Fn(\001)g Fs(f)5 b Fu(\))10 b Fn(\000)g Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fs(m)p Fu(\()p Fs(f)5 b Fu(\))p Fn(j)40 b Fu(=)1010 2325 y Fi(\014)1010 2352 y(\014)1010 2379 y(\014)1010 2406 y(\014)1025 2329 y(Z)1078 2391 y Fs(g)r Fu([)p Fn(L)1146 2372 y Fr(n)1146 2402 y Fp(\010)1173 2391 y Fs(f)15 b Fn(\000)c Fs(h)1290 2329 y Fi(Z)1342 2391 y Fs(f)5 b(dm)p Fu(])p Fs(dm)1510 2325 y Fi(\014)1510 2352 y(\014)1510 2379 y(\014)1510 2406 y(\014)933 2512 y Fn(\024)42 b(kL)1064 2493 y Fr(n)1064 2523 y Fp(\010)1091 2512 y Fs(f)16 b Fn(\000)10 b Fs(h)1208 2450 y Fi(Z)1261 2512 y Fs(f)5 b(dm)p Fn(k)1375 2519 y Fp(1)1394 2512 y Fs(:)-12 2615 y Fu(Let)16 b(us)f(set)g(some)g(notations.)-12 2672 y(F)l(or)g Fs(s)g Fu(and)h Fs(t)f Fu(\014xed)h(in)o(tegers,)f(w)o(e)g(will)i (denote)e(b)o(y)g Fn(P)911 2679 y Fr(s;t)967 2672 y Fu(the)g(\014nite)h (partition)g(of)f(\006)g(de\014ned)h(b)o(y:)939 2796 y(8)p eop %%Page: 9 9 9 8 bop 56 1 a Fn(\017)23 b(P)134 8 y Fr(s;t)187 1 y Fu(=)13 b Fn(P)267 8 y Fp(1)296 1 y Fn([)e(P)369 8 y Fp(2)56 95 y Fn(\017)23 b(P)134 102 y Fp(1)168 95 y Fu(is)16 b(the)f(partition)h(in)g Fs(s)p Fu(-cylinders)h(of)e(the)g(set)g([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(t)p Fn(j)-8 b Fu(])12 b(:=)g Fn(f)p Fs(x)h Fn(2)g Fu(\006)i Fs(=)f(x)1347 102 y Fp(0)1380 95 y Fn(\024)f Fs(t)p Fn(g)p Fu(.)56 189 y Fn(\017)23 b(P)134 196 y Fp(2)166 189 y Fu(=)13 b Fn(f)p Fu([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(t)p Fn(j)-8 b Fu(])333 172 y Fr(c)349 189 y Fn(g)12 b Fu(:=)h Fn(f)p Fs(P)497 196 y Fp(2)516 189 y Fn(g)p Fu(.)-12 282 y(W)l(e)i(will)i(denote)f(b)o(y)f Fs(D)400 289 y Fp(1)434 282 y Fu(the)h(diameter)f(for)g(the)g(distance) h Fs(d)f Fu(of)g Fn(P)1150 289 y Fp(1)1184 282 y Fu(:)604 385 y Fs(D)642 392 y Fp(1)674 385 y Fu(=)e(max)p Fn(f)p Fu(diam\()p Fs(P)6 b Fu(\))p Fs(;)i(P)18 b Fn(2)13 b(P)1142 392 y Fp(1)1161 385 y Fn(g)g Fu(=)g Fs(r)1267 366 y Fr(s)1285 385 y Fs(;)-12 487 y Fu(and)j(b)o(y)f Fs(D)178 494 y Fp(2)212 487 y Fu(the)g(measure)h(of)e Fn(P)551 494 y Fp(2)586 487 y Fu(:)792 543 y Fs(D)830 550 y Fp(2)862 543 y Fu(=)f Fs(m)p Fu(\([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(t)p Fn(j)-8 b Fu(])1064 524 y Fr(c)1079 543 y Fu(\))p Fs(:)-12 627 y Fu(Since)17 b(the)e(measure)g Fs(m)g Fu(is)h(\014nite,)g Fs(D)630 634 y Fp(2)664 627 y Fu(can)g(b)q(e)g(c)o(hosen)f(as)g(small)h (as)f(w)o(e)f(w)o(an)o(t)h(pro)o(vided)h Fs(t)f Fu(is)h(large)f (enough.)-12 683 y(W)l(e)g(will)i(also)e(use)h(the)f(follo)o(wing)h (con)o(v)o(en)o(tions:)56 777 y Fn(\017)23 b Fs(a)12 b Fu(=)h Fs(b)d Fn(\006)h Fs(c)j Fu(means)h(that)g Fs(b)10 b Fn(\000)g Fs(c)i Fn(\024)h Fs(a)g Fn(\024)g Fs(b)d Fu(+)g Fs(c)p Fu(,)k(where)i Fs(a)p Fu(,)f Fs(b)f Fu(and)i Fs(c)f Fu(are)f(real)i(n)o(um)o(b)q(ers.)56 871 y Fn(\017)23 b Fu(If)c Fs(x)f Fu(and)h Fs(y)h Fu(b)q(elong)g(to)e(the)g(same)h (1-cylinder,)h(for)e(an)o(y)h Fs(k)g Fn(2)f Fj(N)g Fu(their)h (preimages)g(under)g Fs(\033)1757 854 y Fr(k)1796 871 y Fu(are)g(in)102 927 y(bijection.)25 b(If)16 b Fs(x)381 911 y Fq(0)409 927 y Fu(is)h(a)f(preimage)h(of)f Fs(x)p Fu(,)g(w)o(e)g(will)i(denote)f(b)o(y)f Fs(y)1190 911 y Fq(0)1218 927 y Fu(the)h(preimage)f(of)g Fs(y)j Fu(b)q(elonging)f(to) d(the)102 984 y(same)g Fs(k)q Fu(-cylinder.)56 1077 y Fn(\017)23 b Fu(for)14 b(an)o(y)h Fs(k)f Fn(2)f Fj(N)p Fu(,)h(let)h Fs(g)486 1084 y Fr(k)522 1077 y Fu(b)q(e)h(de\014ned)h(b)o (y:)505 1184 y Fn(8)p Fs(x)c Fn(2)g Fu(\006)p Fs(;)22 b Fn(8)p Fs(f)c Fn(2)13 b Fs(C)821 1191 y Fr(u)843 1184 y Fu(\(\006\))p Fs(;)22 b Fn(L)978 1165 y Fr(k)978 1195 y Fp(\010)1006 1184 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))12 b(=)1181 1141 y Fi(X)1155 1243 y Fr(\033)1176 1234 y Fl(k)1195 1243 y Fr(x)1215 1234 y Ff(0)1226 1243 y Fp(=)p Fr(x)1281 1184 y Fs(f)5 b Fu(\()p Fs(x)1352 1165 y Fq(0)1364 1184 y Fu(\))p Fs(g)1404 1191 y Fr(k)1424 1184 y Fu(\()p Fs(x)1468 1165 y Fq(0)1480 1184 y Fu(\))p Fs(:)-12 1351 y Fu(Finally)l(,)16 b(let)e(us)g(remark)f(that)h(for)f Fs(\013)h Fu(and)g Fs(\013)751 1335 y Fq(0)777 1351 y Fu(suc)o(h)g(that)f Fs(\013)g(<)g Fu(1)g Fs(<)g(\013)1178 1335 y Fq(0)1189 1351 y Fu(,)h(since)h(the)f(partition)h Fn(P)1624 1358 y Fr(s;t)1678 1351 y Fu(is)f(\014nite,)h(the)-12 1408 y(mixing)h(\(2.1\))e(implies)j(that)e(there)g(exists)h(an)f(in)o (teger)g Fs(k)979 1415 y Fp(0)1014 1408 y Fu(suc)o(h)g(that:)461 1539 y Fn(8)p Fs(k)f Fn(\025)f Fs(k)596 1546 y Fp(0)615 1539 y Fs(;)22 b Fn(8)p Fs(P)6 b(;)i(P)766 1520 y Fq(0)791 1539 y Fn(2)13 b(P)866 1546 y Fr(s;t)907 1539 y Fs(;)22 b(\013)13 b Fn(\024)1037 1508 y Fs(m)p Fu(\()p Fs(\033)1123 1492 y Fq(\000)p Fr(k)1171 1508 y Fs(P)j Fn(\\)11 b Fs(P)1292 1492 y Fq(0)1304 1508 y Fu(\))p 1037 1529 285 2 v 1069 1570 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))p Fs(m)p Fu(\()p Fs(P)1260 1557 y Fq(0)1272 1570 y Fu(\))1339 1539 y Fn(\024)13 b Fs(\013)1416 1520 y Fq(0)1428 1539 y Fs(:)379 b Fu(\(2.2\))-12 1684 y Fh(2.1)56 b(A)19 b(family)d(of)j(cones.)-12 1770 y Fu(Let)e(us)g(b)q(egin)h(the)f(pro)q(of)g(of)f(the)h(\014rst)g(part)f (of)h(theorem)f(1.3.)24 b(T)l(o)17 b(this)g(aim,)g(w)o(e)g(will)h (construct)f(a)f(cone)i Fs(C)-12 1826 y Fu(and)e(an)f(in)o(teger)g Fs(k)h Fu(suc)o(h)g(that)e Fn(L)562 1810 y Fr(k)562 1840 y Fp(\010)590 1826 y Fs(C)i Fn(\032)d Fs(C)k Fu(and)f(the)f(pro)s (jectiv)o(e)g(diameter)h(of)e Fn(L)1387 1810 y Fr(k)1387 1840 y Fp(\010)1415 1826 y Fs(C)k Fu(in)e Fs(C)i Fu(is)e(\014nite.)-12 1883 y(Let)g(us)g(consider)g(the)g(follo)o(wing)h(family)f(of)f(cones.) 22 b(F)l(or)15 b(giv)o(en)h(real)g(p)q(ositiv)o(e)g(n)o(um)o(b)q(ers)g Fs(a;)8 b(b;)g(c)13 b Fu(and)j(in)o(tegers)-12 1939 y Fs(s;)8 b(t)p Fu(,)15 b Fs(C)110 1917 y Fr(s;t)107 1954 y(a;b;c)192 1939 y Fu(is)h(the)f(set)g(of)g(functions)h Fs(f)k Fu(in)c Fs(L)f Fu(whic)o(h)h(satisfy:)56 2064 y Fn(\017)23 b(8)p Fs(P)c Fn(2)13 b(P)250 2071 y Fr(s;t)291 2064 y Fs(;)8 b Fu(0)j Fs(<)437 2033 y Fu(1)p 399 2053 99 2 v 399 2095 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))510 2002 y Fi(Z)509 2133 y Fr(P)563 2064 y Fs(f)f(dm)12 b Fn(\024)h Fs(a)746 2002 y Fi(Z)799 2064 y Fs(f)5 b(dm)p Fu(,)56 2241 y Fn(\017)23 b Fs(K)s Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\024)h Fs(b)295 2179 y Fi(Z)347 2241 y Fs(f)5 b(dm)p Fu(,)56 2383 y Fn(\017)30 b Fu(sup)102 2423 y Fr(x)p Fq(2)p Fr(P)168 2428 y Fk(2)192 2383 y Fn(j)p Fs(f)5 b Fu(\()p Fs(x)p Fu(\))p Fn(j)12 b(\024)h Fs(c)395 2322 y Fi(Z)447 2383 y Fs(f)5 b(dm)p Fu(.)-12 2516 y(When)23 b(there)f(w)o(on't)e(b)q(e)j(an)o(y)f(am)o(biguit)o(y)l(,)h(w)o(e)f (will)h(simply)h(note)d Fs(C)k Fu(instead)e(of)e Fs(C)1516 2494 y Fr(s;t)1513 2530 y(a;b;c)1583 2516 y Fu(.)40 b(The)23 b(follo)o(wing)-12 2572 y(prop)q(erties)16 b(follo)o(w)f(straigh)o (tforw)o(ard)e(from)i(the)g(de\014nition)i(of)e Fs(C)j Fu(:)56 2666 y Fn(\017)23 b Fs(C)13 b Fn(\\)d(\000)p Fs(C)16 b Fu(=)d Fn(;)p Fu(,)939 2796 y(9)p eop %%Page: 10 10 10 9 bop 56 1 a Fn(\017)23 b Fs(C)18 b Fu(is)d(a)g(con)o(v)o(ex)g (cone,)56 95 y Fn(\017)23 b Fs(C)18 b Fu(is)d(closed)i(for)d(the)h (uniform)h(top)q(ology)l(,)f(in)h(particular,)f(it)h(is)f(in)o (tegrally)h(closed.)-12 189 y(Moreo)o(v)o(er,)e(the)h(follo)o(wing)h (result)f(is)h(easily)g(v)o(eri\014ed.)-12 295 y Fo(Lemma)i(2.1)k Fg(An)o(y)15 b Fs(')e Fn(2)g Fs(L)i Fg(satis\014es:)431 415 y Fn(8)p Fs(P)k Fn(2)13 b(P)579 422 y Fp(1)599 415 y Fs(;)22 b Fn(8)p Fs(x)13 b Fn(2)g Fs(P)6 b(;)23 b(')p Fu(\()p Fs(x)p Fu(\))11 b(=)1012 385 y(1)p 968 405 111 2 v 968 447 a Fs(m)p Fu(\()p Fs(P)6 b Fu(\))1092 354 y Fi(Z)1091 485 y Fr(P)1145 415 y Fs('dm)j Fn(\006)i Fs(K)s Fu(\()p Fs(')p Fu(\))p Fs(D)1440 422 y Fp(1)1458 415 y Fs(;)349 b Fu(\(2.3\))-12 571 y Fg(and)16 b(for)e Fs(x)f Fn(2)g Fs(P)257 578 y Fp(2)277 571 y Fg(,)616 646 y Fs(')p Fu(\()p Fs(x)p Fu(\))f(=)824 615 y(1)p 773 635 125 2 v 773 677 a Fs(m)p Fu(\()p Fs(P)860 684 y Fp(2)879 677 y Fu(\))910 584 y Fi(Z)902 715 y Fr(P)924 720 y Fk(2)963 646 y Fs('dm)d Fn(\006)h Fu(2)e(sup)1156 685 y Fr(P)1178 690 y Fk(2)1218 646 y Fn(j)p Fs(')p Fn(j)p Fs(:)533 b Fu(\(2.4\))-12 787 y Fg(So)15 b(that)g(if)g Fs(')g Fg(b)q(elongs)h(to)f Fs(C)s Fg(,)g(for)f(an)o(y)h Fs(x)e Fn(2)g Fu(\006)p Fg(,)316 907 y Fu(min)q([)p Fn(\000)p Fs(c;)8 b(M)d(a)k Fn(\000)h Fs(bD)666 914 y Fp(1)685 907 y Fu(])706 846 y Fi(Z)758 907 y Fs('dm)i Fn(\024)h Fs(')p Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fu(max)o([)p Fs(c;)8 b(M)d(a)k Fu(+)i Fs(bD)1388 914 y Fp(1)1406 907 y Fu(])1427 846 y Fi(Z)1480 907 y Fs('dm:)233 b Fu(\(2.5\))-12 1039 y(In)17 b(order)g(to)f(use)h (the)g(cone)g Fs(C)525 1017 y Fr(s;t)522 1054 y(a;b;c)608 1039 y Fu(and)g(its)g(pro)s(jectiv)o(e)f(metric,)h(w)o(e)g(shall)h (need)f(an)g(adapted)f(norm)h(and)f(an)-12 1101 y(adapted)g (homogeneous)g(form.)22 b(Of)16 b(course)g Fs(\032)p Fu(\()p Fs(f)5 b Fu(\))13 b(=)924 1064 y Fi(R)962 1101 y Fs(f)5 b(dm)16 b Fu(is)g(an)g(adapted)g(homogeneous)g(form.)21 b(F)l(or)16 b(an)o(y)-12 1157 y Fs(d)c(>)h Fu(0,)i(let)h(us)f(consider) h(the)f(norm)340 1323 y Fn(k)p Fs(f)5 b Fn(k)413 1330 y Fr(d)446 1323 y Fu(=)13 b(max)586 1218 y Fi(0)586 1298 y(B)586 1327 y(@)626 1323 y Fs(d)h Fn(j)685 1261 y Fi(Z)738 1323 y Fs(f)5 b(dm)p Fn(j)p Fs(;)j Fu(2)894 1216 y Fi(\014)894 1243 y(\014)894 1270 y(\014)894 1298 y(\014)894 1325 y(\014)894 1352 y(\014)894 1380 y(\014)912 1220 y(R)909 1300 y Fr(P)949 1256 y Fs(f)d(dm)p 912 1312 129 2 v 921 1354 a(m)p Fu(\()p Fs(P)h Fu(\))1045 1216 y Fi(\014)1045 1243 y(\014)1045 1270 y(\014)1045 1298 y(\014)1045 1325 y(\014)1045 1352 y(\014)1045 1380 y(\014)1083 1323 y Fs(P)19 b Fn(2)13 b(P)1206 1330 y Fr(s;t)1247 1323 y Fs(;)8 b Fn(k)p Fs(f)d Fn(k)1341 1330 y Fq(1)1377 1323 y Fs(;)j(K)s Fu(\()p Fs(f)d Fu(\))1503 1218 y Fi(1)1503 1298 y(C)1503 1327 y(A)1549 1323 y Fs(;)-12 1491 y Fu(the)15 b(norm)g Fn(k)g(k)246 1498 y Fr(d)281 1491 y Fu(is)h(equiv)m(alen)o(t)h (to)d(the)i(norm)f Fn(k)f(k)h Fu(of)g Fs(L)p Fu(.)-12 1597 y Fo(Lemma)j(2.2)k Fg(If)c Fs(d)f Fn(\025)g Fu(max)o(\()p Fs(b;)8 b(c;)g Fu(2)p Fs(D)649 1604 y Fp(1)666 1597 y Fs(b)p Fu(\))17 b Fg(then)h(the)g(norm)f Fn(k)g(k)1092 1604 y Fr(d)1130 1597 y Fg(is)h(adapted)g(to)f Fs(C)k Fg(in)d(the)g(sense)g(of)g(section)-12 1653 y(1.2.)-12 1743 y Fo(Pro)q(of)5 b Fu(:)40 b(Let)16 b Fs(f)k Fu(and)15 b Fs(g)i Fu(b)q(e)f(suc)o(h)f(that)g Fs(f)g Fu(+)10 b Fs(g)17 b Fu(and)e Fs(f)g Fn(\000)c Fs(g)16 b Fu(b)q(elong)h(to)d Fs(C)s Fu(,)h(\(2.3\))e(giv)o(es)j(for)e Fs(x)f Fn(2)g Fs(P)6 b Fu(,)15 b Fs(P)k Fn(2)13 b(P)1839 1750 y Fp(1)1858 1743 y Fu(,)383 1866 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))k Fn(\000)i Fs(g)r Fu(\()p Fs(x)p Fu(\))40 b(=)779 1835 y(1)p 735 1856 111 2 v 735 1897 a Fs(m)p Fu(\()p Fs(P)6 b Fu(\))858 1804 y Fi(Z)903 1866 y Fu(\()p Fs(f)15 b Fn(\000)c Fs(g)r Fu(\))p Fs(dm)d Fn(\006)j Fs(bD)1222 1873 y Fp(1)1248 1804 y Fi(Z)1247 1936 y Fr(P)1294 1866 y Fu(\()p Fs(f)k Fn(\000)10 b Fs(g)r Fu(\))p Fs(dm;)383 2019 y(f)5 b Fu(\()p Fs(x)p Fu(\))k(+)i Fs(g)r Fu(\()p Fs(x)p Fu(\))40 b(=)779 1988 y(1)p 735 2008 V 735 2050 a Fs(m)p Fu(\()p Fs(P)6 b Fu(\))858 1957 y Fi(Z)857 2088 y Fr(P)903 2019 y Fu(\()p Fs(f)15 b Fu(+)c Fs(g)r Fu(\))p Fs(dm)d Fn(\006)j Fs(bD)1222 2026 y Fp(1)1248 1957 y Fi(Z)1294 2019 y Fu(\()p Fs(f)k Fu(+)10 b Fs(g)r Fu(\))p Fs(dm:)-12 2174 y Fu(By)16 b(substracting)f(these)g(inequations,)h(w)o (e)f(obtain,)495 2294 y Fn(j)p Fs(g)r Fu(\()p Fs(x)p Fu(\))p Fn(j)c(\024)i Fu(2)8 b(max)o(\(1)p Fs(=m)p Fu(\()p Fs(P)e Fu(\))p Fn(j)977 2233 y Fi(Z)976 2364 y Fr(P)1028 2294 y Fs(g)r(dm)p Fn(j)p Fs(;)i(bD)1208 2301 y Fp(1)1232 2233 y Fi(Z)1285 2294 y Fs(f)d(dm)p Fu(\))p Fs(:)-12 2480 y Fu(Since)19 b Fs(f)e Fn(\000)c Fs(g)18 b Fn(2)f Fs(C)j Fu(and)e Fs(f)f Fu(+)12 b Fs(g)19 b Fn(2)e Fs(C)s Fu(,)g(w)o(e)h(ha)o(v)o(e)f Fn(j)865 2418 y Fi(Z)864 2549 y Fr(P)917 2480 y Fs(g)r(dm)p Fn(j)f(\024)1086 2418 y Fi(Z)1085 2549 y Fr(P)1139 2480 y Fs(f)5 b(dm)18 b Fu(for)f(an)o(y)g Fs(P)23 b Fn(2)18 b(P)1540 2487 y Fr(s;t)1580 2480 y Fu(,)g(so)g(that)f(for)g(an)o(y)-12 2620 y Fs(x)e Fn(2)g(P)106 2627 y Fp(1)125 2620 y Fu(,)h Fn(j)p Fs(g)r Fu(\()p Fs(x)p Fu(\))p Fn(j)d(\024)i(k)p Fs(f)5 b Fn(k)402 2627 y Fr(d)438 2620 y Fu(if)17 b Fs(d)d Fn(\025)h Fu(2)p Fs(bD)650 2627 y Fp(1)669 2620 y Fu(.)23 b(Moreo)o(v)o(er,)15 b(if)i Fs(x)e Fn(2)g Fs(P)1077 2627 y Fp(2)1097 2620 y Fu(,)h(the)g(inequalit)o(y)i Fn(j)p Fs(g)r Fu(\()p Fs(x)p Fu(\))p Fn(j)13 b(\024)i Fs(c)1620 2558 y Fi(Z)1672 2620 y Fs(f)5 b(dm)16 b Fu(follo)o(ws)928 2796 y(10)p eop %%Page: 11 11 11 10 bop -12 17 a Fu(from)14 b(\(2.4\))f(in)j(the)e(same)h(w)o(a)o(y)l (.)k(W)l(e)14 b(also)h(ha)o(v)o(e)f Fn(j)846 -45 y Fi(Z)898 17 y Fs(g)r(dm)p Fn(j)d(\024)1058 -45 y Fi(Z)1111 17 y Fs(f)5 b(dm)p Fu(.)20 b(It)15 b(remains)g(to)f(tak)o(e)f(care)i(of)f (the)h(part)-12 92 y(of)f(the)g(norm)g(whic)o(h)h(is)f(giv)o(en)h(b)o (y)f(the)g(Lipsc)o(hitz)i(constan)o(t.)j(Since)c Fs(f)e Fn(\000)c Fs(g)14 b Fn(2)f Fs(C)j Fu(and)f Fs(f)e Fu(+)8 b Fs(g)14 b Fn(2)f Fs(C)s Fu(,)g(w)o(e)h(ha)o(v)o(e)g(for)-12 149 y Fs(x)h Fu(and)h Fs(y)h Fu(in)f(the)f(same)g(1-cylinder,)373 269 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))10 b Fn(\000)g Fs(g)r Fu(\()p Fs(x)p Fu(\))f Fn(\000)i Fu(\()p Fs(f)5 b Fu(\()p Fs(y)r Fu(\))k Fn(\000)h Fs(g)r Fu(\()p Fs(y)r Fu(\)\))40 b(=)i Fn(\006)p Fs(b)15 b(d)p Fu(\()p Fs(x;)8 b(y)r Fu(\))1258 207 y Fi(Z)1303 269 y Fu(\()p Fs(f)15 b Fn(\000)c Fs(g)r Fu(\))p Fs(dm;)373 387 y(f)5 b Fu(\()p Fs(x)p Fu(\))10 b(+)g Fs(g)r Fu(\()p Fs(x)p Fu(\))f Fn(\000)i Fu(\()p Fs(f)5 b Fu(\()p Fs(y)r Fu(\))k(+)h Fs(g)r Fu(\()p Fs(y)r Fu(\)\))40 b(=)i Fn(\006)p Fs(b)15 b(d)p Fu(\()p Fs(x;)8 b(y)r Fu(\))1258 325 y Fi(Z)1303 387 y Fu(\()p Fs(f)15 b Fu(+)c Fs(g)r Fu(\))p Fs(dm)-12 507 y Fu(By)16 b(substracting)f (these)g(t)o(w)o(o)f(inequalities,)k(w)o(e)c(get)778 628 y Fs(K)s Fu(\()p Fs(g)r Fu(\))e Fn(\024)h Fs(b)968 566 y Fi(Z)1020 628 y Fs(f)5 b(dm:)-12 751 y Fu(Finally)l(,)17 b(if)e Fs(d)e Fn(\025)g Fu(max)o(\()p Fs(b;)8 b(c;)g Fu(2)p Fs(bD)542 758 y Fp(1)559 751 y Fu(\),)15 b(w)o(e)f(ha)o(v)o(e)h Fn(k)p Fs(g)r Fn(k)846 758 y Fr(d)878 751 y Fn(\024)e(k)p Fs(f)5 b Fn(k)999 758 y Fr(d)1019 751 y Fu(.)841 b Fn(4)-12 929 y Fh(2.2)56 b(Con)n(traction)19 b(of)g(the)f(cone.)-12 1015 y Fu(W)l(e)d(are)f(going)g(to)g(pro)o(v)o(e)g(that)f(the)i(cone)f Fs(C)751 993 y Fr(s;t)748 1029 y(a;b;c)833 1015 y Fu(is)h(strictly)g (con)o(tracted)e(b)o(y)i Fn(L)1349 998 y Fr(k)1349 1028 y Fp(\010)1391 1015 y Fu(pro)o(vided)g Fs(k)q Fu(,)f Fs(a)p Fu(,)g Fs(b)p Fu(,)g Fs(c)p Fu(,)g Fs(s)g Fu(and)h Fs(t)-12 1071 y Fu(are)g(large)g(enough.)-12 1177 y Fo(Lemma)j(2.3)k Fg(There)15 b(exist)h Fs(k)e Fn(2)f Fj(N)617 1161 y Fq(\003)635 1177 y Fg(,)i Fs(a)e(>)g Fu(0)p Fg(,)h Fs(b)e(>)h Fu(0)p Fg(,)i Fs(c)d(>)h Fu(0)p Fg(,)i Fs(s)e Fn(2)f Fj(N)1170 1161 y Fq(\003)1189 1177 y Fg(,)j Fs(t)e Fn(2)g Fj(N)1323 1161 y Fq(\003)1356 1177 y Fg(and)j Fu(0)c Fs(<)h(\015)i(<)e Fu(1)i Fg(suc)o(h)g(that)751 1280 y Fn(L)782 1261 y Fr(k)782 1291 y Fp(\010)810 1280 y Fs(C)846 1257 y Fr(s;t)843 1294 y(a;b;c)926 1280 y Fn(\022)e Fs(C)1010 1257 y Fr(s;t)1007 1294 y(\015)r(a;\015)r(b;\015)r(c)1138 1280 y Fs(:)-12 1382 y Fo(Pro)q(of)5 b Fu(:)52 b(Let)18 b(us)g(\014x)g Fs(\013)g(<)g Fu(1)f Fs(<)g(\013)618 1365 y Fq(0)648 1382 y Fu(and)h Fs(\014)767 1365 y Fq(0)797 1382 y Fs(<)f(\014)j(<)e Fu(1.)28 b(The)18 b(parameters)f Fs(a;)8 b(b;)g(c)p Fu(,)16 b Fs(s)i Fu(and)h Fs(t)f Fu(are)g(c)o(hosen)g(to)-12 1438 y(v)o(erify:)44 1532 y(1.)k Fs(a)12 b(>)h(\013)215 1515 y Fq(0)227 1532 y Fu(\(1)d(+)343 1514 y Fr(\013)p 328 1521 52 2 v 328 1548 a Fp(2)p Fr(\013)369 1538 y Ff(0)385 1532 y Fu(\))p Fs(\014)431 1515 y Fq(\000)p Fp(1)478 1532 y Fu(,)44 1626 y(2.)22 b Fs(c)12 b Fn(\025)h Fs(M)5 b(a)10 b Fu(+)h(1,)44 1720 y(3.)22 b Fs(b)12 b(>)207 1702 y Fr(Rc)p 187 1709 82 2 v 187 1735 a(\014)r Fq(\000)p Fr(\014)258 1726 y Ff(0)274 1720 y Fu(,)44 1813 y(4.)22 b Fs(D)140 1820 y Fp(1)172 1813 y Fs(<)13 b Fu(1)p Fs(=b)304 1795 y Fr(\013)p 291 1802 52 2 v 291 1829 a Fp(4)p Fr(\013)332 1820 y Ff(0)346 1813 y Fu(,)44 1907 y(5.)22 b Fs(t)15 b Fu(is)h(suc)o(h)g(that)e Fs(t)f Fn(\025)g Fs(n)484 1914 y Fp(1)519 1907 y Fu(and)j Fs(D)646 1914 y Fp(2)678 1907 y Fs(<)753 1889 y Fr(\013)p 731 1896 67 2 v 731 1923 a Fp(8)p Fr(\013)772 1913 y Ff(0)782 1923 y Fr(c)802 1907 y Fu(.)-12 2006 y(Moreo)o(v)o(er,)10 b(let)h Fs(k)282 2013 y Fp(0)313 2006 y Fu(satis\014es)g(\()p Fo(Exp1)o Fu(\),)g(b)q(e)h(suc)o(h)f(that)f(\(2.2\))g(is)h(v)o(eri\014ed)h(for)e Fs(\013)h Fu(and)g Fs(\013)1421 1990 y Fq(0)1444 2006 y Fu(and)g Fn(8)p Fs(k)j(>)f(k)1663 2013 y Fp(0)1682 2006 y Fu(,)f Fs(M)5 b(r)1778 1990 y Fr(k)1811 2006 y Fs(<)13 b(\014)1887 1978 y Ff(0)1901 2006 y Fu(.)-12 2063 y(Let)j(us)f(\014x)g Fs(k)f(>)f(k)302 2070 y Fp(0)321 2063 y Fu(,)i Fs(\015)g Fu(=)e(max)o(\()p Fs(\014)r(;)8 b(\032)611 2070 y Fr(k)631 2063 y Fu(\))13 b Fs(<)g Fu(1)h(and)i Fs(f)k Fu(in)c Fs(C)967 2041 y Fr(s;t)964 2078 y(a;b;c)1034 2063 y Fu(.)k(Let)c Fs(P)j Fn(2)13 b(P)t Fu(,)519 2165 y(1)p 481 2185 99 2 v 481 2227 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))592 2134 y Fi(Z)591 2265 y Fr(P)645 2196 y Fn(L)676 2177 y Fr(k)676 2207 y Fp(\010)704 2196 y Fs(f)f(dm)41 b Fu(=)956 2165 y(1)p 918 2185 V 918 2227 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))1052 2134 y Fi(Z)1019 2269 y Fr(\033)1040 2260 y Ff(\000)p Fl(k)1083 2269 y Fr(P)1128 2196 y Fs(f)f(dm)836 2352 y Fu(=)956 2321 y(1)p 918 2342 V 918 2383 a Fs(\026)p Fu(\()p Fs(P)h Fu(\))1058 2309 y Fi(X)1029 2408 y Fr(P)1056 2399 y Ff(0)1067 2408 y Fq(2P)1116 2412 y Fl(s;t)1215 2290 y Fi(Z)1151 2426 y Fr(\033)1172 2416 y Ff(\000)p Fl(k)1215 2426 y Fr(P)f Fq(\\)p Fr(P)1293 2416 y Ff(0)1322 2352 y Fs(f)g(dm;)-12 2514 y Fu(using)16 b(\(2.3\))e(and)h(\(2.4\),)31 2569 y(1)p -7 2589 V -7 2631 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))104 2538 y Fi(Z)103 2669 y Fr(P)157 2600 y Fn(L)188 2581 y Fr(k)188 2611 y Fp(\010)216 2600 y Fs(f)f(dm)12 b Fu(=)397 2557 y Fi(X)367 2656 y Fr(P)394 2646 y Ff(0)406 2656 y Fq(2P)455 2660 y Fl(s;t)504 2569 y Fs(m)p Fu(\()p Fs(\033)590 2553 y Fq(\000)p Fr(k)638 2569 y Fs(P)17 b Fn(\\)10 b Fs(P)759 2553 y Fq(0)772 2569 y Fu(\))p 504 2589 285 2 v 536 2631 a Fs(m)p Fu(\()p Fs(P)629 2618 y Fq(0)641 2631 y Fu(\))p Fs(\026)p Fu(\()p Fs(P)c Fu(\))802 2538 y Fi(Z)795 2670 y Fr(P)822 2660 y Ff(0)855 2600 y Fs(f)f(dm)15 b Fn(\006)928 2796 y Fu(11)p eop %%Page: 12 12 12 11 bop -12 -43 a Fi(0)-12 39 y(@)28 49 y Fs(D)66 56 y Fp(1)85 49 y Fs(K)s Fu(\()p Fs(f)5 b Fu(\))216 5 y Fi(X)198 104 y Fr(P)225 95 y Ff(0)236 104 y Fq(2P)285 109 y Fk(1)314 18 y Fs(m)p Fu(\()p Fs(\033)400 1 y Fq(\000)p Fr(k)448 18 y Fs(P)16 b Fn(\\)11 b Fs(P)569 1 y Fq(0)581 18 y Fu(\))p 314 38 285 2 v 346 80 a Fs(m)p Fu(\()p Fs(P)439 67 y Fq(0)451 80 y Fu(\))p Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))604 49 y Fs(m)p Fu(\()p Fs(P)697 30 y Fq(0)708 49 y Fu(\))25 b(+)h(2)840 18 y Fs(m)p Fu(\()p Fs(\033)926 1 y Fq(\000)p Fr(k)973 18 y Fs(P)17 b Fn(\\)10 b Fs(P)1088 25 y Fp(2)1108 18 y Fu(\))p 840 38 287 2 v 872 80 a Fs(m)p Fu(\()p Fs(P)959 87 y Fp(2)978 80 y Fu(\))p Fs(\026)p Fu(\()p Fs(P)c Fu(\))1131 49 y Fs(m)p Fu(\()p Fs(P)1218 56 y Fp(2)1238 49 y Fu(\))i(sup)1277 88 y Fr(P)1299 93 y Fk(2)1339 49 y Fn(j)p Fs(f)d Fn(j)1392 -43 y Fi(1)1392 39 y(A)1439 49 y Fs(:)-12 180 y Fu(Since)17 b Fs(f)j Fu(b)q(elongs)c(to)f Fs(C)404 157 y Fr(s;t)401 194 y(a;b;c)471 180 y Fu(,)g(this)h(leads)f(to)g(\(using)h(\(2.2\)\):)444 278 y(1)p 406 299 99 2 v 406 340 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))517 247 y Fi(Z)516 379 y Fr(P)570 309 y Fn(L)601 290 y Fr(k)601 320 y Fp(\010)629 309 y Fs(f)f(dm)41 b Fn(\024)h Fs(\013)867 290 y Fq(0)887 247 y Fi(Z)940 309 y Fs(f)5 b(dm)j Fu([)o(1)i(+)g Fs(D)1167 316 y Fp(1)1202 309 y Fs(b)f Fu(+)i(2)p Fs(D)1338 316 y Fp(2)1372 309 y Fs(c)p Fu(])22 b(and)444 431 y(1)p 406 451 V 406 493 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))517 400 y Fi(Z)516 531 y Fr(P)570 462 y Fn(L)601 443 y Fr(k)601 473 y Fp(\010)629 462 y Fs(f)f(dm)41 b Fn(\025)838 425 y Fi(\002)857 462 y Fs(\013)11 b Fn(\000)f Fs(\013)971 443 y Fq(0)983 462 y Fu(\()p Fs(D)1039 469 y Fp(1)1058 462 y Fs(b)g Fu(+)g(2)p Fs(D)1194 469 y Fp(2)1213 462 y Fs(c)p Fu(\))1251 425 y Fi(\003)1277 400 y(Z)1330 462 y Fs(f)5 b(dm;)-12 616 y Fu(b)o(y)15 b(the)h(c)o(hoices)g(\(4)p Fs(:)e Fu(and)h(5)p Fs(:)p Fu(\))f(of)h Fs(a;)8 b(b;)g(c)13 b Fu(and)i Fn(P)798 623 y Fr(s;t)854 616 y Fu(w)o(e)g(obtain,)383 738 y Fs(\013=)p Fu(2)466 676 y Fi(Z)519 738 y Fs(f)5 b(dm)12 b Fn(\024)713 707 y Fu(1)p 675 727 V 675 769 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))786 676 y Fi(Z)785 807 y Fr(P)839 738 y Fn(L)870 719 y Fr(k)870 749 y Fp(\010)898 738 y Fs(f)f(dm)12 b Fn(\024)h Fs(\013)1078 719 y Fq(0)1090 738 y Fu(\(1)c(+)i Fs(\013=)p Fu(\(2)p Fs(\013)1308 719 y Fq(0)1319 738 y Fu(\)\))1363 676 y Fi(Z)1415 738 y Fs(f)5 b(dm:)301 b Fu(\(2.6\))-12 893 y(F)l(or)15 b(an)o(y)g Fs(')e Fn(2)h Fs(L)h Fu(and)h Fs(k)e Fn(2)g Fj(N)p Fu(,)g(the)i(fact)f(that)f(\010)i (v)o(eri\014es)g(\()p Fo(BD)p Fu(\))f(and)h(\()p Fo(K)p Fu(\))f(leads)h(to)f(the)g(follo)o(wing)i(standard)-12 949 y(inequalit)o(y)617 1005 y Fs(K)s Fu(\()p Fn(L)708 987 y Fr(k)708 1017 y Fp(\010)736 1005 y Fs(')p Fu(\))12 b Fn(\024)h Fs(M)5 b(r)915 987 y Fr(k)936 1005 y Fs(K)s Fu(\()p Fs(')p Fu(\))k(+)h Fs(R)e Fu(sup)g Fn(j)p Fs(')p Fn(j)p Fs(;)534 b Fu(\(2.7\))-12 1087 y(Com)o(bined)16 b(with)g(\(2.5\))d(and)j(the)f(de\014nition)i(of)e(the)g(cone,)g(this)h (giv)o(es)f(for)g Fs(f)i Fn(2)c Fs(C)1419 1065 y Fr(s;t)1416 1102 y(a;b;c)439 1217 y Fs(K)s Fu(\()p Fn(L)530 1198 y Fr(k)530 1228 y Fp(\010)558 1217 y Fs(f)5 b Fu(\))12 b Fn(\024)663 1155 y Fi(Z)716 1217 y Fs(f)5 b(dm)814 1166 y Fi(h)836 1217 y Fs(bM)g(r)927 1198 y Fr(k)973 1217 y Fu(+)25 b Fs(R)8 b Fu(max)o(\()p Fs(c;)g(M)d(a)k Fu(+)h Fs(bD)1404 1224 y Fp(1)1423 1217 y Fu(\))1441 1166 y Fi(i)-12 1334 y Fu(and)16 b(with)f(the)g(c)o(hoices)h(2)p Fs(:)f Fu(and)g(4)p Fs(:)g Fu(ab)q(o)o(v)o(e:)597 1443 y Fs(K)s Fu(\()p Fn(L)688 1424 y Fr(k)688 1454 y Fp(\010)715 1443 y Fs(f)5 b Fu(\))12 b Fn(\024)h Fs(\014)848 1424 y Fq(0)860 1443 y Fs(b)888 1381 y Fi(Z)940 1443 y Fs(f)5 b(dm)10 b Fu(+)h Fs(cR)1149 1381 y Fi(Z)1202 1443 y Fs(f)5 b(dm;)514 b Fu(\(2.8\))-12 1590 y(so,)15 b Fs(K)s Fu(\()p Fn(L)148 1574 y Fr(k)148 1604 y Fp(\010)175 1590 y Fs(f)5 b Fu(\))13 b Fn(\024)f Fs(\014)r(b)328 1528 y Fi(Z)381 1590 y Fs(f)5 b(dm)15 b Fu(if)h Fs(b)c(>)634 1572 y Fr(cR)p 614 1579 82 2 v 614 1606 a(\014)r Fq(\000)p Fr(\014)685 1596 y Ff(0)716 1590 y Fu(\(whic)o(h)j(is)h(3.\).)-12 1665 y(Finally)l(,)h(for)d Fs(x)f Fn(2)g Fs(P)332 1672 y Fp(2)352 1665 y Fu(,)127 1814 y Fn(jL)171 1795 y Fr(k)171 1825 y Fp(\010)199 1814 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))p Fn(j)40 b Fu(=)418 1720 y Fi(\014)418 1748 y(\014)418 1775 y(\014)418 1802 y(\014)418 1830 y(\014)418 1857 y(\014)448 1770 y(X)433 1870 y Fr(x)453 1860 y Ff(0)465 1870 y Fq(2)p Fr(P)511 1875 y Fk(2)535 1814 y Fs(g)557 1821 y Fr(k)578 1814 y Fu(\()p Fs(x)622 1795 y Fq(0)634 1814 y Fu(\))p Fs(f)5 b Fu(\()p Fs(x)723 1795 y Fq(0)734 1814 y Fu(\))k(+)821 1770 y Fi(X)807 1870 y Fr(x)827 1860 y Ff(0)838 1870 y Fq(62)p Fr(P)884 1875 y Fk(2)909 1814 y Fs(g)931 1821 y Fr(k)952 1814 y Fu(\()p Fs(x)996 1795 y Fq(0)1007 1814 y Fu(\))p Fs(f)c Fu(\()p Fs(x)1096 1795 y Fq(0)1107 1814 y Fu(\))1125 1720 y Fi(\014)1125 1748 y(\014)1125 1775 y(\014)1125 1802 y(\014)1125 1830 y(\014)1125 1857 y(\014)341 1963 y Fn(\024)42 b Fs(c)446 1901 y Fi(Z)499 1963 y Fs(f)5 b(dm)611 1919 y Fi(X)598 2018 y Fr(x)618 2009 y Ff(0)628 2018 y Fq(2)p Fr(P)674 2023 y Fk(2)699 1963 y Fs(g)721 1970 y Fr(k)742 1963 y Fu(\()p Fs(x)786 1944 y Fq(0)797 1963 y Fu(\))10 b(+)g(\()p Fs(M)5 b(a)10 b Fu(+)h Fs(bD)1075 1970 y Fp(1)1093 1963 y Fu(\))1119 1901 y Fi(Z)1172 1963 y Fs(f)5 b(dm)1284 1919 y Fi(X)1271 2018 y Fr(x)1291 2009 y Ff(0)1301 2018 y Fq(62)p Fr(P)1347 2023 y Fk(2)1372 1963 y Fs(g)1394 1970 y Fr(k)1415 1963 y Fu(\()p Fs(x)1459 1944 y Fq(0)1470 1963 y Fu(\))p Fs(;)22 b Fu(using)16 b(\(2.3\))341 2108 y Fn(\024)42 b Fu(max\()p Fs(c;)8 b(M)d(a)k Fu(+)h Fs(bD)747 2115 y Fp(1)766 2108 y Fu(\))792 2046 y Fi(Z)844 2108 y Fs(f)5 b(dm)15 b Fu([)7 b(sup)963 2147 y Fr(x)p Fq(2)p Fr(P)1029 2152 y Fk(2)1053 2108 y Fn(L)1084 2089 y Fr(k)1084 2119 y Fp(\010)1112 2108 y Fo(1)p Fu(\()p Fs(x)p Fu(\)])p Fs(;)341 2235 y Fn(\024)42 b Fs(\032)442 2242 y Fr(k)463 2235 y Fs(c)491 2173 y Fi(Z)543 2235 y Fs(f)5 b(;)23 b Fu(b)o(y)15 b(the)g(c)o(hoices)h(2)p Fs(:)p Fu(,)e(4)p Fs(:)h Fu(and)g(5)p Fs(:)g Fu(ab)q(o)o(v)o(e)g(and)g(h)o(yp)q(othesis)h (\()p Fo(Exp1)p Fu(\))f(,)-12 2352 y(this)h(conclude)h(the)e(pro)q(of)g (of)g(the)g(lemma.)1138 b Fn(4)-12 2530 y Fh(2.3)56 b(Computation)18 b(of)g(the)h(pro)s(jectiv)n(e)e(diameter)f(and)j(conclusion.)-12 2615 y Fu(In)c(order)f(to)f(obtain)i(the)f(sp)q(eed)h(of)f(con)o(v)o (ergence)g(of)g(the)g(iterates)g(of)g(the)g(transfer)g(op)q(erator)f (to)g(the)i(sp)q(ectral)-12 2672 y(pro)s(jection,)g(it)g(remains)h(to)e (estimate)i(the)f(pro)s(jectiv)o(e)g(diameter)g(of)g Fn(L)1231 2655 y Fr(k)1231 2685 y Fp(\010)1259 2672 y Fu(\()p Fs(C)s Fu(\))f(in)i Fs(C)s Fu(.)928 2796 y(12)p eop %%Page: 13 13 13 12 bop -12 8 a Fo(Lemma)18 b(2.4)k Fg(The)15 b(pro)s(jectiv)o(e)g (diameter)h(of)f Fn(L)840 -9 y Fr(k)840 21 y Fp(\010)868 8 y Fs(C)j Fg(in)e Fs(C)h Fg(is)f(b)q(ounded)h(b)o(y)e Fu(2)8 b(log)g(max)1506 -43 y Fi(\020)1538 -13 y Fp(1+)p Fr(\015)p 1538 -3 66 2 v 1538 23 a Fp(1)p Fq(\000)p Fr(\015)1608 8 y Fs(;)1633 -10 y Fp(2)p Fr(\013)1674 -22 y Ff(0)1685 -10 y Fp(+)p Fr(\013)p 1633 -3 102 2 v 1673 23 a(\013)1740 -43 y Fi(\021)1767 8 y Fg(.)-12 112 y Fo(Pro)q(of)d Fu(:)40 b(Let)16 b Fs(f)k Fu(and)15 b Fs(g)i Fu(b)q(e)f(in)g Fn(L)570 96 y Fr(k)570 126 y Fp(\010)598 112 y Fs(C)h Fu(and)f Fs(\021)e(>)f Fu(0)h(suc)o(h)i(that)e Fs(\021)r(f)h Fn(\000)10 b Fs(g)17 b Fu(b)q(elongs)f(to)f Fs(C)s Fu(,)f Fs(\021)i Fu(should)h(v)o(erify:)44 224 y(1.)22 b Fn(8)p Fs(P)d Fn(2)13 b(P)250 231 y Fr(s;t)291 224 y Fu(,)h(0)f Fs(<)444 193 y(\021)p 407 213 99 2 v 407 255 a(\026)p Fu(\()p Fs(P)6 b Fu(\))518 162 y Fi(Z)517 293 y Fr(P)571 224 y Fs(f)f(dm)k Fn(\000)760 193 y Fu(1)p 722 213 V 722 255 a Fs(\026)p Fu(\()p Fs(P)d Fu(\))833 162 y Fi(Z)832 293 y Fr(P)886 224 y Fs(g)r(dm)11 b Fn(\024)i Fs(a\021)1089 162 y Fi(Z)1142 224 y Fs(f)5 b(dm)10 b Fn(\000)g Fs(a)1320 162 y Fi(Z)1373 224 y Fs(g)r(dm)p Fu(,)44 369 y(2.)22 b(for)14 b(an)o(y)h Fs(x)g Fu(and)h Fs(y)h Fu(in)f(the)f(same)g (1-cylinder,)400 486 y Fn(\000)p Fs(b\021)502 424 y Fi(Z)555 486 y Fs(f)5 b(dm)10 b Fu(+)g Fs(b)743 424 y Fi(Z)796 486 y Fs(g)r(dm)41 b Fn(\024)1007 455 y Fs(\021)r Fu(\()p Fs(f)5 b Fu(\()p Fs(x)p Fu(\))j Fn(\000)i Fs(f)5 b Fu(\()p Fs(y)r Fu(\)\))10 b Fn(\000)g Fu(\()p Fs(g)r Fu(\()p Fs(x)p Fu(\))f Fn(\000)h Fs(g)r Fu(\()p Fs(y)r Fu(\)\))p 1007 475 604 2 v 1244 517 a Fs(d)p Fu(\()p Fs(x;)e(y)r Fu(\))925 607 y Fn(\024)42 b Fs(b\021)1053 545 y Fi(Z)1106 607 y Fs(f)5 b(dm)10 b Fn(\000)g Fs(b)1294 545 y Fi(Z)1347 607 y Fs(g)r(dm;)44 739 y Fu(3.)22 b(for)14 b(an)o(y)h Fs(x)e Fn(2)g Fs(P)368 746 y Fp(2)388 739 y Fu(,)371 853 y Fn(\000)p Fs(c)p Fu(\()p Fs(\021)475 791 y Fi(Z)528 853 y Fs(f)5 b(dm)10 b Fn(\000)674 791 y Fi(Z)727 853 y Fs(g)r(dm)p Fu(\))i Fn(\024)h Fs(\021)r(f)5 b Fu(\()p Fs(x)p Fu(\))j Fn(\000)j Fs(g)r Fu(\()p Fs(x)p Fu(\))g Fn(\024)i Fs(c)p Fu(\()p Fs(\021)1275 791 y Fi(Z)1328 853 y Fs(f)5 b(dm)10 b Fn(\000)1474 791 y Fi(Z)1527 853 y Fs(g)r(dm)p Fu(\))p Fs(:)-12 984 y Fu(T)l(o)15 b(ha)o(v)o(e)g(1)p Fs(:)p Fu(,)f Fs(\021)i Fu(should)h(v)o(erify:)332 1190 y Fs(\021)d Fn(\025)35 b Fu(sup)417 1229 y Fr(P)5 b Fq(2P)493 1233 y Fl(s;t)547 1098 y Fs(a)579 1036 y Fi(Z)631 1098 y Fs(g)r(dm)k Fn(\000)i Fu(1)p Fs(=\026)p Fu(\()p Fs(P)6 b Fu(\))926 1036 y Fi(Z)925 1168 y Fr(P)978 1098 y Fs(g)r(dm)p 543 1180 526 2 v 543 1247 a(a)575 1185 y Fi(Z)627 1247 y Fs(f)f(dm)10 b Fn(\000)h Fu(1)p Fs(=\026)p Fu(\()p Fs(P)6 b Fu(\))926 1185 y Fi(Z)925 1316 y Fr(P)978 1247 y Fs(f)f(dm)1089 1190 y(;)j Fu(and)14 b Fs(\021)g Fn(\025)36 b Fu(sup)1282 1229 y Fr(P)5 b Fq(2P)1358 1233 y Fl(s;t)1410 1036 y Fi(Z)1409 1168 y Fr(P)1463 1098 y Fs(g)r(dm)p 1408 1180 144 2 v 1408 1185 a Fi(Z)1407 1316 y Fr(P)1461 1247 y Fs(f)g(dm)1557 1190 y(;)-12 1398 y Fu(By)16 b(\(2.6\),)d(it)i (is)h(su\016cien)o(t)g(to)e(ha)o(v)o(e:)592 1572 y Fs(\021)f Fn(\025)683 1453 y Fi(Z)736 1515 y Fs(g)r(dm)p 681 1561 V 681 1567 a Fi(Z)734 1629 y Fs(f)5 b(dm)838 1572 y Fu(max)930 1508 y Fi(\022)1009 1541 y Fu(1)p 968 1561 105 2 v 968 1603 a(1)10 b Fn(\000)h Fs(\015)1078 1572 y(;)1103 1541 y Fu(2)p Fs(\013)1155 1524 y Fq(0)1177 1541 y Fu(+)f Fs(\013)p 1103 1561 149 2 v 1162 1603 a(\013)1256 1508 y Fi(\023)1297 1572 y Fs(:)-12 1743 y Fu(T)l(o)15 b(ha)o(v)o(e)g(2)p Fs(:)f Fu(and)i(3)p Fs(:)p Fu(,)e(b)o(y)h(lemma)h(2.3,)d(it)j (su\016ces)f(that)g Fs(\021)h Fu(satis\014es:)768 1914 y Fs(\021)e Fn(\025)860 1795 y Fi(Z)913 1857 y Fs(g)r(dm)p 858 1903 144 2 v 858 1909 a Fi(Z)911 1970 y Fs(f)5 b(dm)1012 1883 y Fu(1)k(+)i Fs(\015)p 1012 1903 105 2 v 1012 1945 a Fu(1)e Fn(\000)i Fs(\015)1121 1914 y(:)-12 2085 y Fu(Similarly)l(,)17 b(let)f Fs(\020)g(>)d Fu(0)i(b)q(e)h(suc)o(h)f(that)g Fs(g)c Fn(\000)f Fs(\020)s(f)18 b Fn(2)13 b Fs(C)s Fu(.)20 b(It)15 b(su\016ces)h(that)e Fs(\020)19 b Fu(v)o(eri\014es:)544 2255 y Fs(\020)d Fn(\024)634 2137 y Fi(Z)688 2199 y Fs(g)r(dm)p 633 2245 144 2 v 633 2250 a Fi(Z)686 2312 y Fs(f)5 b(dm)789 2255 y Fu(min)872 2191 y Fi(\024)896 2255 y Fu(1)10 b Fn(\000)g Fs(\015)s(;)1026 2225 y Fu(1)g Fn(\000)g Fs(\015)p 1026 2245 105 2 v 1026 2287 a Fu(1)g(+)g Fs(\015)1135 2255 y(;)1220 2225 y(\013)p 1160 2245 149 2 v 1160 2287 a Fu(2)p Fs(\013)1212 2274 y Fq(0)1234 2287 y Fu(+)g Fs(\013)1314 2191 y Fi(\025)1345 2255 y Fs(:)-12 2447 y Fu(So,)15 b(the)g(diameter)h(\001)f(of)g Fn(L)468 2431 y Fr(k)468 2461 y Fp(\010)495 2447 y Fs(C)j Fu(in)e Fs(C)i Fu(is)e(b)q(ounded)h(b)o(y)e(2)8 b(log)f(max)1133 2397 y Fi(\020)1166 2427 y Fp(1+)p Fr(\015)p 1166 2437 66 2 v 1166 2463 a Fp(1)p Fq(\000)p Fr(\015)1236 2447 y Fs(;)1261 2429 y Fp(2)p Fr(\013)1302 2417 y Ff(0)1312 2429 y Fp(+)p Fr(\013)p 1261 2436 102 2 v 1300 2463 a(\013)1368 2397 y Fi(\021)1395 2447 y Fu(.)465 b Fn(4)3 2512 y Fu(The)16 b(follo)o(wing)g(lemma)f(sho)o(ws)g(that)f(an)o(y)h(function)h(in)g Fs(L)f Fu(can)h(b)q(e)f(pushed)i(in)o(to)e(the)g(cone)h Fs(C)s Fu(.)-12 2615 y Fo(Lemma)i(2.5)k Fg(F)l(or)13 b(an)o(y)g Fs(f)18 b Fn(2)13 b Fs(L)p Fg(,)g(if)h Fs(a)f(>)g Fu(1)p Fg(,)g Fs(b)f(>)h(K)s Fu(\()p Fs(h)p Fu(\))g Fg(and)g Fs(c)f(>)h Fu(sup)1173 2626 y Fr(x)p Fq(2)p Fr(P)1239 2631 y Fk(2)1266 2615 y Fn(j)p Fs(h)p Fu(\()p Fs(x)p Fu(\))p Fn(j)f Fg(then)i(there)f(exists)h Fs(C)s Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\025)h Fu(0)-12 2672 y Fg(suc)o(h)j(that)e Fs(f)h Fu(+)c Fs(C)s Fu(\()p Fs(f)5 b Fu(\))p Fs(h)14 b Fg(b)q(elong)j(to)d Fs(C)s Fg(,)h(moreo)o(v)o(er)f Fs(C)s Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\024)h Fg(Ct)h Fn(k)p Fs(f)5 b Fn(k)p Fg(.)20 b(In)15 b(particular,)h Fs(h)f Fg(b)q(elongs)h(to)f Fs(C)928 2796 y Fu(13)p eop %%Page: 14 14 14 13 bop -12 1 a Fo(Pro)q(of)5 b Fu(:)40 b(Let)16 b Fs(f)h Fn(2)c Fs(L)p Fu(,)i Fs(C)s Fu(\()p Fs(f)5 b Fu(\))14 b(should)j(satis\014es)56 207 y Fn(\017)23 b Fs(C)s Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\025)h Fu(max)353 62 y Fi(0)353 142 y(B)353 169 y(B)353 196 y(B)353 223 y(B)353 252 y(@)393 207 y Fn(\000)433 54 y Fi(Z)432 185 y Fr(P)486 116 y Fs(f)5 b(dm)p 433 197 144 2 v 456 239 a(\026)p Fu(\()p Fs(P)h Fu(\))582 207 y Fs(;)622 148 y Fu(1)p Fs(=\026)p Fu(\()p Fs(P)g Fu(\))774 87 y Fi(Z)799 190 y Fr(P)835 148 y Fs(f)f(d\026)11 b Fn(\000)f Fs(a)1001 87 y Fi(Z)1054 148 y Fs(f)5 b(d\026)p 622 197 511 2 v 826 239 a(a)10 b Fn(\000)g Fu(1)1152 207 y Fs(P)19 b Fn(2)13 b Fs(P)1272 214 y Fp(1)1292 62 y Fi(1)1292 142 y(C)1292 169 y(C)1292 196 y(C)1292 223 y(C)1292 252 y(A)1332 207 y Fu(,)56 493 y Fn(\017)23 b Fs(C)s Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\025)266 436 y Fs(K)s Fu(\()p Fs(f)5 b Fu(\))k Fn(\000)i Fs(b)454 374 y Fi(Z)506 436 y Fs(f)p 266 482 268 2 v 310 524 a(b)f Fn(\000)g Fs(K)s Fu(\()p Fs(h)p Fu(\))538 493 y(,)56 695 y Fn(\017)23 b Fs(C)s Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\025)20 b Fu(sup)261 735 y Fr(x)p Fq(2)p Fr(P)327 740 y Fk(2)356 639 y Fn(j)p Fs(f)5 b Fu(\()p Fs(x)p Fu(\))p Fn(j)k(\000)i Fs(c)554 577 y Fi(Z)606 639 y Fs(f)5 b(dm)p 356 685 341 2 v 432 727 a(c)10 b Fn(\000)h(j)p Fs(h)p Fu(\()p Fs(x)p Fu(\))p Fn(j)702 695 y Fu(.)1873 817 y Fn(4)3 873 y Fu(W)l(e)16 b(will)g(alw)o(a)o(ys)f (assume)g(that)f Fs(a)p Fu(,)h Fs(b)g Fu(and)g Fs(c)g Fu(satisfy)g(the)g(h)o(yp)q(othesis)h(of)f(lemma)h(2.5.)-12 930 y(Let)i Fs(\024)g Fu(=)f(\(tanh\(\001)p Fs(=)p Fu(4\)\))415 913 y Fp(1)p Fr(=k)470 930 y Fu(.)27 b(Using)19 b(the)f(fact)f(that)g Fs(h)h Fu(b)q(elongs)h(to)e Fs(C)s Fu(,)h(the)g(results)g(of)g(section) g(1.2)f(giv)o(e)h(for)-12 986 y Fs(d)12 b Fn(\025)h Fu(max\()p Fs(b;)8 b Fu(2)p Fs(bD)297 993 y Fp(1)314 986 y Fs(;)g(c)p Fu(\):)505 1064 y Fn(8)p Fs(f)18 b Fn(2)13 b Fs(C)s(;)22 b Fn(kL)738 1045 y Fr(n)738 1075 y Fp(\010)766 1064 y Fs(f)15 b Fn(\000)c Fs(h)883 1002 y Fi(Z)935 1064 y Fs(f)5 b(dm)p Fn(k)1049 1071 y Fr(d)1082 1064 y Fn(\024)13 b Fu(Ct)h Fs(\024)1221 1045 y Fr(n)1253 1002 y Fi(Z)1306 1064 y Fs(f)5 b(dm)-12 1168 y Fu(since)17 b(the)e(norms)g Fn(k)f(k)h Fu(and)h Fn(k)f(k)540 1175 y Fr(d)575 1168 y Fu(are)g(equiv)m(alen)o(t,)515 1291 y Fn(8)p Fs(f)j Fn(2)13 b Fs(C)s(;)22 b Fn(kL)748 1272 y Fr(n)748 1302 y Fp(\010)776 1291 y Fs(f)15 b Fn(\000)c Fs(h)893 1229 y Fi(Z)946 1291 y Fs(f)5 b(dm)p Fn(k)12 b(\024)h Fu(Ct)h Fs(\024)1211 1272 y Fr(n)1242 1229 y Fi(Z)1296 1291 y Fs(f)5 b(dm)-12 1414 y Fu(Let)16 b Fs(f)h Fn(2)c Fs(L)i Fu(and)h Fs(f)309 1421 y Fr(C)351 1414 y Fu(=)d Fs(f)i Fu(+)c Fs(C)s Fu(\()p Fs(f)5 b Fu(\))p Fs(h)p Fu(.)244 1537 y Fn(kL)298 1518 y Fr(n)298 1548 y Fp(\010)326 1537 y Fs(f)15 b Fn(\000)10 b Fs(h)442 1475 y Fi(Z)495 1537 y Fs(f)5 b(dm)p Fn(k)41 b(\024)h(kL)781 1518 y Fr(n)781 1548 y Fp(\010)809 1537 y Fs(f)831 1544 y Fr(C)871 1537 y Fn(\000)10 b Fs(h)950 1475 y Fi(Z)1003 1537 y Fs(f)1025 1544 y Fr(C)1055 1537 y Fs(dm)p Fn(k)f Fu(+)i Fs(C)s Fu(\()p Fs(f)5 b Fu(\))p Fn(kL)1350 1518 y Fr(n)1350 1548 y Fp(\010)1376 1537 y Fs(h)11 b Fn(\000)f Fs(h)1492 1475 y Fi(Z)1545 1537 y Fs(hdm)p Fn(k)650 1655 y Fu(=)42 b Fn(kL)781 1636 y Fr(n)781 1666 y Fp(\010)809 1655 y Fs(f)831 1662 y Fr(C)871 1655 y Fn(\000)10 b Fs(h)950 1593 y Fi(Z)1003 1655 y Fs(f)1025 1662 y Fr(C)1055 1655 y Fs(dm)p Fn(k)650 1772 y(\024)42 b Fu(Ct)30 b Fs(\024)834 1754 y Fr(n)865 1710 y Fi(Z)918 1772 y Fs(f)940 1779 y Fr(C)970 1772 y Fs(dm)15 b Fu(since)h Fs(f)1182 1779 y Fr(C)1225 1772 y Fn(2)c Fs(C)650 1862 y Fn(\024)42 b Fu(Ct)30 b Fs(\024)834 1843 y Fr(n)858 1862 y Fn(k)p Fs(f)5 b Fn(k)p Fs(:)876 b Fu(\(2.9\))-12 1964 y(\(2.9\))14 b(implies)j(exp)q(onen)o(tial)g(mixing)f(for)f Fs(g)f Fn(2)e Fs(L)824 1948 y Fp(1)844 1964 y Fu(\()p Fs(m)p Fu(\))j(and)g Fs(f)20 b Fu(suc)o(h)c(that)e Fs(f)5 b(h)16 b Fu(b)q(elong)g(to)e Fs(L)i Fu(\(recall)g(\(1.1\)\):)490 2066 y Fn(j)o Fs(\026)p Fu(\()p Fs(g)c Fn(\016)e Fs(\033)642 2048 y Fr(n)665 2066 y Fs(f)5 b Fu(\))10 b Fn(\000)g Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fs(\026)p Fu(\()p Fs(f)5 b Fu(\))p Fn(j)11 b(\024)i Fu(Ct)i Fs(\024)1106 2048 y Fr(n)1130 2066 y Fn(k)p Fs(f)5 b(h)p Fn(kk)p Fs(g)r Fn(k)1299 2076 y Fr(L)1323 2067 y Fk(1)1338 2076 y Fp(\()p Fr(m)p Fp(\))1399 2066 y Fs(;)-12 2168 y Fu(it)16 b(also)f(implies)i (that)e Fn(L)409 2175 y Fp(\010)452 2168 y Fu(is)g(quasi-compact)h(on)f Fs(L)p Fu(.)20 b(This)c(concludes)g(the)g(pro)q(of)e(of)h(theorem)g (1.3.)128 b Fn(4)-12 2312 y Fv(3)67 b(Dynamics)23 b(without)g(big)g (branc)n(hes)g(at)f(in\014nit)n(y)-12 2413 y Fu(In)17 b(this)f(section,)g(w)o(e)f(pro)o(v)o(e)g(theorem)h(1.4.)k(So)c(w)o(e)f (don't)g(assume)h(\()p Fo(Exp1)p Fu(\))f(an)o(ymore)g(but)h(w)o(e)f (assume)h(that)-12 2470 y(\006)j(has)f(no)g(big)i(branc)o(hes)f(at)e (in\014nit)o(y)j(and)f(that)f(\010)g(satis\014es)h(\()p Fo(SA)p Fu(\))f(and)g(\()p Fo(S-Exp1)p Fu(\))h(\(de\014nition)h(page)e (5\).)-12 2526 y(Let)e(us)f(recall)h(that)f(\()p Fo(S-Exp1)p Fu(\))g(implies)i(\()p Fo(K)p Fu(\).)d(So,)h(let)g(us)h(note)f Fs(\026)e Fu(=)g Fs(hm)i Fu(where)g Fs(h)g Fu(is)h(the)f(normalized)i (\014xed)-12 2583 y(p)q(oin)o(t)f(for)e Fn(L)207 2590 y Fp(\010)251 2583 y Fu(giv)o(en)h(b)o(y)g(theorem)g(1.1.)928 2796 y(14)p eop %%Page: 15 15 15 14 bop -12 1 a Fo(Remark)17 b(3.1)22 b Fu(Moreo)o(v)o(er,)d(w)o(e)f (will)j(assume)e(that)30 b(lim)917 31 y Fr(j)r Fq(!1)1012 1 y Fs(\016)1034 -18 y Fq(0)1032 13 y Fr(k)q(;j)1098 1 y Fu(=)20 b(1.)31 b(Indeed,)21 b(if)e(lim)8 b(sup)1456 39 y Fr(j)r Fq(!1)1577 1 y Fs(\016)1599 -18 y Fq(0)1597 13 y Fr(k)q(;j)1663 1 y Fs(<)20 b Fu(1,)f(then)h(\010)-12 89 y(satis\014es)14 b(\()p Fo(Exp1)p Fu(\))f(and)h(the)f(results)h(of)f (the)h(preceding)h(section)f(sho)o(w)f(that)g(the)g(con)o(v)o(ergence)h (of)f(the)h(iterates)-12 145 y(of)h Fn(L)71 152 y Fp(\010)114 145 y Fu(to)g(the)g(sp)q(ectral)h(pro)s(jection)f(is)g(uniform)h(on)f (\006)g(and)h(exp)q(onen)o(tial.)-12 235 y(F)l(or)h Fs(N)k Fn(2)c Fj(N)p Fu(,)f Fn(k)h(k)303 242 y Fr(N)354 235 y Fu(is)h(the)g(uniform)g(norm)f(on)g([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)d Fn(j)-8 b Fu(])15 b(\(notation)i(page)h(9\).)26 b(The)17 b(cones)h(of)f(the)h(preceding)-12 291 y(section)e(ma)o(y)e(b) q(e)i(adapted)g(to)e(giv)o(e)i(the)f(follo)o(wing)h(result.)-12 397 y Fo(Prop)q(osition)j(3.1)j Fg(If)h Fu(\006)f Fg(has)g(no)g(big)h (branc)o(hes)g(at)e(in\014nit)o(y)j(and)f Fu(\010)f Fg(v)o(eri\014es)h (\()p Fo(S-Exp1)p Fg(\))f(then,)i Fn(8)p Fs(N)29 b Fn(\025)-12 454 y Fs(n)15 461 y Fp(1)35 454 y Fs(;)8 b Fn(8)p Fs(f)17 b Fn(2)c Fs(L)p Fg(,)i Fn(8)p Fs(k)f Fn(\025)f Fs(k)357 461 y Fp(1)376 454 y Fg(,)i(there)g(exists)h(a)f(sequence)h Fu(\()p Fs(\013)919 461 y Fr(j)937 454 y Fu(\()p Fs(N)5 b Fu(\)\))1033 461 y Fr(j)r Fq(2)p Fm(N)1098 454 y Fg(,)14 b Fs(\013)1154 461 y Fr(j)1173 454 y Fu(\()p Fs(N)5 b Fu(\))11 b Fn(!)i Fu(0)i Fg(suc)o(h)h(that)485 579 y Fn(kL)539 557 y Fr(k)q(j)539 594 y Fp(\010)577 579 y Fs(f)f Fn(\000)10 b Fs(h)693 517 y Fi(Z)746 579 y Fs(f)5 b Fn(k)796 586 y Fr(N)842 579 y Fn(\024)13 b Fs(\013)919 586 y Fr(j)953 579 y Fn(k)p Fs(f)5 b Fn(k)k Fu(+)i Fs(m)p Fu(\([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)d Fn(j)-8 b Fu(])1261 560 y Fr(c)1276 579 y Fu(\))8 b(sup)f Fs(f)e(;)403 b Fu(\(3.1\))-12 712 y Fg(Moreo)o(v)o(er,)18 b Fs(\013)234 719 y Fr(j)270 712 y Fg(can)h(b)q(e)g(expressed)g(in)g(terms)f(of)870 678 y Fi(Q)913 690 y Fr(j)913 727 y(`)p Fp(=0)982 712 y Fs(\016)1004 696 y Fq(0)1002 726 y Fr(k)q(;`)1048 712 y Fg(.)29 b(A)18 b(suitable)i(c)o(hoice)f(of)f Fs(N)23 b Fg(with)c(resp)q(ect)f(to)g Fs(j)-12 769 y Fg(giv)o(es)e(theorem)e (1.4.)-12 858 y Fu(The)20 b(pro)q(of)e(of)h(this)g(prop)q(osition)h (follo)o(ws)f(theorem)g(1.3's)f(pro)q(of.)31 b(The)19 b(p)q(oin)o(t)h(is)g(that)e(for)h Fs(k)h Fu(and)f Fs(n)g Fu(large)-12 915 y(enough,)g Fn(L)192 898 y Fr(k)192 928 y Fp(\010)220 915 y Fo(1)p Fu(\()p Fs(x)p Fu(\),)e Fs(x)g Fn(2)g Fu([)p Fs(n)p Fu(],)h(is)g(strictly)h(smaller)f(than)g(1) g(but)g(this)g(b)q(ound)h(is)f(not)f(uniform)i(in)f Fs(n)p Fu(.)28 b(This)19 b(is)-12 971 y(wh)o(y)12 b(w)o(e)f(shall)i(use)f(a)g (family)g(of)f(cones)h(sp)q(eci\014ed)i(only)f(on)e(the)h(set)g([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)e Fu(+)s Fs(k)q(K)s(j)s Fn(j)-8 b Fu(],)12 b(that)f(is)h(a)o(w)o(a)o(y)e(from)h(in\014nit)o(y)l(.)-12 1084 y(F)l(or)20 b(\014xed)h Fs(N)26 b Fn(\025)c Fs(n)337 1091 y Fp(1)377 1084 y Fu(and)f Fs(k)i Fn(\025)e Fs(k)598 1091 y Fp(1)618 1084 y Fu(,)g(let)g Fs(D)q Fu(\()p Fs(j)s Fu(\))e(denotes)i(the)f(set)h([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)16 b Fu(+)e Fs(k)q(K)s(j)s Fn(j)-8 b Fu(])20 b(and)g(for)g Fs(f)27 b Fn(2)22 b Fs(L)p Fu(,)f Fs(K)1833 1091 y Fr(j)1851 1084 y Fu(\()p Fs(f)5 b Fu(\))-12 1141 y(denotes)16 b(the)f(Lipsc)o (hitz)i(constan)o(t)d(of)h(the)g(function)h Fs(f)i Fu(:)12 b Fs(D)q Fu(\()p Fs(j)s Fu(\))g Fn(!)h Fj(R)m Fu(.)-12 1197 y(Let)k(us)f(consider)h(the)f(follo)o(wing)h(family)g(of)f(cones.) 23 b(Let)16 b Fs(a;)8 b(b;)g(c)14 b Fu(b)q(e)j(p)q(ositiv)o(e)g(real)f (n)o(um)o(b)q(ers,)h Fs(j;)8 b(s;)g(t)13 b Fn(2)h Fj(N)h Fu(and)-12 1254 y Fn(P)20 1261 y Fr(s;t)76 1254 y Fu(the)g(\014nite)h (partition)g(of)f(\006)g(de\014ned)h(page)f(9,)g Fs(C)909 1231 y Fr(j)906 1268 y(N)939 1254 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))k(is)k(the)f(set)g(of)g(functions)h Fs(f)k Fu(of)15 b Fs(L)g Fu(suc)o(h)g(that:)44 1371 y(1.)22 b Fn(8)p Fs(P)d Fn(2)13 b(P)250 1378 y Fr(s;t)291 1371 y Fs(;)8 b(P)18 b Fn(\032)13 b Fs(D)q Fu(\()p Fs(j)s Fu(\))p Fs(;)8 b Fu(0)i Fs(<)649 1341 y Fu(1)p 611 1361 99 2 v 611 1402 a Fs(\026)p Fu(\()p Fs(P)c Fu(\))722 1309 y Fi(Z)721 1441 y Fr(P)775 1371 y Fs(f)f(dm)12 b Fn(\024)h Fs(a)972 1309 y Fi(Z)947 1443 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1039 1371 y Fs(f)5 b(dm)p Fu(,)44 1559 y(2.)22 b Fs(K)141 1566 y Fr(j)159 1559 y Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\024)h Fs(b)323 1497 y Fi(Z)299 1631 y Fr(D)q Fp(\()p Fr(j)r Fp(\))390 1559 y Fs(f)5 b(dm)p Fu(,)44 1747 y(3.)78 b(sup)102 1788 y Fr(x)p Fq(2)p Fr(P)168 1793 y Fk(2)185 1788 y Fq(\\)p Fr(D)q Fp(\()p Fr(j)r Fp(\))290 1747 y Fn(j)p Fs(f)5 b Fu(\()p Fs(x)p Fu(\))p Fn(j)11 b(\024)i Fs(c)505 1685 y Fi(Z)481 1819 y Fr(D)q Fp(\()p Fr(j)r Fp(\))572 1747 y Fs(f)5 b(dm)p Fu(.)-12 1905 y(The)16 b(argumen)o(ts)e(of)h(the)g (pro)q(of)g(of)g(lemma)g(2.2)f(pro)o(v)o(e)h(that)f(the)i(norm)293 2070 y Fn(k)p Fs(f)5 b Fn(k)366 2077 y Fr(d;j)424 2070 y Fu(=)13 b(max\()p Fs(d)h Fn(j)648 2008 y Fi(Z)623 2142 y Fr(D)q Fp(\()p Fr(j)r Fp(\))715 2070 y Fs(f)5 b(dm)p Fn(j)p Fs(;)j Fu(2)871 1963 y Fi(\014)871 1990 y(\014)871 2018 y(\014)871 2045 y(\014)871 2072 y(\014)871 2100 y(\014)871 2127 y(\014)889 1967 y(R)886 2048 y Fr(P)926 2004 y Fs(f)d(dm)p 889 2060 129 2 v 898 2101 a(m)p Fu(\()p Fs(P)h Fu(\))1022 1963 y Fi(\014)1022 1990 y(\014)1022 2018 y(\014)1022 2045 y(\014)1022 2072 y(\014)1022 2100 y(\014)1022 2127 y(\014)1060 2070 y Fs(P)19 b Fn(2)13 b(P)1183 2077 y Fr(s;t)1234 2070 y Fn(\\)d Fs(D)q Fu(\()p Fs(j)s Fu(\))p Fs(;)e Fn(k)p Fs(f)d Fn(k)1465 2077 y Fr(N)t Fp(+)p Fr(k)q(K)r(j)1591 2070 y Fu(\))211 b(\(3.2\))-12 2248 y(is)17 b(adapted)e(to)h(the)g(cone)g Fs(C)485 2226 y Fr(j)482 2263 y(N)515 2248 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))13 b(pro)o(vided)j Fs(d)e Fn(\025)g Fu(max\(2)p Fs(bD)1128 2255 y Fp(1)1146 2248 y Fs(;)8 b(c)p Fu(\).)20 b(Let)c(us)g(remark)g(that)f(for)g(an)o(y)h Fs(d)d(>)h Fu(0,)-12 2311 y(the)g(norm)f Fn(k)h(k)242 2318 y Fr(d;j)301 2311 y Fu(is)h(equiv)m(alen)o(t)g(to)e(the)h(uniform)g(norm)f(on)h Fs(D)q Fu(\()p Fs(j)s Fu(\).)k(Moreo)o(v)o(er,)12 b Fs(C)1414 2289 y Fr(j)1411 2325 y(N)1444 2311 y Fu(\()p Fs(a;)c(b;)g(c)p Fu(\))j(is)j(a)f(con)o(v)o(ex)h(cone)-12 2391 y(whic)o(h)g(is)g(closed) g(for)e(the)h(norm)g Fn(k)f(k)610 2400 y Fr(D)q Fp(\()p Fr(j)r Fp(\))699 2391 y Fu(and)h Fs(C)821 2369 y Fr(j)818 2406 y(N)851 2391 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))e Fn(\\)g(\000)p Fs(C)1106 2369 y Fr(j)1103 2406 y(N)1133 2391 y Fu(\()p Fs(a;)i(b;)g(c)p Fu(\))i(=)j Fn(;)p Fu(.)19 b(Of)13 b(course,)g Fs(f)18 b Fn(!)1712 2329 y Fi(Z)1688 2463 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1779 2391 y Fs(f)5 b(dm)13 b Fu(is)-12 2524 y(also)h(adapted)g(to)g Fs(C)342 2502 y Fr(j)339 2538 y(N)372 2524 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\).)17 b(When)d(there)g(w)o(on't)f(b)q(e)i(an)o(y)f(am)o(biguit)o(y) l(,)g(w)o(e)g(will)i(simply)f(write)g Fs(C)1728 2502 y Fr(j)1725 2538 y(N)1772 2524 y Fu(instead)-12 2586 y(of)g Fs(C)76 2564 y Fr(j)73 2601 y(N)106 2586 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\).)928 2796 y(15)p eop %%Page: 16 16 16 15 bop -12 1 a Fo(Outline)18 b(of)e(the)g(pro)q(of)g(of)g(prop)q (osition)i(3.1)13 b Fg(W)l(e)h(will)i(pro)o(v)o(e)d(that)g(pro)o(vided) i Fs(a)p Fg(,)f Fs(b)p Fg(,)f Fs(c)p Fg(,)g Fs(s)p Fg(,)h Fs(t)p Fg(,)g Fs(N)19 b Fg(and)14 b Fs(k)h Fg(are)-12 57 y(large)g(enough)h(and)f(w)o(ell)i(c)o(hosen,)e(for)f(all)i Fs(j)f Fn(2)e Fj(N)p Fg(,)475 160 y Fn(L)506 141 y Fr(k)506 171 y Fp(\010)534 160 y Fs(C)570 137 y Fr(j)567 174 y(N)600 160 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))h Fn(\032)k Fs(C)835 137 y Fr(j)r Fq(\000)p Fp(1)832 174 y Fr(N)899 160 y Fu(\()p Fs(\015)s(a;)8 b(\015)s(b)o(;)f(\016)1075 167 y Fr(j)1091 160 y Fs(c)p Fu(\))12 b Fn(\032)h Fs(C)1225 137 y Fr(j)r Fq(\000)p Fp(1)1222 174 y Fr(N)1288 160 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))-12 270 y Fg(where)16 b Fu(0)c Fs(<)h(\015)i(<)e Fu(1)p Fg(,)h Fs(\016)360 277 y Fr(j)394 270 y Fg(satis\014es)h(for)g(an)o(y)g Fs(`)e Fn(2)g Fj(N)p Fg(,)852 236 y Fi(Q)895 249 y Fr(`)895 284 y(j)921 270 y Fs(\016)941 277 y Fr(j)972 270 y Fn(\024)g Fg(Ct)1093 236 y Fi(Q)1136 249 y Fr(`)1136 284 y(j)1162 270 y Fs(\016)1184 254 y Fq(0)1182 284 y Fr(k)q(;j)1245 270 y Fg(and)i Fu(tanh)1437 246 y Fp(\001)1466 251 y Fl(j)p 1437 260 46 2 v 1451 286 a Fp(4)1500 270 y Fn(\024)e Fs(\016)1568 277 y Fr(j)1602 270 y Fg(where)i Fu(\001)1771 277 y Fr(j)1805 270 y Fg(is)g(the)-12 339 y(diameter)h(of)e Fn(L)260 322 y Fr(k)260 352 y Fp(\010)288 339 y Fs(C)324 317 y Fr(j)321 353 y(N)354 339 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))13 b Fg(in)j Fs(C)598 317 y Fr(j)r Fq(\000)p Fp(1)595 353 y Fr(N)661 339 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))p Fg(.)-12 401 y(No)o(w,)19 b(let)h Fn(C)i Fu(=)280 367 y Fi(T)318 415 y Fr(j)344 401 y Fs(C)380 379 y Fr(j)377 416 y(N)410 401 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))p Fg(.)29 b(Pro)o(vided)20 b Fs(a)p Fg(,)g Fs(b)f Fg(and)g Fs(c)g Fg(are)g(large)g(enough,)i Fs(h)e Fg(b)q(elong)h(to)f Fn(C)s Fg(.)32 b(Then,)20 b(the)-12 458 y(results)c(of)f(section)g(1.2) g(giv)o(e)g(for)g(an)o(y)f Fs(f)k Fn(2)13 b(C)s Fg(:)612 602 y Fs(\022)633 614 y Fr(C)660 603 y Fk(0)658 626 y Fl(N)687 614 y Fp(\()p Fr(a;b;c)p Fp(\))785 602 y Fu(\()p Fn(L)834 580 y Fr(k)q(j)834 617 y Fp(\010)872 602 y Fs(f)r(;)8 b(h)p Fu(\))k Fn(\024)1021 525 y Fi( )1057 543 y Fr(j)r Fq(\000)p Fp(1)1059 559 y Fi(Y)1058 658 y Fr(`)p Fp(=1)1126 602 y Fs(\016)1146 609 y Fr(`)1163 525 y Fi(!)1221 602 y Fu(\001)1259 609 y Fr(j)1277 602 y Fs(;)-12 756 y Fg(\()p Fo(S-Exp1)p Fg(\))18 b(giv)o(es)h(that)e Fu(\()439 722 y Fi(Q)482 734 y Fr(j)r Fq(\000)p Fp(1)482 771 y Fr(`)p Fp(=1)553 756 y Fs(\016)573 763 y Fr(`)589 756 y Fu(\))h(\001)663 763 y Fr(j)700 756 y Fg(go)q(es)g(to)f Fu(0)h Fg(when)h Fs(j)h Fg(go)q(es)e(to)g(in\014nit)o(y)l(.)30 b(Using)19 b Fn(k)f(k)1599 763 y Fr(d;)p Fp(0)1665 756 y Fg(and)1756 720 y Fi(R)1777 772 y Fr(D)q Fp(\(0\))1873 756 y Fg(as)-12 813 y(adapted)d(norm)g(and)h(homogeneous)f(form,)f(w)o(e)h(obtain)g (prop)q(osition)h(3.1)e(and)i(theorem)f(1.4.)3 894 y Fu(Let)h(us)f(b)q(egin)i(the)e(pro)q(of)g(of)f(prop)q(osition)i(3.1)f (with)g(the)h(follo)o(wing)f(simple)i(prop)q(ert)o(y)e(of)g(the)g(sets) g Fs(D)q Fu(\()p Fs(j)s Fu(\).)-12 1000 y Fo(Lemma)j(3.2)k Fg(Since)17 b Fu(\006)e Fg(has)g(no)g(big)h(branc)o(hes)f(at)g (in\014nit)o(y)l(,)h(for)f(an)o(y)g Fs(`)d Fn(2)h Fj(N)h Fg(w)o(e)h(ha)o(v)o(e,)744 1103 y Fs(\033)772 1084 y Fq(\000)p Fr(k)820 1103 y Fs(D)q Fu(\()p Fs(`)10 b Fn(\000)g Fu(1\))i Fn(\032)h Fs(D)q Fu(\()p Fs(`)p Fu(\))p Fs(:)-12 1205 y Fg(Moreo)o(v)o(er,)h(the)h(sequence)h Fs(m)p Fu(\()p Fs(D)q Fu(\()p Fs(`)p Fu(\))9 b Fn(n)h Fs(\033)689 1188 y Fq(\000)p Fr(k)737 1205 y Fs(D)q Fu(\()p Fs(`)g Fn(\000)g Fu(1\)\))15 b Fg(is)g(summable.)-12 1294 y Fo(Pro)q(of)5 b Fu(:)38 b(The)14 b(fact)f(that)g(\006)g(is)h(without)g(big)g(branc)o (hes)g(at)f(in\014nit)o(y)i(directly)g(leads)f(to)f Fs(\033)1534 1278 y Fq(\000)p Fr(k)1582 1294 y Fs(D)q Fu(\()p Fs(`)7 b Fn(\000)g Fu(1\))12 b Fn(\032)h Fs(D)q Fu(\()p Fs(`)p Fu(\).)-12 1351 y(Let)j(us)f(pro)o(v)o(e)g(that)f(the)h(sequence)i Fs(u)642 1358 y Fr(j)673 1351 y Fu(=)c Fs(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)s Fu(\))8 b Fn(n)i Fs(\033)945 1334 y Fq(\000)p Fr(k)993 1351 y Fs(D)q Fu(\()p Fs(j)i Fn(\000)f Fu(1\)\))j(is)h(summable.)21 b(W)l(e)15 b(ha)o(v)o(e)-12 1407 y Fs(u)14 1414 y Fr(j)45 1407 y Fu(=)e Fs(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)s Fu(\)\))8 b Fn(\000)i Fs(m)p Fu(\()p Fs(\033)405 1391 y Fq(\000)p Fr(k)453 1407 y Fs(D)q Fu(\()p Fs(j)i Fn(\000)f Fu(1\)\))h(=)h Fs(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)s Fu(\)\))7 b Fn(\000)k Fu(\(1)e Fn(\000)i Fs(m)p Fu(\()p Fs(\033)1114 1391 y Fq(\000)p Fr(k)1162 1407 y Fs(D)q Fu(\()p Fs(j)h Fn(\000)e Fu(1\))1336 1391 y Fr(c)1353 1407 y Fu(\)\))15 b(and)526 1532 y Fs(m)p Fu(\()p Fs(\033)612 1513 y Fq(\000)p Fr(k)660 1532 y Fs(D)q Fu(\()p Fs(j)d Fn(\000)e Fu(1\))834 1513 y Fr(c)851 1532 y Fu(\)\))41 b(=)1005 1470 y Fi(Z)1058 1532 y Fo(1)1084 1541 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))1203 1532 y Fl(c)14 b Fn(\016)9 b Fs(\033)1291 1513 y Fr(k)1312 1532 y Fs(dm)928 1650 y Fu(=)1049 1588 y Fi(Z)994 1721 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))1113 1712 y Fl(c)1147 1650 y Fn(L)1178 1631 y Fr(k)1178 1661 y Fp(\010)1206 1650 y Fo(1)p Fs(dm:)-12 1824 y Fu(\()p Fo(S-Exp1)p Fu(\))15 b(implies)j Fn(L)388 1807 y Fr(k)388 1837 y Fp(\010)415 1824 y Fo(1)p Fu(\()p Fs(x)p Fu(\))13 b Fn(\024)g Fu(1)h(on)i Fs(D)q Fu(\()p Fs(j)c Fn(\000)e Fu(1\))839 1807 y Fr(c)868 1824 y Fn(\032)j Fu([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)d Fn(j)-8 b Fu(])1038 1807 y Fr(c)1053 1824 y Fu(,)15 b(so)g(that)601 1926 y Fs(m)p Fu(\()p Fs(\033)687 1907 y Fq(\000)p Fr(k)735 1926 y Fs(D)q Fu(\()p Fs(j)d Fn(\000)e Fu(1\))909 1907 y Fr(c)926 1926 y Fu(\)\))i Fn(\024)h Fs(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)e Fn(\000)g Fu(1\))1254 1907 y Fr(c)1271 1926 y Fu(\))p Fs(:)-12 2028 y Fu(This)16 b(leads)g(to)e Fs(u)287 2035 y Fr(j)318 2028 y Fn(\024)f Fs(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)s Fu(\)\))8 b Fn(\000)j Fs(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)g Fn(\000)g Fu(1\)\))j(and)h Fs(u)971 2035 y Fr(j)1005 2028 y Fu(is)g(summable.)611 b Fn(4)-12 2206 y Fh(3.1)56 b(Con)n(traction)19 b(of)g(the)f(cone.)-12 2292 y Fu(W)l(e)d(are)g(going)h(to)e(pro)o(v)o(e)h(that)f(for)h(an)o(y) g Fs(\015)f(<)f Fu(1)i(and)h Fs(N)r(;)8 b(k)q(;)g(a;)g(b;)f(c;)g(s;)g (t)13 b Fu(w)o(ell)j(c)o(hosen,)-12 2348 y Fn(L)19 2332 y Fr(k)19 2362 y Fp(\010)47 2348 y Fs(C)83 2326 y Fr(j)80 2363 y(N)113 2348 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))i Fn(\032)j Fs(C)349 2326 y Fr(j)r Fq(\000)p Fp(1)346 2363 y Fr(N)412 2348 y Fu(\()p Fs(\015)s(a;)8 b(\015)s(b)o(;)g(\016)589 2355 y Fr(j)604 2348 y Fs(c)p Fu(\))13 b(and)h(the)f(diameter)h(of)g Fn(L)1088 2332 y Fr(k)1088 2362 y Fp(\010)1116 2348 y Fs(C)1152 2326 y Fr(j)1149 2363 y(N)1182 2348 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))i(in)15 b Fs(C)1422 2326 y Fr(j)r Fq(\000)p Fp(1)1419 2363 y Fr(N)1485 2348 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))i(is)k(b)q(ounded)h(b)o(y)-12 2414 y(2)8 b(log)90 2390 y Fp(1+)p Fr(\016)151 2395 y Fl(j)p 90 2403 77 2 v 90 2429 a Fp(1)p Fq(\000)p Fr(\016)151 2434 y Fl(j)187 2414 y Fu(where)15 b Fs(\016)338 2421 y Fr(j)369 2414 y Fu(=)e Fs(\016)439 2397 y Fq(0)437 2428 y Fr(j;k)483 2414 y Fu(\(1)c Fn(\000)i Fs(m)p Fu(\()p Fs(j)s Fu(\)\))695 2397 y Fq(\000)p Fp(1)755 2414 y Fu(and)16 b Fs(m)p Fu(\()p Fs(j)s Fu(\))11 b(=)i Fs(c)i(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)s Fu(\))8 b Fn(n)i Fs(\033)1260 2397 y Fq(\000)p Fr(k)1308 2414 y Fs(D)q Fu(\()p Fs(j)i Fn(\000)f Fu(1\)\).)-12 2527 y Fo(Remark)17 b(3.2)22 b Fu(F)l(or)14 b(an)o(y)g(\014xed)h Fs(c)p Fu(,)e(it)i(exists)f Fs(j)785 2534 y Fp(0)819 2527 y Fu(suc)o(h)g(that)g(for)f(an)o(y)h Fs(j)h Fn(\025)e Fs(j)1272 2534 y Fp(0)1291 2527 y Fu(,)h Fs(m)p Fu(\()p Fs(j)s Fu(\))d Fs(<)i Fu(1)h(and)h Fs(\016)1620 2534 y Fr(j)1651 2527 y Fs(<)e Fu(1.)19 b(In)c(order)-12 2584 y(to)g(mak)o(e)f(the)i(reading)f(easier,)h(w)o(e)f(will)i(alw)o(a)o(ys) d(assume)h(that)g Fs(j)1115 2591 y Fp(0)1147 2584 y Fu(=)e(1.)928 2796 y(16)p eop %%Page: 17 17 17 16 bop -12 1 a Fu(Let)16 b(us)f(note)g Fs(m)e Fu(=)f(sup)398 12 y Fr(j)424 1 y Fs(m)p Fu(\()p Fs(j)s Fu(\),)h(since)j Fs(m)p Fu(\()p Fs(j)s Fu(\))c Fs(<)h Fu(1)i(for)f(all)i Fs(j)f Fn(\025)e Fu(1)i(and)g Fs(m)p Fu(\()p Fs(j)s Fu(\))d Fn(!)h Fu(0,)h(w)o(e)h(ha)o(v)o(e)g Fs(m)d(<)h Fu(1.)-12 114 y(Let)20 b(us)g(\014x)g Fs(\013)g(<)h Fu(1)e Fs(<)i(\013)438 97 y Fq(0)450 114 y Fu(,)f(0)g Fs(<)g(\014)i(<)f Fu(\(1)12 b Fn(\000)i Fs(m)p Fu(\),)20 b Fs(\015)i Fu(=)1012 93 y Fr(\014)p 984 103 77 2 v 984 130 a Fp(1)p Fq(\000)p Fr(m)1085 114 y Fu(and)e Fs(\014)1206 97 y Fq(0)1238 114 y Fs(<)g(\014)r Fu(.)34 b(The)20 b(parameters)e Fs(a;)8 b(b;)g(c)p Fu(,)18 b Fs(s)p Fu(,)j Fs(t)-12 170 y Fu(and)16 b Fs(k)g Fu(are)f(c)o(hosen)g(in)h(the)f(follo)o(wing)h(w)o(a)o(y:)44 258 y(1.)22 b Fs(a)12 b(>)h(\013)215 242 y Fq(0)227 258 y Fu(\(1)d(+)343 240 y Fr(\013)p 328 247 52 2 v 328 274 a Fp(2)p Fr(\013)369 265 y Ff(0)385 258 y Fu(\))p Fs(\014)431 242 y Fq(\000)p Fp(1)478 258 y Fu(,)44 350 y(2.)22 b Fs(k)126 357 y Fp(0)160 350 y Fu(is)16 b(suc)o(h)g(that)e Fn(8)p Fs(k)g(>)f(k)542 357 y Fp(0)561 350 y Fu(,)i Fs(M)5 b(r)660 333 y Fr(k)694 350 y Fs(<)13 b(\014)770 333 y Fq(0)782 350 y Fu(,)44 441 y(3.)22 b Fs(c)12 b Fn(\025)h Fs(M)5 b(a)10 b Fu(+)h(1,)44 533 y(4.)22 b Fs(b)12 b(>)207 515 y Fr(Rc)p 187 522 82 2 v 187 549 a(\014)r Fq(\000)p Fr(\014)258 539 y Ff(0)274 533 y Fu(,)44 624 y(5.)22 b Fs(D)140 631 y Fp(1)172 624 y Fs(<)13 b Fu(1)p Fs(=b)304 607 y Fr(\013)p 291 614 52 2 v 291 640 a Fp(4)p Fr(\013)332 631 y Ff(0)346 624 y Fu(,)44 716 y(6.)22 b(let)16 b Fs(t)184 723 y Fp(0)220 716 y Fu(b)q(e)g(suc)o(h)g(that)f(for)g(all)i Fs(t)d(>)g(t)713 723 y Fp(0)749 716 y Fu(w)o(e)h(ha)o(v)o(e)g Fs(D)959 723 y Fp(2)992 716 y Fs(<)1077 698 y Fr(\013)p 1046 705 85 2 v 1046 732 a Fp(16)p Fr(\013)1105 722 y Ff(0)1115 732 y Fr(c)1136 716 y Fu(.)21 b(Let)16 b Fs(N)k Fu(and)c Fs(t)e Fn(\024)g Fs(N)20 b Fu(b)q(e)c(suc)o(h)g(that)f Fs(t)f(>)g(t)1893 723 y Fp(0)102 772 y Fu(and)819 861 y Fs(D)857 868 y Fp(2)889 861 y Fn(\024)f Fu(2)1000 804 y Fr(N)982 817 y Fi(X)985 914 y Fr(n)p Fp(=)p Fr(t)1056 861 y Fs(m)p Fu(\([)p Fs(n)p Fu(]\))p Fs(;)622 b Fu(\(3.3\))44 997 y(7.)22 b Fs(k)126 1004 y Fp(1)160 997 y Fu(is)16 b(suc)o(h)g(that)e(\(2.2\))g(is)h(satis\014ed)h(for)f Fn(P)838 1004 y Fr(s;t)893 997 y Fu(and)h Fs(k)1006 1004 y Fp(1)1040 997 y Fu(v)o(eri\014es)g(\()p Fo(S-Exp1)p Fu(\),)44 1089 y(8.)22 b Fs(k)14 b Fn(\025)f Fu(max)o(\()p Fs(k)314 1096 y Fp(0)333 1089 y Fs(;)8 b(k)378 1096 y Fp(1)397 1089 y Fu(\).)-12 1177 y(Let)15 b Fs(f)j Fn(2)13 b Fs(C)188 1155 y Fr(j)185 1191 y(N)218 1177 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))j(and)k Fs(P)k Fn(\032)13 b Fs(D)q Fu(\()p Fs(j)e Fn(\000)e Fu(1\),)14 b(lemma)g(3.2)g(implies)j (that)c Fs(\033)1251 1160 y Fq(\000)p Fr(k)1300 1177 y Fs(P)19 b Fn(\032)13 b Fs(D)q Fu(\()p Fs(j)s Fu(\).)18 b(Let)d(us)g(remark)e(that,)-12 1233 y(b)o(y)i(the)h(c)o(hoice)g(6.)j (ab)q(o)o(v)o(e,)220 1330 y Fs(m)p Fu(\()p Fs(\033)306 1314 y Fq(\000)p Fr(k)354 1330 y Fs(P)d Fn(\\)11 b Fs(P)469 1337 y Fp(2)499 1330 y Fn(\\)f Fs(D)q Fu(\()p Fs(j)s Fu(\)\))p 220 1350 433 2 v 252 1392 a Fs(\026)p Fu(\()p Fs(P)c Fu(\))p Fs(m)p Fu(\()p Fs(P)437 1399 y Fp(2)466 1392 y Fn(\\)11 b Fs(D)q Fu(\()p Fs(j)s Fu(\)\))670 1361 y(=)723 1330 y Fs(m)p Fu(\()p Fs(\033)809 1314 y Fq(\000)p Fr(k)857 1330 y Fs(P)16 b Fn(\\)11 b Fs(P)972 1337 y Fp(2)992 1330 y Fu(\))p 723 1350 287 2 v 755 1392 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))p Fs(m)p Fu(\()p Fs(P)940 1399 y Fp(2)960 1392 y Fu(\))1148 1316 y(_)1092 1330 y Fs(m)p Fu(\()p Fs(P)1179 1337 y Fp(2)1199 1330 y Fu(\))p 1019 1350 271 2 v 1019 1392 a Fs(m)p Fu(\()p Fs(P)1106 1399 y Fp(2)1136 1392 y Fn(\\)k Fs(D)q Fu(\()p Fs(j)s Fu(\)\))1307 1361 y Fn(\024)j Fu(2)1383 1330 y Fs(m)p Fu(\()p Fs(\033)1469 1314 y Fq(\000)p Fr(k)1517 1330 y Fs(P)j Fn(\\)11 b Fs(P)1632 1337 y Fp(2)1652 1330 y Fu(\))p 1383 1350 287 2 v 1415 1392 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))p Fs(m)p Fu(\()p Fs(P)1600 1399 y Fp(2)1620 1392 y Fu(\))1674 1361 y Fs(:)-12 1480 y Fu(Using)17 b(this)f(and)g(follo)o(wing)g(the)g(pro)q(of)g(of)f (lemma)h(2.3,)f(w)o(e)g(obtain)h(\(using)g(\(2.2\))e(and)i(the)g (de\014nition)i(of)d(the)-12 1537 y(cone\):)464 1622 y(1)p 426 1642 99 2 v 426 1684 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))537 1591 y Fi(Z)536 1722 y Fr(P)590 1653 y Fn(L)621 1634 y Fr(k)621 1664 y Fp(\010)649 1653 y Fs(f)f(dm)41 b Fn(\024)h Fs(\013)887 1634 y Fq(0)920 1591 y Fi(Z)896 1724 y Fr(D)q Fp(\()p Fr(j)r Fp(\))987 1653 y Fs(f)5 b(dm)j Fu([)o(1)i(+)g Fs(D)1214 1660 y Fp(1)1249 1653 y Fs(b)g Fu(+)g(4)p Fs(D)1385 1660 y Fp(2)1419 1653 y Fs(c)p Fu(])464 1785 y(1)p 426 1805 V 426 1847 a Fs(\026)p Fu(\()p Fs(P)c Fu(\))537 1754 y Fi(Z)536 1885 y Fr(P)590 1816 y Fn(L)621 1797 y Fr(k)621 1827 y Fp(\010)649 1816 y Fs(f)f(dm)41 b Fn(\025)858 1779 y Fi(\002)877 1816 y Fs(\013)10 b Fn(\000)g Fs(\013)990 1797 y Fq(0)1002 1816 y Fu(\()p Fs(D)1058 1823 y Fp(1)1077 1816 y Fs(b)g Fu(+)g(4)p Fs(D)1213 1823 y Fp(2)1232 1816 y Fs(c)p Fu(\))1270 1779 y Fi(\003)1310 1754 y(Z)1286 1887 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1377 1816 y Fs(f)5 b(dm:)-12 1977 y Fu(whic)o(h)16 b(leads,)g(b)o(y)f(the)g(c)o(hoices)h(\(5.)j(and)d(6.\))j(ab)q(o)o(v)o (e,)14 b(to)355 2093 y Fs(\013=)p Fu(2)451 2032 y Fi(Z)427 2165 y Fr(D)q Fp(\()p Fr(j)r Fp(\))519 2093 y Fs(f)5 b(dm)12 b Fn(\024)713 2063 y Fu(1)p 675 2083 V 675 2125 a Fs(\026)p Fu(\()p Fs(P)6 b Fu(\))786 2032 y Fi(Z)785 2163 y Fr(P)839 2093 y Fn(L)870 2075 y Fr(k)870 2105 y Fp(\010)898 2093 y Fs(f)f(dm)12 b Fn(\024)h Fs(\013)1078 2075 y Fq(0)1090 2093 y Fu(\(1)c(+)i Fs(\013=)p Fu(\(2)p Fs(\013)1308 2075 y Fq(0)1319 2093 y Fu(\)\))1376 2032 y Fi(Z)1352 2165 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1443 2093 y Fs(f)5 b(dm:)-12 2255 y Fu(Moreo)o(v)o(er,)14 b(since)i(\006)f(is)h (without)f(big)h(branc)o(hes)f(at)g(in\014nit)o(y)l(,)h(\(2.7\))e(b)q (ecomes)581 2350 y Fs(K)620 2357 y Fr(j)r Fq(\000)p Fp(1)683 2350 y Fu(\()p Fn(L)732 2331 y Fr(k)732 2361 y Fp(\010)760 2350 y Fs(f)5 b Fu(\))12 b Fn(\024)h Fs(M)5 b(r)936 2331 y Fr(k)957 2350 y Fs(K)996 2357 y Fr(j)1014 2350 y Fu(\()p Fs(f)g Fu(\))10 b(+)g Fs(R)g Fu(sup)1174 2391 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1256 2350 y Fn(j)p Fs(f)5 b Fn(j)p Fs(;)-12 2491 y Fu(and,)15 b(for)g(an)o(y)g Fs(f)i Fn(2)c Fs(C)363 2469 y Fr(j)360 2505 y(N)393 2491 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))13 b(and)i Fs(x)e Fn(2)f Fs(D)q Fu(\()p Fs(j)s Fu(\))i(w)o(e)h(ha)o(v)o(e,)g(\(as)f(in)i(lemma)g (2.1\),)298 2610 y(min)q([)p Fn(\000)p Fs(c;)8 b(M)d(a)k Fn(\000)h Fs(bD)648 2617 y Fp(1)667 2610 y Fu(])701 2548 y Fi(Z)677 2682 y Fr(D)q Fp(\()p Fr(j)r Fp(\))768 2610 y Fs(f)5 b(dm)13 b Fn(\024)g Fs(f)5 b Fu(\()p Fs(x)p Fu(\))12 b Fn(\024)h Fu(max)o([)p Fs(c;)8 b(M)d(a)k Fu(+)i Fs(bD)1393 2617 y Fp(1)1411 2610 y Fu(])1446 2548 y Fi(Z)1421 2682 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1513 2610 y Fs(f)5 b(dm)928 2796 y Fu(17)p eop %%Page: 18 18 18 17 bop -12 17 a Fu(This)18 b(giv)o(es)g Fs(K)247 24 y Fr(j)r Fq(\000)p Fp(1)310 17 y Fu(\()p Fn(L)359 -2 y Fr(k)359 28 y Fp(\010)387 17 y Fs(f)5 b Fu(\))16 b Fn(\024)515 -45 y Fi(Z)490 89 y Fr(D)q Fp(\()p Fr(j)r Fp(\))582 17 y Fs(f)5 b(dm)681 -33 y Fi(h)701 17 y Fs(bM)g(r)792 -2 y Fr(k)823 17 y Fu(+)11 b Fs(R)c Fu(max\()p Fs(c;)h(M)d(a)j Fu(+)j Fs(bD)1240 24 y Fp(1)1259 17 y Fu(\))1277 -33 y Fi(i)1315 17 y Fu(so,)18 b(b)o(y)g(the)f(c)o(hoices)i(\(3.)26 b(and)18 b(5.\))-12 137 y(ab)q(o)o(v)o(e,)554 212 y Fs(K)593 219 y Fr(j)r Fq(\000)p Fp(1)656 212 y Fu(\()p Fn(L)705 193 y Fr(k)705 223 y Fp(\010)732 212 y Fs(f)5 b Fu(\))13 b Fn(\024)g Fs(\014)r(b)907 150 y Fi(Z)883 284 y Fr(D)q Fp(\()p Fr(j)r Fp(\))974 212 y Fs(f)5 b(dm)15 b Fu(if)h Fs(b)c(>)1241 181 y(cR)p 1207 202 124 2 v 1207 243 a(\014)g Fn(\000)e Fs(\014)1318 230 y Fq(0)1335 212 y Fs(:)-12 350 y Fu(Finally)l(,)17 b(for)d Fs(x)f Fn(2)g Fs(P)332 357 y Fp(2)362 350 y Fn(\\)d Fs(D)q Fu(\()p Fs(j)i Fn(\000)f Fu(1\),)j(\()p Fo(S-Exp1)p Fu(\))h(giv)o(es:)696 448 y Fn(jL)740 430 y Fr(k)740 460 y Fp(\010)767 448 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))p Fn(j)12 b(\024)h Fs(\016)951 430 y Fq(0)949 460 y Fr(j;k)994 448 y Fs(c)1036 386 y Fi(Z)1011 520 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1103 448 y Fs(f)5 b(dm:)-12 620 y Fu(T)l(o)10 b(pro)o(v)o(e)g(that)f Fn(L)292 604 y Fr(k)292 633 y Fp(\010)320 620 y Fs(C)356 598 y Fr(j)353 634 y(N)386 620 y Fu(\()p Fs(a;)f(b;)g(c)p Fu(\))i Fn(\032)j Fs(C)622 598 y Fr(j)r Fq(\000)p Fp(1)619 634 y Fr(N)685 620 y Fu(\()p Fs(\015)s(a;)8 b(\015)s(b)o(;)f(\016)861 627 y Fr(j)877 620 y Fs(c)p Fu(\),)j(it)g(remains)h(to)e(compare)1384 558 y Fi(Z)1360 692 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1451 620 y Fs(f)c(dm)10 b Fu(and)1672 558 y Fi(Z)1625 692 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))1762 620 y Fn(L)1793 601 y Fr(k)1793 631 y Fp(\010)1821 620 y Fs(f)5 b(dm)p Fu(.)-12 808 y Fo(Lemma)18 b(3.3)k Fg(F)l(or)15 b(an)o(y)f Fs(f)k Fn(2)13 b Fs(C)549 786 y Fr(j)546 822 y(N)579 808 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))p Fg(,)783 746 y Fi(Z)739 880 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))872 808 y Fn(L)903 789 y Fr(k)903 819 y Fp(\010)931 808 y Fs(f)d(dm)12 b Fu(=)1097 746 y Fi(Z)1072 880 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1164 808 y Fs(f)5 b(dm)p Fu([1)k Fn(\006)h Fs(m)p Fu(\()p Fs(j)s Fu(\)])p Fg(.)-12 968 y Fo(Pro)q(of)5 b Fu(:)40 b(Let)16 b Fs(f)h Fn(2)c Fs(C)373 946 y Fr(j)370 982 y(N)403 968 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\),)k(follo)o(wing)k(lemma)g(3.2,)d(w)o(e)i(ha)o(v)o(e)g Fs(\033)1194 951 y Fq(\000)p Fr(k)1242 968 y Fs(D)q Fu(\()p Fs(j)d Fn(\000)f Fu(1\))h Fn(\032)h Fs(D)q Fu(\()p Fs(j)s Fu(\),)342 1008 y Fi(Z)295 1141 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))432 1069 y Fn(L)463 1051 y Fr(k)463 1081 y Fp(\010)490 1069 y Fs(f)5 b(dm)13 b Fu(=)711 1008 y Fi(Z)632 1143 y Fr(\033)653 1133 y Ff(\000)p Fl(k)696 1143 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))832 1069 y Fs(f)5 b(dm)13 b Fu(=)998 1008 y Fi(Z)974 1141 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1065 1069 y Fs(f)5 b(dm)10 b Fn(\000)1325 1008 y Fi(Z)1201 1143 y Fr(D)q Fp(\()p Fr(j)r Fp(\))p Fq(n)p Fr(\033)1314 1133 y Ff(\000)p Fl(k)1357 1143 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))1493 1069 y Fs(f)5 b(dm:)-12 1221 y Fu(If)16 b Fs(x)f Fu(b)q(elongs)h(to)e Fs(D)q Fu(\()p Fs(j)s Fu(\))9 b Fn(n)h Fs(\033)460 1204 y Fq(\000)p Fr(k)508 1221 y Fs(D)q Fu(\()p Fs(j)i Fn(\000)f Fu(1\))j(then)284 1319 y(min)q([)p Fn(\000)p Fs(c;)8 b(M)d(a)k Fn(\000)h Fs(bD)634 1326 y Fp(1)653 1319 y Fu(])688 1257 y Fi(Z)663 1391 y Fr(D)q Fp(\()p Fr(j)r Fp(\))755 1319 y Fs(f)5 b(dm)12 b Fn(\024)h Fs(f)5 b Fu(\()p Fs(x)p Fu(\))12 b Fn(\024)1069 1257 y Fi(Z)1045 1391 y Fr(D)q Fp(\()p Fr(j)r Fp(\))1136 1319 y Fs(f)5 b(dm)23 b Fu(max)o([)p Fs(c;)8 b(M)d(a)k Fu(+)h Fs(bD)1573 1326 y Fp(1)1592 1319 y Fu(])p Fs(;)-12 1491 y Fu(this)17 b(giv)o(es)226 1429 y Fi(Z)180 1563 y Fr(D)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))316 1491 y Fn(L)347 1472 y Fr(k)347 1502 y Fp(\010)375 1491 y Fs(f)5 b(dm)14 b Fu(=)543 1429 y Fi(Z)519 1563 y Fr(D)q Fp(\()p Fr(j)r Fp(\))610 1491 y Fs(f)5 b(dm)p Fu([1)10 b Fn(\006)h Fs(c)16 b(m)p Fu(\()p Fs(D)q Fu(\()p Fs(j)s Fu(\))9 b Fn(n)h Fs(\033)1054 1472 y Fq(\000)p Fr(k)1102 1491 y Fs(D)q Fu(\()p Fs(j)j Fn(\000)e Fu(1\)\)])k(\(w)o(e)g(ha)o(v)o(e)h(c)o(hosen)g Fs(bD)1719 1498 y Fp(1)1752 1491 y Fs(<)e Fu(1)i(and)-12 1614 y Fs(c)c Fn(\025)h Fs(aM)j Fu(+)10 b(1\))15 b(and)g(the)g(lemma)h(is)f (pro)o(v)o(en.)1116 b Fn(4)3 1670 y Fu(So)15 b(w)o(e)g(ha)o(v)o(e)g Fn(L)269 1654 y Fr(k)269 1684 y Fp(\010)297 1670 y Fs(C)333 1648 y Fr(j)330 1684 y(N)363 1670 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))i Fn(\032)j Fs(C)599 1648 y Fr(j)r Fq(\000)p Fp(1)596 1684 y Fr(N)662 1670 y Fu(\()p Fs(\015)s(a;)8 b(\015)s(b)o(;)g(\016)839 1677 y Fr(j)854 1670 y Fs(c)p Fu(\).)19 b(It)c(remains)h(to)f(estimate) g(the)g(pro)s(jectiv)o(e)g(diameter.)-12 1773 y Fo(Lemma)j(3.4)k Fg(The)13 b(pro)s(jectiv)o(e)f(diameter)h Fu(\001)787 1780 y Fr(j)817 1773 y Fg(of)f Fn(L)897 1757 y Fr(k)897 1787 y Fp(\010)925 1773 y Fs(C)961 1751 y Fr(j)958 1788 y(N)991 1773 y Fu(\()p Fs(a;)c(b;)g(c)p Fu(\))i Fg(in)j Fs(C)1229 1751 y Fr(j)r Fq(\000)p Fp(1)1226 1788 y Fr(N)1292 1773 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))i Fg(is)j(b)q(ounded)g(b)o(y)g Fu(2)8 b(log)1831 1749 y Fp(1+)p Fr(\016)1892 1754 y Fl(j)p 1831 1763 77 2 v 1831 1789 a Fp(1)p Fq(\000)p Fr(\016)1892 1794 y Fl(j)-12 1847 y Fg(and)16 b(for)e Fs(f)20 b Fg(and)c Fs(g)g Fg(in)g Fs(C)404 1825 y Fr(j)401 1861 y(N)434 1847 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))p Fg(,)677 1925 y Fs(\022)698 1932 y Fr(j)r Fq(\000)p Fp(1)761 1925 y Fu(\()p Fn(L)810 1906 y Fr(k)810 1936 y Fp(\010)838 1925 y Fs(f)r(;)g Fn(L)914 1906 y Fr(k)914 1936 y Fp(\010)942 1925 y Fs(g)r Fu(\))j Fn(\024)i Fs(\016)1063 1932 y Fr(j)1082 1925 y Fs(\022)1103 1932 y Fr(j)1121 1925 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))-12 2007 y Fg(where)16 b Fs(\022)141 2014 y Fr(j)174 2007 y Fg(denote)g(the)f(pro)s(jectiv)o(e)g(metric)h(of)e (the)i(cone)f Fs(C)1021 1985 y Fr(j)1018 2022 y(N)1051 2007 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))p Fg(.)-12 2100 y Fo(Pro)q(of)d Fu(:)61 b(F)l(ollo)o(wing)22 b(the)e(pro)q(of)g(of)g (lemma)h(2.4,)g(w)o(e)f(obtain)h(\001)1170 2107 y Fr(j)1209 2100 y Fn(\024)h Fu(2)8 b(log)g(max)o(\()1470 2076 y Fp(1+)p Fr(\016)1531 2081 y Fl(j)p 1470 2090 V 1470 2116 a Fp(1)p Fq(\000)p Fr(\016)1531 2121 y Fl(j)1552 2100 y Fs(;)1577 2082 y Fp(2)p Fr(\013)1618 2070 y Ff(0)1629 2082 y Fp(+)p Fr(\013)p 1577 2089 102 2 v 1617 2116 a(\013)1684 2100 y Fs(;)1709 2079 y Fp(1+)p Fr(\015)p 1709 2090 66 2 v 1709 2116 a Fp(1)p Fq(\000)p Fr(\015)1779 2100 y Fu(\))20 b(since)-12 2177 y(lim)9 b Fs(\016)79 2184 y Fr(j)120 2177 y Fu(=)22 b(1)f(\(see)g(remark)f(3.1\),)h(w)o(e)f(ma)o(y) h(assume)f(that)1039 2153 y Fp(1+)p Fr(\016)1100 2158 y Fl(j)p 1039 2166 77 2 v 1039 2192 a Fp(1)p Fq(\000)p Fr(\016)1100 2197 y Fl(j)1143 2177 y Fn(\025)1205 2159 y Fp(2)p Fr(\013)1246 2147 y Ff(0)1257 2159 y Fp(+)p Fr(\013)p 1205 2166 102 2 v 1245 2192 a(\013)1333 2177 y Fu(and)1432 2153 y Fp(1+)p Fr(\016)1493 2158 y Fl(j)p 1432 2166 77 2 v 1432 2192 a Fp(1)p Fq(\000)p Fr(\016)1493 2197 y Fl(j)1536 2177 y Fn(\025)1598 2156 y Fp(1+)p Fr(\015)p 1598 2166 66 2 v 1598 2192 a Fp(1)p Fq(\000)p Fr(\015)1690 2177 y Fu(so)g(w)o(e)h(ha)o(v)o(e)-12 2259 y(\001)26 2266 y Fr(j)57 2259 y Fn(\024)13 b Fu(2)8 b(log)207 2235 y Fp(1+)p Fr(\016)268 2240 y Fl(j)p 207 2248 77 2 v 207 2274 a Fp(1)p Fq(\000)p Fr(\016)268 2279 y Fl(j)304 2259 y Fu(and)15 b(tanh)495 2235 y Fp(\001)524 2240 y Fl(j)p 495 2248 46 2 v 509 2274 a Fp(4)559 2259 y Fn(\024)e Fu(tanh)705 2208 y Fi(h)727 2259 y Fu(2)8 b(log)828 2235 y Fp(1+)p Fr(\016)889 2240 y Fl(j)p 828 2248 77 2 v 828 2274 a Fp(1)p Fq(\000)p Fr(\016)889 2279 y Fl(j)910 2208 y Fi(i)944 2259 y Fu(=)13 b Fs(\016)1012 2266 y Fr(j)1031 2259 y Fu(.)-12 2337 y(Let)j Fs(f)k Fu(and)15 b Fs(g)i Fu(b)q(elong)f(to)f Fs(C)476 2314 y Fr(j)473 2351 y(N)506 2337 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\),)j(prop)q(osition)16 b(1.5)f(and)g(the)g(estimate)g(of)g(\001)1423 2344 y Fr(j)1456 2337 y Fu(imply)544 2414 y Fs(\022)565 2431 y Fr(C)592 2415 y Fl(j)q Ff(\000)p Fk(1)590 2443 y Fl(N)649 2431 y Fp(\()p Fr(a;b;c)p Fp(\))747 2414 y Fu(\()p Fn(L)796 2395 y Fr(k)796 2425 y Fp(\010)824 2414 y Fs(f)r(;)8 b Fn(L)900 2395 y Fr(k)900 2425 y Fp(\010)927 2414 y Fs(g)r Fu(\))k Fn(\024)h Fs(\016)1049 2421 y Fr(j)1067 2414 y Fs(\022)1088 2431 y Fr(C)1115 2415 y Fl(j)1113 2443 y(N)1143 2431 y Fp(\()p Fr(a;b;c)p Fp(\))1241 2414 y Fu(\()p Fs(f)r(;)8 b(g)r Fu(\))p Fs(:)1873 2500 y Fn(4)1 2556 y Fu(Finally)l(,)15 b(let)e(us)g(remark)f(that)h(since)641 2522 y Fi(P)689 2570 y Fr(j)714 2556 y Fs(m)p Fu(\()p Fs(j)s Fu(\))f Fs(<)h Fn(1)p Fu(,)g(the)g(pro)q(duct)g(of)g(\(1)6 b Fn(\000)g Fs(m)p Fu(\()p Fs(j)s Fu(\)\))1441 2540 y Fq(\000)p Fp(1)1498 2556 y Fu(go)q(es)13 b(to)f(some)g(p)q(ositiv)o(e) -12 2628 y(limit)17 b(so)e(that)f(,)312 2585 y Fi(Y)279 2684 y Fr(`)p Fp(=0)p Fr(;:::;j)411 2628 y Fs(\016)431 2635 y Fr(`)460 2628 y Fn(\024)f Fu(Ct)600 2585 y Fi(Y)566 2684 y Fr(`)p Fp(=0)p Fr(;:::;j)699 2628 y Fs(\016)721 2598 y Ff(0)719 2640 y Fr(k)q(;`)765 2628 y Fu(.)928 2796 y(18)p eop %%Page: 19 19 19 18 bop -12 1 a Fh(3.2)56 b(Estimate)16 b(of)j(the)f(deca)n(y)h(of)f (correlations.)-12 87 y Fu(In)g(this)g(section,)g(w)o(e)f(conclude)i (the)f(pro)q(of)f(of)g(prop)q(osition)h(3.1)e(and)i(w)o(e)f(sho)o(w)g (ho)o(w)g(\(3.1\))e(can)j(b)q(e)g(used)g(to)-12 143 y(estimate)13 b(the)g(sp)q(eed)h(of)f(con)o(v)o(ergence)g(of)g Fn(L)747 127 y Fr(n)747 157 y Fp(\010)788 143 y Fu(to)f(the)h(sp)q(ectral)h(pro) s(jection,)f(on)g(compact)f(sets)h(and)g(the)g(deca)o(y)-12 205 y(of)i(correlations.)20 b(Let)c Fn(C)f Fu(=)472 171 y Fi(T)510 218 y Fr(j)r(>)p Fp(0)581 205 y Fs(C)617 183 y Fr(j)614 219 y(N)647 205 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\).)16 b(The)g(cone)f Fn(C)j Fu(is)e(non)f(empt)o(y)g(indeed,)i(w)o (e)e(ha)o(v)o(e)f(the)i(follo)o(wing)-12 261 y(result.)-12 361 y Fo(Lemma)i(3.5)k Fg(If)d Fs(a)p Fg(,)h Fs(b)e Fg(and)h Fs(c)f Fg(are)h(large)g(enough,)h(then)f(for)f(an)o(y)h Fs(f)24 b Fn(2)19 b Fs(L)p Fg(,)g(it)g(exists)g Fs(R)p Fu(\()p Fs(f)5 b Fu(\))18 b Fn(\025)h Fu(0)g Fg(suc)o(h)g(that)-12 418 y Fs(f)c Fu(+)c Fs(R)p Fu(\()p Fs(f)5 b Fu(\))p Fs(h)12 b Fn(2)h(C)s Fg(,)h(moreo)o(v)o(er)g Fs(R)p Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\024)h Fg(Ct)i Fn(k)p Fs(f)5 b Fn(k)-12 504 y Fo(Pro)q(of)g Fu(:)40 b(It)15 b(su\016ces)h(that)e Fs(R)p Fu(\()p Fs(f)5 b Fu(\))15 b(v)o(erify)g(\(recall)h(that)f Fn(8)p Fs(j)g Fn(2)e Fj(N)p Fu(,)h([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(n)1204 511 y Fp(1)1222 504 y Fn(j)-8 b Fu(])12 b Fn(\032)h Fs(D)q Fu(\(0\))f Fn(\032)h Fs(D)q Fu(\()p Fs(j)s Fu(\)\))56 660 y Fn(\017)23 b Fs(R)p Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\025)265 583 y Fs(M)g Fu(sup)c Fs(f)16 b Fn(\000)10 b Fs(a)543 546 y Fi(R)507 629 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;n)570 634 y Fk(1)587 629 y Fq(j)g Fp(])613 583 y Fs(f)p 265 650 376 2 v 266 692 a(a)329 655 y Fi(R)293 738 y Fp([)g Fq(j)p Fp(0)p Fr(;n)356 743 y Fk(1)373 738 y Fq(j)g Fp(])399 692 y Fs(h)10 b Fn(\000)g Fs(M)j Fu(sup)8 b Fs(h)645 660 y Fu(,)56 902 y Fn(\017)23 b Fs(R)p Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\025)265 825 y Fs(K)s Fu(\()p Fs(f)5 b Fu(\))k Fn(\000)i Fs(b)483 788 y Fi(R)448 871 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;n)511 876 y Fk(1)527 871 y Fq(j)g Fp(])553 825 y Fs(f)p 265 891 316 2 v 266 933 a(b)324 897 y Fi(R)289 980 y Fp([)g Fq(j)p Fp(0)p Fr(;n)352 985 y Fk(1)368 980 y Fq(j)g Fp(])394 933 y Fs(h)10 b Fn(\000)h Fs(K)s Fu(\()p Fs(h)p Fu(\))585 902 y(,)56 1143 y Fn(\017)23 b Fs(R)p Fu(\()p Fs(f)5 b Fu(\))12 b Fn(\025)265 1066 y Fu(sup)333 1077 y Fr(P)355 1082 y Fk(2)383 1066 y Fs(f)j Fn(\000)10 b Fs(c)524 1030 y Fi(R)488 1113 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;n)551 1118 y Fk(1)568 1113 y Fq(j)g Fp(])594 1066 y Fs(f)p 265 1133 357 2 v 266 1175 a(c)325 1138 y Fi(R)289 1221 y Fp([)g Fq(j)p Fp(0)p Fr(;n)352 1226 y Fk(1)368 1221 y Fq(j)g Fp(])394 1175 y Fs(h)10 b Fn(\000)h Fu(sup)544 1186 y Fr(P)566 1191 y Fk(2)594 1175 y Fs(h)626 1143 y Fu(.)-12 1297 y(The)16 b(parameters)e Fs(a)p Fu(,)h Fs(b)g Fu(and)h Fs(c)f Fu(are)g(c)o(hosen)h(in)g(order)f(to)g(ensure)h(that)e(the)i (three)f(denominators)h(are)f(strictly)-12 1353 y(p)q(ositiv)o(e.)1720 b Fn(4)5 1410 y Fu(In)16 b(what)g(follo)o(ws,)h Fs(a;)8 b(b;)g(c)14 b Fu(are)i(large)g(enough)h(to)e(guaran)o(t)o(y)g(that)h (lemma)h(3.5)e(is)i(v)o(eri\014ed.)24 b(In)17 b(particular,)g Fs(h)-12 1466 y Fu(b)q(elongs)f(to)f Fn(C)s Fu(.)-12 1522 y(Let)h Fs(f)h Fn(2)c(C)18 b Fu(and)d Fs(j)g Fn(2)e Fj(N)p Fu(,)h Fs(f)20 b Fu(b)q(elongs)c(to)f Fs(C)717 1500 y Fr(j)714 1537 y(N)747 1522 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))k(and)k Fn(L)1021 1506 y Fr(k)q(`)1021 1536 y Fp(\010)1057 1522 y Fs(f)k Fu(b)q(elongs)c(to)f Fs(C)1354 1500 y Fr(j)r Fq(\000)p Fr(`)1351 1537 y(N)1413 1522 y Fu(\()p Fs(a;)8 b(b;)g(c;)g Fu(\))k(for)i Fs(`)f Fu(=)g(0)p Fs(;)8 b(:)g(:)g(:)t(;)g(j) s Fu(.)580 1623 y Fs(\022)601 1635 y Fr(C)628 1623 y Fk(0)626 1646 y Fl(N)658 1623 y Fu(\()p Fn(L)707 1601 y Fr(k)q(j)707 1637 y Fp(\010)744 1623 y Fs(f)d(;)j(h)p Fu(\))41 b Fn(\024)h Fs(\016)974 1630 y Fp(1)994 1623 y Fs(\022)1015 1635 y Fr(C)1042 1623 y Fk(1)1040 1646 y Fl(N)1071 1623 y Fu(\()p Fn(L)1120 1599 y Fr(k)q Fp(\()p Fr(j)r Fq(\000)p Fp(1\))1120 1637 y(\010)1230 1623 y Fs(f)5 b(;)j(h)p Fu(\))782 1748 y Fs(:)g(:)g(:)40 b Fn(\024)i Fu(\001)992 1755 y Fr(j)1017 1689 y(j)r Fq(\000)p Fp(1)1019 1705 y Fi(Y)1018 1804 y Fr(`)p Fp(=1)1086 1748 y Fs(\016)1106 1755 y Fr(`)1123 1748 y Fs(:)-12 1908 y Fu(The)16 b(norm)g Fn(k)f(k)263 1915 y Fr(d;)p Fp(0)326 1908 y Fu(de\014ned)i(b)o(y)f (\(3.2\))e(and)i Fs(\032)p Fu(\()p Fs(f)5 b Fu(\))13 b(=)909 1846 y Fi(Z)884 1979 y Fr(D)q Fp(\(0\))977 1908 y Fs(f)5 b(dm)13 b Fu(=)1150 1846 y Fi(Z)1120 1979 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])1224 1908 y Fs(f)5 b(dm)15 b Fu(are)h(adapted)g(to)f Fs(C)1674 1891 y Fp(0)1671 1921 y Fr(N)1704 1908 y Fu(,)h(moreo)o(v)o(er)-12 2030 y(the)f(norms)g Fn(k)g(k)264 2037 y Fr(\013;)p Fp(0)331 2030 y Fu(and)h Fn(k)e(k)480 2037 y Fr(N)529 2030 y Fu(are)h(equiv)m (alen)o(t,)h(so)f(for)g Fs(f)i Fn(2)c(C)s Fu(,)267 2166 y Fn(kL)321 2144 y Fr(k)q(j)321 2181 y Fp(\010)358 2166 y Fs(f)j Fn(\000)493 2136 y Fs(h)p 446 2156 121 2 v 474 2161 a Fi(R)442 2244 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])540 2198 y Fs(h)599 2105 y Fi(Z)569 2238 y Fp([)g Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])673 2166 y Fn(L)704 2144 y Fr(k)q(j)704 2181 y Fp(\010)742 2166 y Fs(f)5 b Fn(k)792 2173 y Fr(N)838 2166 y Fn(\024)13 b Fu(\001)924 2173 y Fr(j)965 2107 y(j)r Fq(\000)p Fp(1)966 2123 y Fi(Y)965 2222 y Fr(`)p Fp(=1)1033 2166 y Fs(\016)1053 2173 y Fr(`)1093 2166 y Fu(exp)1170 2089 y Fi( )1206 2166 y Fu(\001)1244 2173 y Fr(j)1285 2107 y(j)r Fq(\000)p Fp(1)1286 2123 y Fi(Y)1286 2222 y Fr(`)p Fp(=1)1354 2166 y Fs(\016)1374 2173 y Fr(`)1390 2089 y Fi(!)1464 2105 y(Z)1439 2238 y Fr(D)q Fp(\(0\))1532 2166 y Fs(f)5 b(dm:)-12 2326 y Fu(If)16 b Fs(p)c Fu(=)h Fs(k)q(j)g Fu(+)d Fs(r)q Fu(,)283 2434 y Fn(kL)337 2412 y Fr(p)337 2448 y Fp(\010)365 2434 y Fs(f)15 b Fn(\000)500 2403 y Fs(h)p 453 2424 V 481 2429 a Fi(R)449 2512 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])547 2466 y Fs(h)606 2372 y Fi(Z)575 2506 y Fp([)g Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])679 2434 y Fn(L)710 2412 y Fr(p)710 2448 y Fp(\010)738 2434 y Fs(f)5 b Fn(k)788 2441 y Fr(N)863 2434 y Fn(\024)42 b(kL)994 2415 y Fr(r)994 2445 y Fp(\010)1022 2434 y Fn(k)15 b(kL)1114 2412 y Fr(k)q(j)1114 2448 y Fp(\010)1151 2434 y Fs(f)h Fn(\000)1286 2403 y Fs(h)p 1239 2424 V 1267 2429 a Fi(R)1235 2512 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])1333 2466 y Fs(h)1392 2372 y Fi(Z)1362 2506 y Fp([)g Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])1466 2434 y Fn(L)1497 2412 y Fr(k)q(j)1497 2448 y Fp(\010)1535 2434 y Fs(f)5 b Fn(k)1585 2441 y Fr(N)863 2604 y Fn(\024)42 b Fs(M)20 b Fn(kL)1058 2582 y Fr(k)q(j)1058 2618 y Fp(\010)1096 2604 y Fs(f)15 b Fn(\000)1231 2573 y Fs(h)p 1184 2593 V 1212 2599 a Fi(R)1180 2682 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])1278 2635 y Fs(h)1337 2542 y Fi(Z)1306 2675 y Fp([)g Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])1410 2604 y Fn(L)1441 2582 y Fr(k)q(j)1441 2618 y Fp(\010)1479 2604 y Fs(f)5 b Fn(k)1529 2611 y Fr(N)1563 2604 y Fs(:)928 2796 y Fu(19)p eop %%Page: 20 20 20 19 bop -12 1 a Fu(It)15 b(is)h(easy)f(to)g(pro)o(v)o(e)f(that)h (\(recall)h(that)697 -36 y Fi(R)735 1 y Fs(hdm)d Fu(=)g(1\),)335 127 y Fn(k)441 96 y Fs(h)p 363 117 184 2 v 391 122 a Fi(R)359 205 y Fp([)-6 b Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])457 159 y Fs(hdm)579 65 y Fi(Z)549 199 y Fp([)g Fq(j)p Fp(0)p Fr(;N)t Fq(j)g Fp(])653 127 y Fn(L)684 105 y Fr(p)684 141 y Fp(\010)712 127 y Fs(f)5 b(dm)10 b Fn(\000)g Fs(h)892 65 y Fi(Z)945 127 y Fs(f)5 b(dm)p Fn(k)1059 134 y Fr(N)1105 127 y Fn(\024)13 b Fu(Ct)22 b(sup)8 b Fn(j)p Fs(f)d Fn(j)p Fs(m)p Fu(\([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)d Fn(j)-8 b Fu(])1535 108 y Fr(c)1549 127 y Fu(\))-12 299 y(Finally)l(,)17 b(for)d Fs(f)k Fn(2)13 b(C)18 b Fu(and)d Fs(p)e Fu(=)g Fs(k)q(j)f Fu(+)f Fs(r)q Fu(,)j Fs(r)f(<)g(j)18 b Fu(w)o(e)d(ha)o(v)o(e)63 444 y Fn(kL)117 421 y Fr(p)117 458 y Fp(\010)144 444 y Fs(f)g Fn(\000)c Fs(h)261 382 y Fi(Z)314 444 y Fs(f)5 b Fn(k)364 451 y Fr(N)410 444 y Fn(\024)13 b Fu(Ct)i(\001)562 451 y Fr(j)602 384 y(j)r Fq(\000)p Fp(1)604 400 y Fi(Y)603 499 y Fr(`)p Fp(=1)671 444 y Fs(\016)691 451 y Fr(`)731 444 y Fu(exp)808 366 y Fi( )844 444 y Fu(\001)882 451 y Fr(j)923 384 y(j)r Fq(\000)p Fp(1)924 400 y Fi(Y)923 499 y Fr(`)p Fp(=1)991 444 y Fs(\016)1011 451 y Fr(`)1028 366 y Fi(!)1087 444 y Fn(j)1108 382 y Fi(Z)1160 444 y Fs(f)5 b(dm)p Fn(j)k Fu(+)i(Ct)k Fs(m)p Fu(\([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)d Fn(j)-8 b Fu(])1565 425 y Fr(c)1579 444 y Fu(\))8 b(sup)g Fn(j)p Fs(f)d Fn(j)p Fs(:)73 b Fu(\(3.4\))-12 643 y Fo(Remark)17 b(3.3)22 b Fu(\001)314 650 y Fr(j)356 584 y(j)r Fq(\000)p Fp(1)358 600 y Fi(Y)357 699 y Fr(`)p Fp(=1)425 643 y Fs(\016)445 650 y Fr(`)486 643 y Fu(exp)563 566 y Fi( )599 643 y Fu(\001)637 650 y Fr(j)678 584 y(j)r Fq(\000)p Fp(1)679 600 y Fi(Y)678 699 y Fr(`)p Fp(=1)746 643 y Fs(\016)766 650 y Fr(`)783 566 y Fi(!)835 643 y Fu(dep)q(ends)c(on)e Fs(N)21 b Fu(and)c Fs(k)q Fu(,)f(\()p Fo(S-Exp1)p Fu(\))g(implies)i(that)e(for)g(\014xed) -12 739 y Fs(N)k Fu(and)15 b Fs(k)q Fu(,)g(this)h(expression)g(go)q(es) f(to)f(zero)h(when)h Fs(j)i Fu(go)q(es)d(to)f(in\014nit)o(y)l(.)-12 829 y(Lemma)h(3.5)g(and)g(\(3.4\))f(imply)i(for)f Fs(f)j Fn(2)12 b Fs(L)p Fu(,)452 952 y Fn(kL)506 930 y Fr(k)q(j)506 966 y Fp(\010)544 952 y Fs(f)j Fn(\000)10 b Fs(h)660 890 y Fi(Z)713 952 y Fs(f)5 b Fn(k)763 959 y Fr(N)809 952 y Fn(\024)13 b Fu(Ct)i Fs(\013)952 959 y Fr(j)986 952 y Fn(k)p Fs(f)5 b Fn(k)k Fu(+)i Fs(m)p Fu(\([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)d Fn(j)-8 b Fu(])1294 933 y Fr(c)1308 952 y Fu(\))8 b(sup)g Fs(f)d(;)-12 1127 y Fu(with)16 b Fs(\013)121 1134 y Fr(j)139 1127 y Fu(\()p Fs(N)5 b Fu(\))12 b(=)g(\001)314 1134 y Fr(j)355 1093 y Fi(Q)398 1105 y Fr(j)r Fq(\000)p Fp(1)398 1141 y Fr(`)p Fp(=1)469 1127 y Fs(\016)489 1134 y Fr(`)529 1127 y Fu(exp)606 1049 y Fi( )642 1127 y Fu(\001)680 1134 y Fr(j)721 1068 y(j)r Fq(\000)p Fp(1)722 1084 y Fi(Y)721 1182 y Fr(`)p Fp(=1)789 1127 y Fs(\016)809 1134 y Fr(`)826 1049 y Fi(!)862 1127 y Fu(.)20 b(This)c(conclude)h(the)e(pro)q(of)g(of)f(prop)q (osition)i(3.1.)128 b Fn(4)-12 1225 y Fu(No)o(w,)12 b(w)o(e)h(are)f (going)h(to)f(sho)o(w)g(ho)o(w)g(\(3.4\))f(leads)j(to)e(the)h(estimate) f(of)g(the)h(sp)q(eed)h(of)e(con)o(v)o(ergence)h(on)g(compact)-12 1282 y(sets)18 b(of)f(\006)g(and)h(to)f(the)h(deca)o(y)f(of)h (correlations.)27 b(Let)18 b Fs(q)h Fn(2)e Fj(N)f Fu(and)i Fs(f)23 b Fu(b)q(elongs)18 b(to)f Fn(C)s Fu(,)g(so)h(that)f Fs(f)22 b Fu(b)q(elongs)d(to)-12 1344 y Fs(C)24 1320 y Fr(k)q Fp(\()p Fr(j)r Fp(+2)p Fr(q)q Fp(\))21 1358 y Fr(N)168 1344 y Fu(and)f Fn(L)290 1320 y Fr(k)q Fp(\()p Fr(j)r Fp(+)p Fr(q)q Fp(\))290 1358 y(\010)399 1344 y Fs(f)k Fn(2)16 b Fs(C)525 1321 y Fr(k)q(q)522 1358 y(N)563 1344 y Fu(.)26 b(This)18 b(leads)g(to)f(the)g(follo)o(wing)h(estimate)g (whic)o(h)g(will)h(b)q(e)f(used)g(to)e(b)q(ound)-12 1400 y(the)f(sp)q(eed)i(of)d(con)o(v)o(ergence)i(on)f(compacts.)218 1520 y Fn(kL)272 1497 y Fr(k)q Fp(\()p Fr(j)r Fp(+)p Fr(q)q Fp(\))272 1535 y(\010)381 1520 y Fs(f)g Fn(\000)c Fs(h)498 1459 y Fi(Z)551 1520 y Fs(f)5 b Fn(k)601 1527 y Fr(N)t Fp(+)p Fr(K)r(k)q(q)758 1520 y Fn(\024)301 1662 y Fu(exp[\001)421 1669 y Fr(j)r Fp(+)p Fr(q)491 1603 y(j)r Fq(\000)p Fp(1)493 1619 y Fi(Y)492 1718 y Fr(`)p Fp(=1)560 1662 y Fs(\016)580 1669 y Fr(`)p Fp(+)p Fr(q)641 1662 y Fu(]\001)692 1669 y Fr(j)r Fp(+)p Fr(q)792 1600 y Fi(Z)767 1734 y Fr(D)q Fp(\(0\))859 1662 y Fs(f)g(dm)973 1603 y Fr(j)r Fq(\000)p Fp(1)974 1619 y Fi(Y)974 1718 y Fr(`)p Fp(=1)1042 1662 y Fs(\016)1062 1669 y Fr(`)p Fp(+)p Fr(q)1133 1662 y Fu(+)10 b Fs(m)p Fu(\([)-8 b Fn(j)p Fu(0)p Fs(;)8 b(N)13 b Fu(+)e Fs(K)s(k)q(q)r Fn(j)-8 b Fu(])1501 1643 y Fr(c)1517 1662 y Fu(\))8 b(sup)g Fn(j)p Fs(f)d Fn(j)p Fs(:)135 b Fu(\(3.5\))-12 1851 y(W)l(e)15 b(c)o(ho)q(ose)h(a)f(sequence)h Fs(q)r Fu(\()p Fs(j)s Fu(\))e(suc)o(h)h(that)g Fs(q)r Fu(\()p Fs(j)s Fu(\))823 1823 y Fr(j)r Fq(!1)830 1851 y Fn(\000)-8 b(!)20 b(1)c Fu(and)377 1998 y(\001)415 2007 y Fr(j)r Fp(+)p Fr(q)q Fp(\()p Fr(j)r Fp(\))544 1939 y Fr(j)r Fq(\000)p Fp(1)546 1955 y Fi(Y)545 2053 y Fr(`)p Fp(=1)613 1998 y Fs(\016)633 2007 y Fr(`)p Fp(+)p Fr(q)q Fp(\()p Fr(j)r Fp(\))760 1998 y Fu(exp)837 1920 y Fi( )873 1998 y Fu(\001)911 2007 y Fr(j)r Fp(+)p Fr(q)q Fp(\()p Fr(j)r Fp(\))1040 1939 y Fr(j)r Fq(\000)p Fp(1)1042 1955 y Fi(Y)1041 2053 y Fr(`)p Fp(=1)1109 1998 y Fs(\016)1129 2007 y Fr(`)p Fp(+)p Fr(q)q Fp(\()p Fr(j)r Fp(\))1234 1920 y Fi(!)1283 1998 y Fu(:=)d Fi(f)-46 b Fs(\013)1372 2005 y Fr(j)1403 1998 y Fn(\000)-7 b(!)13 b Fu(0)p Fs(:)-12 2155 y Fu(F)l(or)i(example,) g(if)h Fs(\016)324 2138 y Fq(0)322 2169 y Fr(k)q(;j)382 2155 y Fn(\024)d Fu(\(1)c Fn(\000)584 2137 y Fp(1)p 530 2144 127 2 v 530 2171 a Fr(N)t Fp(+)p Fr(K)r(k)q(j)661 2155 y Fu(\))679 2138 y Fr(\013)704 2155 y Fu(,)14 b(since)j(\001)881 2162 y Fr(j)911 2155 y Fn(\024)c Fu(2)8 b(log)1061 2131 y Fp(1+)p Fr(\016)1122 2136 y Fl(j)p 1061 2144 77 2 v 1061 2171 a Fp(1)p Fq(\000)p Fr(\016)1122 2176 y Fl(j)1143 2155 y Fu(,)14 b(for)h(an)o(y)g(0)d Fs(<)h(")g(<)g(")1512 2138 y Fq(0)1536 2155 y Fs(<)g Fu(1,)h(w)o(e)h(can)g(c)o(ho)q(ose)-12 2224 y Fs(q)r Fu(\()p Fs(j)s Fu(\))c(=)i Fs(j)149 2207 y Fr(")167 2224 y Fu(,)i(then)g(w)o(e)g(ha)o(v)o(e,)h Fi(f)-46 b Fs(\013)512 2231 y Fr(j)543 2224 y Fn(\024)13 b Fs(C)s Fu(\()p Fs(N)r(;)8 b(";)g(")768 2207 y Fq(0)777 2224 y Fu(\))835 2206 y Fp(1)p 800 2213 89 2 v 800 2246 a Fr(j)816 2236 y Fl(\013)p Ff(\000)p Fl(")875 2229 y Ff(0)893 2224 y Fu(.)-12 2387 y Fo(Remark)17 b(3.4)22 b Fu(The)16 b(condition)h(\()p Fo(S-Exp1)p Fu(\))e(ma)o(y)g(b)q(e)h (replaced)h(b)o(y:)j(it)c(exists)f(a)h(sequence)g Fs(q)r Fu(\()p Fs(j)s Fu(\))e(whic)o(h)j(go)q(es)-12 2443 y(to)e(in\014nit)o (y)h(with)g Fs(j)h Fu(and)f(suc)o(h)f(that)697 2588 y(\001)735 2597 y Fr(j)r Fp(+)p Fr(q)q Fp(\()p Fr(j)r Fp(\))864 2528 y Fr(j)r Fq(\000)p Fp(1)865 2544 y Fi(Y)865 2643 y Fr(`)p Fp(=1)933 2588 y Fs(\016)953 2597 y Fr(`)p Fp(+)p Fr(q)q Fp(\()p Fr(j)r Fp(\))1070 2559 y Fr(j)r Fq(!1)1077 2588 y Fn(\000)-8 b(!)20 b Fu(0)p Fs(:)928 2796 y Fu(20)p eop %%Page: 21 21 21 20 bop -12 1 a Fu(No)o(w,)14 b(let)i Fs(x)f Fu(b)q(elongs)h(to)f (some)g(compact)g Fs(Q)p Fu(,)f(it)i(exists)f Fs(j)979 8 y Fp(0)1014 1 y Fu(suc)o(h)g(that)-12 57 y Fs(Q)e Fn(\032)g Fu([0)p Fs(;)8 b(N)i Fu(+)d Fs(K)s(k)q(q)r Fu(\()p Fs(j)357 64 y Fp(0)376 57 y Fu(\)])12 b Fn(\032)h Fu([0)p Fs(;)8 b(N)i Fu(+)d Fs(K)s(k)q(q)r Fu(\()p Fs(j)s Fu(\)])12 b Fn(8)p Fs(j)j Fn(\025)e Fs(j)911 64 y Fp(0)930 57 y Fu(,)h(so,)f(for)g(an)o(y)g Fs(f)18 b Fn(2)13 b Fs(L)p Fu(,)g Fs(j)i Fn(\025)e Fs(j)1417 64 y Fp(0)1450 57 y Fu(and)h Fs(p)e Fu(=)h Fs(k)q Fu(\()p Fs(j)c Fu(+)e Fs(q)r Fu(\()p Fs(j)s Fu(\)\))g(+)g Fs(r)q Fu(,)-12 114 y(lemma)16 b(3.5)e(and)h(\(3.5\))f(giv)o(e,)343 175 y Fi(\014)343 202 y(\014)343 230 y(\014)343 257 y(\014)358 241 y Fn(L)389 219 y Fr(p)389 255 y Fp(\010)417 241 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))10 b Fn(\000)g Fs(h)p Fu(\()p Fs(x)p Fu(\))657 179 y Fi(Z)709 241 y Fs(f)5 b(dm)800 175 y Fi(\014)800 202 y(\014)800 230 y(\014)800 257 y(\014)828 241 y Fn(\024)13 b Fu(Ct)i([)q Fi(f)-46 b Fs(\013)984 248 y Fr(j)1012 241 y Fu(+)10 b Fs(m)p Fu(\([0)p Fs(;)e(N)13 b Fu(+)d Fs(K)s(k)q(q)r Fu(\()p Fs(j)s Fu(\)])1427 222 y Fr(c)1443 241 y Fu(\)])p Fn(k)p Fs(f)5 b Fn(k)p Fs(:)3 368 y Fu(If)16 b Fs(u)75 375 y Fr(j)93 368 y Fu(\()p Fs(N)5 b Fu(\))12 b(=)h Fi(f)-45 b Fs(\013)260 375 y Fr(j)288 368 y Fu(+)10 b Fs(Rm)p Fu([0)p Fs(;)e(N)13 b Fu(+)e Fs(k)q(K)s(q)r Fu(\()p Fs(j)s Fu(\)])721 352 y Fr(c)736 368 y Fu(],)k(this)g(can)h(b)q (e)g(written)f(as)610 493 y Fn(kL)664 471 y Fr(p)664 507 y Fp(\010)692 493 y Fs(f)g Fn(\000)10 b Fs(h)808 431 y Fi(Z)861 493 y Fs(f)5 b(dm)p Fn(k)975 500 y Fr(Q)1017 493 y Fn(\024)13 b Fs(C)s Fu(\()p Fs(j)1138 500 y Fp(0)1157 493 y Fu(\))p Fs(u)1201 500 y Fr(j)1219 493 y Fn(k)p Fs(f)5 b Fn(k)528 b Fu(\(3.6\))-12 616 y(where)16 b Fs(C)s Fu(\()p Fs(j)193 623 y Fp(0)211 616 y Fu(\))f(go)q(es)g(to)g(in\014nit) o(y)h(with)g Fs(j)679 623 y Fp(0)698 616 y Fu(:)318 746 y Fs(C)s Fu(\()p Fs(j)391 753 y Fp(0)410 746 y Fu(\))d(=)186 b(sup)494 781 y Fl(k)q Fk(\()p Fl(j)q Fk(+)p Fl(q)q Fk(\()p Fl(j)q Fk(\)\))p Ff(\024)p Fl(p)f(n)865 2331 y Fp(0)885 2324 y Fs(;)8 b Fu(\()p Fn(L)955 2331 y Fp(\010)981 2324 y Fo(1)p Fu(\))1025 2331 y Fe(n)1062 2324 y Fu(=)21 b(sup)1110 2365 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])1202 2324 y Fn(L)1233 2331 y Fp(\010)1261 2324 y Fo(1)p Fu(\()p Fs(x)p Fu(\))12 b Fn(\024)h Fu(1)p Fs(;)377 b Fu(\()p Fo(H)p Fu(\))-12 2499 y Fo(Lemma)18 b(4.1)k Fg(If)16 b Fu(\010)e Fg(satis\014es)i(\()p Fo(SA)p Fg(\))e(and)h(\()p Fo(H)p Fg(\))g(then)g(it)h(satis\014es)f(\() p Fo(K)p Fg(\).)-12 2589 y Fo(Pro)q(of)5 b Fu(:)44 b(Recall)18 b(that)e(w)o(e)g(ha)o(v)o(e)f(the)i(b)q(ounded)g(distortion)g(for)e Fn(L)1164 2596 y Fp(\010)1192 2589 y Fu(:)22 b(for)16 b Fs(x)g Fu(and)g Fs(y)i Fu(in)f(the)g(same)f(1-cylinder)-12 2645 y(and)g Fs(k)f Fn(2)f Fj(N)p Fu(,)h(w)o(e)g(ha)o(v)o(e)h Fn(L)426 2629 y Fr(k)426 2658 y Fp(\010)454 2645 y Fo(1)p Fu(\()p Fs(x)p Fu(\))d Fn(\024)h Fs(e)625 2629 y Fr(C)r(d)p Fp(\()p Fr(x;y)q Fp(\))749 2645 y Fn(L)780 2629 y Fr(k)780 2658 y Fp(\010)808 2645 y Fo(1)p Fu(\()p Fs(y)r Fu(\).)21 b(If)16 b Fs(x)g Fu(b)q(elongs)h(to)e([)p Fs(n)p Fu(],)g(b)o(y)h(in)o (tegrating)g(on)f(the)h(cylinder)928 2796 y(21)p eop %%Page: 22 22 22 21 bop -12 1 a Fu([)p Fs(n)p Fu(],)15 b(w)o(e)f(get:)278 124 y Fn(L)309 105 y Fr(k)309 135 y Fp(\010)337 124 y Fo(1)p Fu(\()p Fs(x)p Fu(\))41 b Fn(\024)h Fs(R)650 93 y Fu(1)p 598 114 128 2 v 598 155 a Fs(m)p Fu(\([)p Fs(n)p Fu(]\))738 62 y Fi(Z)763 165 y Fp([)p Fr(n)p Fp(])814 124 y Fn(L)845 105 y Fr(k)845 135 y Fp(\010)873 124 y Fo(1)p Fs(dm)466 252 y Fn(\024)594 221 y Fs(R)p 548 242 V 548 283 a(m)p Fu(\([)p Fs(n)p Fu(]\))688 190 y Fi(Z)713 293 y Fp(\006)749 252 y Fn(L)780 233 y Fr(k)780 263 y Fp(\010)807 252 y Fo(1)p Fs(dm)13 b Fu(=)1009 221 y Fs(R)p 963 242 V 963 283 a(m)p Fu(\([)p Fs(n)p Fu(]\))1111 252 y(\(b)q(ecause)j Fs(m)f Fu(is)g(conformal\))p Fs(:)-12 378 y Fu(for)g(some)g(constan)o(t)f Fs(R)e(>)h Fu(0.)20 b(Let)15 b Fs(n)614 385 y Fp(0)650 378 y Fu(b)q(e)g(giv)o(en)h(b)o(y)f (\()p Fo(H)p Fu(\),)f Fs(n)f Fn(\024)g Fs(n)1112 385 y Fp(0)1147 378 y Fu(and)j Fs(k)d Fn(2)g Fj(N)p Fu(.)602 500 y(sup)594 541 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])686 500 y Fn(L)717 481 y Fr(k)717 511 y Fp(\010)745 500 y Fo(1)p Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fs(R)7 b Fu(max)936 529 y Fr(p)p Fq(\024)p Fr(n)1002 534 y Fk(0)1083 469 y Fu(1)p 1033 489 124 2 v 1033 531 a Fs(m)p Fu(\([)p Fs(p)p Fu(]\))1174 500 y(:=)12 b Fs(M)1283 481 y Fq(0)1295 500 y Fs(:)-12 646 y Fu(Let)k(us)f(note)g Fs(M)273 653 y Fr(k)307 646 y Fu(=)e(sup)376 685 y Fp(\006)431 646 y Fn(L)462 627 y Fr(k)462 657 y Fp(\010)490 646 y Fo(1)p Fu(.)20 b(If)15 b Fs(n)e(>)g(n)709 653 y Fp(0)729 646 y Fu(,)i(w)o(e)g(ha)o(v)o(e,)f(for)h(an)o(y)g Fs(x)d Fn(2)h Fu([)p Fs(n)p Fu(],)i Fn(L)1290 653 y Fp(\010)1318 646 y Fo(1)p Fu(\()p Fs(x)p Fu(\))d Fn(\024)h Fu(1)h(and)444 781 y Fn(L)475 762 y Fr(k)q Fp(+1)475 795 y(\010)542 781 y Fo(1)p Fu(\()p Fs(x)p Fu(\))e Fn(\024)h(L)721 788 y Fp(\010)749 781 y Fo(1)p Fu(\()p Fs(x)p Fu(\))8 b(sup)865 820 y Fp(\006)920 781 y Fn(L)951 762 y Fr(k)951 792 y Fp(\010)979 781 y Fo(1)13 b Fn(\024)g Fu(sup)1087 820 y Fp(\006)1142 781 y Fn(L)1173 762 y Fr(k)1173 792 y Fp(\010)1200 781 y Fo(1)g Fn(\024)g Fu(max)7 b Fs(M)1423 788 y Fr(k)1445 781 y Fs(;)-12 910 y Fu(so)13 b(w)o(e)h(get,)f Fs(M)239 917 y Fr(k)q Fp(+1)318 910 y Fn(\024)g Fu(max\()p Fs(M)518 893 y Fq(0)529 910 y Fs(;)8 b(M)594 917 y Fr(k)615 910 y Fu(\))13 b(and)h(b)o(y)f(induction)j Fs(M)1039 917 y Fr(k)1073 910 y Fn(\024)d Fu(max)o(\()p Fs(M)1272 893 y Fq(0)1284 910 y Fs(;)8 b Fu(1\).)18 b(This)c(pro)o(v)o(es)f(that) g(\010)g(satis\014es)-12 966 y(\()p Fo(K)p Fu(\).)1795 b Fn(4)7 1022 y Fu(W)l(e)19 b(will)h(sa)o(y)e(that)g(\006)g(has)h Ft(no)g(jumps)h(to)g(in\014nity)d Fu(if)i(it)g(exists)g(an)g(in)o (teger)g Fs(K)i Fu(suc)o(h)e(that,)f(for)h(all)g Fs(n)g Fn(2)g Fj(N)-12 1079 y Fu(and)d(for)g(all)h Fs(x)d Fn(2)g Fu([)p Fs(n)p Fu(],)i Fs(\033)r(x)d Fn(2)i Fu([)p Fs(p)p Fu(])g(with)h Fs(p)e Fn(\024)h Fs(n)c Fu(+)g Fs(k)q Fu(.)22 b(In)17 b(other)e(w)o(ords,)h(the)g(matrix)f(whic)o(h)i(de\014nes)g (\006)f(has)g(the)-12 1135 y(follo)o(wing)g(form:)610 1150 y Fi(0)610 1230 y(B)610 1257 y(B)610 1285 y(B)610 1312 y(B)610 1341 y(@)671 1188 y Fn(\003)41 b(\001)8 b(\001)g(\001)40 b(\003)56 b Fu(0)h Fn(\001)8 b(\001)g(\001)39 b(\001)8 b(\001)g(\001)40 b(\001)8 b(\001)g(\001)676 1235 y Fu(.)676 1251 y(.)676 1268 y(.)896 1239 y(.)914 1251 y(.)932 1264 y(.)991 1239 y(.)1009 1251 y(.)1026 1264 y(.)671 1324 y Fn(\003)310 b(\003)71 b Fu(0)676 1371 y(.)676 1387 y(.)676 1404 y(.)1086 1375 y(.)1103 1387 y(.)1121 1400 y(.)1180 1375 y(.)1198 1387 y(.)1215 1400 y(.)1251 1150 y Fi(1)1251 1230 y(C)1251 1257 y(C)1251 1285 y(C)1251 1312 y(C)1251 1341 y(A)-12 1486 y Fu(with)16 b Fn(\003)c(2)h(f)p Fu(0)p Fs(;)8 b Fu(1)p Fn(g)p Fu(.)-12 1593 y Fo(Example)18 b(4.1)23 b([Dynamics)17 b(without)h(jumps)f(to)h (in\014nit)o(y)g(satisfying)g(\(Exp1\).])-12 1649 y Fu(If)e(\006)f(has) g(no)g(jumps)g(to)g(in\014nit)o(y)h(and)g(if)f(\010)g(v)o(eri\014es\()p Fo(SA)p Fu(\))g(and:)505 1718 y Fn(9)p Fs(n)557 1725 y Fp(0)593 1718 y Fu(suc)o(h)g(that)g Fn(8)p Fs(n)e(>)g(n)934 1725 y Fp(0)954 1718 y Fs(;)8 b Fu(\()p Fn(L)1024 1725 y Fp(\010)1051 1718 y Fo(1)p Fu(\))1095 1725 y Fe(n)1132 1718 y Fn(\024)13 b Fs(\032)f(<)h Fu(1)p Fs(;)446 b Fu(\()p Fo(Exp2)p Fu(\))3 1787 y(then)16 b(\010)f(v)o(eri\014es)h(\()p Fo(Exp1)p Fu(\))e(.)-12 1877 y(Indeed,)22 b(\()p Fo(Exp2)p Fu(\))d(implies)j(\()p Fo(H)p Fu(\))d(whic)o(h)i(implies)h(\()p Fo(K)p Fu(\))d(b)o(y)g(lemma)h(4.1.)33 b(So,)21 b(it)f(exists)g Fs(M)25 b(>)c Fu(0)e(suc)o(h)h(that)-12 1933 y(for)f(an)o(y)f Fs(n)h Fn(2)h Fj(N)p Fu(,)e Fn(kL)366 1916 y Fr(n)366 1946 y Fp(\010)394 1933 y Fo(1)p Fn(k)443 1940 y Fq(1)499 1933 y Fn(\024)h Fs(M)5 b Fu(.)31 b(If)20 b(\006)e(has)h(no)g(jumps)g (to)g(in\014nit)o(y)h(and)f(\010)g(satis\014es)g(\()p Fo(Exp2)p Fu(\))g(then,)g(b)o(y)-12 1989 y(induction,)e(w)o(e)e(ma)o(y) f(sho)o(w,)483 2092 y Fn(8)p Fs(p)f Fn(\025)g Fu(1)p Fs(;)22 b Fn(8)p Fs(n)13 b(>)g(n)790 2099 y Fp(0)820 2092 y Fu(+)d(\()p Fs(p)g Fn(\000)g Fu(1\))p Fs(K)s(;)15 b Fu(sup)1065 2133 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])1156 2092 y Fn(L)1187 2069 y Fr(p)1187 2106 y Fp(\010)1215 2092 y Fo(1)p Fu(\()p Fs(x)p Fu(\))d Fn(\024)h Fs(\032)1387 2073 y Fr(p)1406 2092 y Fs(:)-12 2230 y Fu(Let)j(us)f(\014x)g Fs(k)216 2237 y Fp(1)248 2230 y Fn(2)e Fj(N)h Fu(and)i Fs(n)455 2237 y Fp(1)487 2230 y Fu(=)d Fs(n)562 2237 y Fp(0)592 2230 y Fu(+)e(\()p Fs(k)g Fn(\000)f Fu(1\))p Fs(K)s Fu(,)k(if)i Fs(k)e Fn(\025)f Fs(k)998 2237 y Fp(1)1032 2230 y Fu(and)i Fs(n)e Fn(\025)g Fs(n)1235 2237 y Fp(1)1271 2230 y Fu(then:)467 2332 y(sup)459 2373 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])551 2332 y Fn(L)582 2313 y Fr(k)582 2343 y Fp(\010)610 2332 y Fo(1)p Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fu(sup\()p Fn(L)875 2311 y Fr(k)q Fq(\000)p Fr(k)939 2316 y Fk(1)875 2346 y Fp(\010)959 2332 y Fo(1)p Fu(\))j(sup)1011 2373 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])1103 2332 y Fn(L)1134 2311 y Fr(k)1152 2316 y Fk(1)1134 2346 y Fp(\010)1172 2332 y Fo(1)p Fu(\()p Fs(x)p Fu(\))c Fn(\024)h Fs(M)5 b(\032)1393 2313 y Fr(k)1411 2318 y Fk(1)1430 2332 y Fs(:)-12 2475 y Fu(Then,)16 b(it)f(su\016ces)h(to)e(c)o(hose)h Fs(k)519 2482 y Fp(1)554 2475 y Fu(suc)o(h)g(that)g Fs(M)5 b(\032)828 2459 y Fr(k)846 2464 y Fk(1)878 2475 y Fn(\024)13 b Fs(\014)h(<)f Fu(1.)823 b Fn(4)-12 2532 y Fu(W)l(e)14 b(will)i(sa)o(y)d(that)g(\006)g(has)h Ft(b)n(ounde)n(d)h(jumps)f Fu(if)g(it)g(exists)h(an)e(in)o(teger)h Fs(K)j Fu(suc)o(h)d(that)f(for) g(all)i Fs(n)e Fn(2)g Fj(N)p Fu(,)f(for)i Fs(x)e Fn(2)h Fu([)p Fs(n)p Fu(],)928 2796 y(22)p eop %%Page: 23 23 23 22 bop -12 1 a Fs(\033)r(x)13 b Fn(2)h Fu([)p Fs(p)p Fu(])h(with)h Fs(n)11 b Fn(\000)f Fs(K)17 b Fn(\024)d Fs(p)f Fn(\024)h Fs(n)d Fu(+)f Fs(K)s Fu(.)22 b(In)16 b(other)f(w)o(ords,)g(the)h(matrix)f(whic)o(h)i(de\014nes)g(\006)e(has) h(the)g(follo)o(wing)-12 57 y(form:)406 64 y Fi(0)406 144 y(B)406 172 y(B)406 199 y(B)406 226 y(B)406 253 y(B)406 281 y(B)406 308 y(B)406 335 y(B)406 362 y(B)406 390 y(B)406 417 y(B)406 446 y(@)467 101 y Fn(\003)41 b(\001)8 b(\001)g(\001)39 b(\001)8 b(\001)g(\001)55 b(\003)72 b Fu(0)56 b Fn(\001)8 b(\001)g(\001)40 b(\001)8 b(\001)g(\001)39 b(\001)8 b(\001)g(\001)40 b(\001)8 b(\001)g(\001)39 b(\001)8 b(\001)g(\001)40 b(\001)8 b(\001)g(\001)472 148 y Fu(.)472 164 y(.)472 181 y(.)817 152 y(.)835 164 y(.)852 177 y(.)912 152 y(.)929 164 y(.)947 177 y(.)467 237 y Fn(\003)434 b(\003)72 b Fu(0)467 294 y(0)56 b Fn(\003)450 b(\003)71 b Fu(0)472 341 y(.)472 357 y(.)472 374 y(.)533 345 y(.)551 357 y(.)569 370 y(.)628 345 y(.)646 357 y(.)663 370 y(.)1101 345 y(.)1118 357 y(.)1136 370 y(.)1195 345 y(.)1213 357 y(.)1231 370 y(.)467 430 y(0)151 b(0)71 b Fn(\003)450 b(\003)71 b Fu(0)472 477 y(.)472 493 y(.)472 510 y(.)723 481 y(.)740 493 y(.)758 506 y(.)817 481 y(.)835 493 y(.)852 506 y(.)1290 481 y(.)1307 493 y(.)1325 506 y(.)1384 481 y(.)1402 493 y(.)1420 506 y(.)1456 64 y Fi(1)1456 144 y(C)1456 172 y(C)1456 199 y(C)1456 226 y(C)1456 253 y(C)1456 281 y(C)1456 308 y(C)1456 335 y(C)1456 362 y(C)1456 390 y(C)1456 417 y(C)1456 446 y(A)-12 592 y Fu(with)16 b Fn(\003)c(2)h(f)p Fu(0)p Fs(;)8 b Fu(1)p Fn(g)p Fu(.)-12 698 y Fo(Example)18 b(4.2)23 b([Dynamics)17 b(satisfying)h(\(S-exp1\).])-12 755 y Fu(If)e(\006)f(is)g(ap)q(erio)q(dic)i(and)f(has)f(b)q(ounded)h(jumps,)f (if)h(\010)f(v)o(eri\014es)h(\()p Fo(SA)p Fu(\))e(and)h(the)h(follo)o (wing)g(t)o(w)o(o)e(prop)q(erties:)44 868 y(1.)22 b Fn(9)p Fs(n)154 875 y Fp(0)189 868 y Fu(suc)o(h)16 b(that)e Fn(8)p Fs(n)f(>)g(n)530 875 y Fp(0)550 868 y Fs(;)23 b Fn(8)p Fs(x)13 b Fn(2)g Fu([)p Fs(n)p Fu(])p Fs(;)21 b Fn(L)811 875 y Fp(\010)839 868 y Fo(1)p Fu(\()p Fs(x)p Fu(\))12 b Fn(\024)h Fu(\(1)d Fn(\000)1091 837 y Fu(1)p 1088 857 28 2 v 1088 899 a Fs(n)1121 868 y Fu(\))1139 849 y Fr(\013)1178 868 y Fs(\013)j(>)g Fu(0)p Fs(;)372 b Fu(\()p Fo(S)10 b Fn(\000)h Fo(Exp2)p Fu(\))44 975 y(2.)22 b(the)17 b(in)o(v)m(arian)o(t)h(densit)o(y)g Fs(h)f Fu(go)q(es)g(to)f(zero)h(at)g(in\014nit)o(y)h(\(that)e(is,)i(if) 26 b(sup)1273 1016 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])1365 975 y Fs(h)p Fu(\()p Fs(x)p Fu(\))15 b(:=)h Fs(h)1558 982 y Fr(n)1599 975 y Fu(then)h Fs(h)1730 982 y Fr(n)1771 975 y Fu(go)q(es)g(to)102 1067 y(zero)e(when)h Fs(n)f Fu(go)q(es)g(to)g(in\014nit)o(y\),)-12 1161 y(then)h(\010)f(v)o (eri\014es)h(\()p Fo(S-Exp1)p Fu(\).)-12 1250 y(First)f(of)g(all,)h (let)g(us)g(remark)e(that)h(\()p Fo(S-Exp2)p Fu(\))g(implies)j(\()p Fo(H)p Fu(\))c(whic)o(h)j(implies)g(\()p Fo(K)p Fu(\).)d(No)o(w,)h(if)h (\006)f(has)g(b)q(ounded)-12 1307 y(jumps)e(and)g(\010)g(v)o(eri\014es) h(\()p Fo(S-Exp2)p Fu(\))f(then,)g(b)o(y)g(induction,)i(w)o(e)e(ha)o(v) o(e:)18 b(for)12 b(an)o(y)h Fs(k)h Fn(\025)f Fu(1)f(and)h Fs(n)g(>)g(n)1662 1314 y Fp(0)1688 1307 y Fu(+)6 b(\()p Fs(k)h Fn(\000)f Fu(1\))p Fs(K)s Fu(,)624 1453 y(sup)616 1494 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])708 1453 y Fn(L)739 1434 y Fr(k)739 1464 y Fp(\010)767 1453 y Fo(1)p Fu(\()p Fs(x)p Fu(\))12 b Fn(\024)915 1396 y Fr(k)q Fq(\000)p Fp(1)918 1409 y Fi(Y)916 1507 y Fr(j)r Fp(=0)979 1453 y Fu(\(1)d Fn(\000)1142 1422 y Fu(1)p 1080 1442 147 2 v 1080 1484 a Fs(n)h Fu(+)h Fs(K)s(j)1231 1453 y Fu(\))1249 1434 y Fr(\013)1273 1453 y Fs(:)534 b Fu(\(4.1\))-12 1607 y(The)14 b(follo)o(wing)g(lemma)f(giv)o(es)h(an)f(estimate)g(of)g Fn(L)846 1591 y Fr(k)846 1621 y Fp(\010)874 1607 y Fo(1)p Fu(\()p Fs(x)p Fu(\))g(for)f Fs(x)h Fn(2)g Fu([)p Fs(n)p Fu(])g(and)g Fs(N)k Fn(\024)c Fs(n)g Fn(\024)g Fs(N)e Fu(+)6 b(\()p Fs(k)h Fn(\000)f Fu(1\))p Fs(K)17 b Fu(pro)o(vided)-12 1664 y Fs(N)j Fu(and)15 b Fs(k)h Fu(are)f(large)h(enough.)-12 1770 y Fo(Lemma)i(4.2)k Fg(F)l(or)15 b(an)o(y)h Fu(\()455 1752 y Fp(1)p 455 1759 18 2 v 455 1786 a(2)477 1770 y Fu(\))495 1754 y Fr(\013)534 1770 y Fs(<)e(\021)h(<)f Fu(1)p Fg(,)i(it)g(exists)g Fs(n)921 1777 y Fp(1)955 1770 y Fn(\025)e Fs(n)1031 1777 y Fp(0)1067 1770 y Fg(suc)o(h)i(that)g (for)f(an)o(y)h Fs(N)i Fn(\025)c Fs(n)1558 1777 y Fp(1)1594 1770 y Fg(it)i(exists)h Fs(k)1791 1777 y Fp(1)1826 1770 y Fg(suc)o(h)-12 1827 y(that)e(for)f Fs(k)g Fn(\025)f Fs(k)266 1834 y Fp(1)285 1827 y Fg(,)745 1883 y Fu(sup)634 1924 y Fr(N)t Fq(\024)p Fr(n)h(k)244 2323 y Fp(0)285 2316 y Fu(suc)o(h)f(that)e(for)h Fs(k)j(>)g(k)706 2299 y Fq(0)718 2316 y Fu(,)753 2252 y Fi(\022)792 2285 y Fs(N)14 b Fu(+)d Fs(K)s Fu(\()p Fs(k)f Fu(+)h(1\))p 792 2305 278 2 v 837 2347 a Fs(N)k Fu(+)10 b(2)p Fs(K)s(k)1074 2252 y Fi(\023)1108 2263 y Fr(\013)1156 2316 y Fs(<)23 b(\021)g Fu(and)e Fs(k)1378 2323 y Fp(1)1421 2316 y Fs(>)i(k)1504 2299 y Fq(0)1537 2316 y Fu(suc)o(h)f(that)f Fs(k)j(>)f(k)1881 2323 y Fp(1)1901 2316 y Fu(,)102 2430 y Fs(u)128 2438 y Fr(k)147 2428 y Ff(0)168 2366 y Fi(\022)206 2400 y Fs(N)15 b Fu(+)10 b(\()p Fs(k)346 2383 y Fq(0)368 2400 y Fu(+)g(1\))p Fs(K)p 206 2420 290 2 v 212 2461 a(N)15 b Fu(+)10 b(\()p Fs(k)h Fn(\000)g Fu(2\))p Fs(K)501 2366 y Fi(\023)534 2377 y Fr(\013)571 2430 y Fs(<)i(\021)k Fu(where)e Fs(u)816 2437 y Fr(k)853 2430 y Fu(is)g(de\014ned)i(b)o(y) 480 2565 y Fs(u)506 2572 y Fr(k)540 2565 y Fu(=)c(sup)p Fn(fL)710 2546 y Fr(k)710 2576 y Fp(\010)738 2565 y Fo(1)p Fu(\()p Fs(x)p Fu(\))h Fs(=)h(x)e Fn(2)g Fu([)p Fs(n)p Fu(])p Fs(;)8 b(N)16 b Fn(\024)d Fs(n)g(<)g(N)h Fu(+)d Fs(K)s Fu(\()p Fs(k)f Fn(\000)h Fu(1\))p Fn(g)p Fs(:)p -12 2607 771 2 v 40 2634 a Fk(1)57 2650 y Fd(Since)k(\006)d(has)i(b)q (ounded)g(jumps,)f(the)h(set)e([)-6 b Fc(j)p Fd(0)p Fb(;)6 b(N)t Fc(j)-6 b Fd(])11 b(is)j(compact.)j(So)d(theorem)f(1.1)g(imply)i (that)e(suc)o(h)g(an)h(in)o(teger)g Fb(k)1738 2654 y Fk(0)1768 2650 y Fd(exists.)928 2796 y Fu(23)p eop %%Page: 24 24 24 23 bop -12 1 a Fu(Let)18 b Fs(k)g(>)f(k)190 8 y Fp(1)210 1 y Fu(,)g Fs(N)22 b Fn(\025)17 b Fs(n)378 8 y Fp(1)416 1 y Fu(and)g Fs(x)g Fn(2)g Fu([)p Fs(n)p Fu(])h(with)g Fs(N)j Fn(\024)c Fs(n)g(<)g(N)f Fu(+)d Fs(k)q(K)s Fu(,)k(if)i Fs(x)1248 -16 y Fq(0)1277 1 y Fu(is)f(the)g(preimage)g(of)f Fs(x)h Fu(b)o(y)f Fs(\033)r Fu(,)h(then)-12 57 y Fs(x)14 41 y Fq(0)38 57 y Fn(2)13 b Fu([)p Fs(n)121 41 y Fq(0)133 57 y Fu(])i(with)g Fs(N)301 64 y Fp(0)333 57 y Fs(<)e(N)i Fn(\000)10 b Fs(K)16 b Fn(\024)d Fs(n)608 41 y Fq(0)632 57 y Fs(<)g(N)i Fu(+)10 b(\()p Fs(k)h Fu(+)f(1\))p Fs(K)s Fu(,)-12 151 y Fn(L)19 132 y Fr(k)q Fp(+1)19 166 y(\010)86 151 y Fo(1)p Fu(\()p Fs(x)p Fu(\))i(=)251 108 y Fi(X)234 207 y Fr(\033)q(x)275 198 y Ff(0)286 207 y Fp(=)p Fr(x)341 151 y Fs(g)363 158 y Fp(0)383 151 y Fu(\()p Fs(x)427 133 y Fq(0)438 151 y Fu(\))p Fn(L)487 133 y Fr(k)487 163 y Fp(\010)515 151 y Fo(1)p Fu(\()p Fs(x)585 133 y Fq(0)596 151 y Fu(\))71 276 y(=)169 233 y Fi(X)119 332 y Fr(N)147 337 y Fk(0)164 332 y Fr()g(k)244 1176 y Fq(0)256 1193 y Fu(,)188 1321 y Fn(L)219 1302 y Fr(k)q Fp(+1)219 1335 y(\010)286 1321 y Fo(1)p Fu(\()p Fs(x)p Fu(\))41 b Fn(\024)h Fu(max)o([)p Fs(")10 b Fu(+)59 b(sup)666 1361 y Fr(N)694 1366 y Fk(0)711 1361 y Fr()i Fu(1)h(for)g(an)o(y)g Fs(x)e Fn(2)740 787 y Fi(S)778 834 y Fr(n)809 821 y Fs(I)829 828 y Fr(n)868 821 y Fu(and)j(that)e Fs(T)22 b Fu(is)16 b(uniformly)g(expanding)h(if)f(it)g(exists)g Fs(D)e(>)g Fu(1)-12 877 y(suc)o(h)j(that)e Fs(T)224 861 y Fq(0)236 877 y Fu(\()p Fs(x)p Fu(\))e Fn(\025)i Fs(D)i Fu(for)f(an)o(y)g Fs(x)e Fn(2)659 843 y Fi(S)697 891 y Fr(n)728 877 y Fs(I)748 884 y Fr(n)771 877 y Fu(.)23 b(If)17 b Fs(T)22 b Fu(is)16 b(uniformly)i(expanding)f(then)g Fs(B)i Fu(injects)e(naturally)g(in)-12 934 y Fs(L)f Fu(and)g(w)o(e)f (ma)o(y)h(w)o(ork)e(on)i(the)g(sym)o(b)q(olic)h(dynamic)g(as)e(w)o(ell) i(as)e(on)h(the)g(in)o(terv)m(al.)23 b(But,)15 b(since)i(w)o(e)f(also)g (wish)-12 990 y(to)e(consider)i(non)f(uniformly)g(expanding)h(maps,)e (it)h(is)g(preferable)h(to)e(w)o(ork)f(directly)j(on)f Fs(I)t Fu(.)k(The)c(tec)o(hniques)-12 1047 y(that)d(w)o(e)h(ha)o(v)o(e) f(dev)o(elop)q(ed)i(in)g(sections)f(2)g(and)g(3)f(are)g(directly)i (applicable)i(to)c(the)h(uniformly)g(expanding)h(case)-12 1103 y(and)i(are)f(applicable)j(with)e(some)f(mo)q(di\014cations)h(to)f (the)g(non)h(uniformly)g(expanding)h(case.)k(In)16 b(the)f(in)o(terv)m (al)-12 1159 y(maps)j(setting,)g(the)g Fs(k)q Fu(-cylinders)i(corresp)q (ond)e(to)g(subin)o(terv)m(als)h(of)f Fs(I)j Fu(of)d(the)g(form)1478 1125 y Fi(T)1516 1138 y Fr(k)q Fq(\000)p Fp(1)1516 1173 y Fr(i)p Fp(=0)1590 1159 y Fs(T)1623 1143 y Fq(\000)p Fr(i)1664 1159 y Fs(I)1684 1166 y Fr(n)1705 1171 y Fl(i)1721 1159 y Fu(.)28 b(W)l(e)18 b(will)-12 1216 y(treat)c(the)i(follo)o(wing) g(example.)-12 1319 y Fo(Example)i(4.3)-12 1375 y Fu(Let)e Fs(T)i Fu(:)12 b Fs(I)17 b Fn(!)c Fs(I)18 b Fu(b)q(e)e(de\014ned)h(in)f (the)f(follo)o(wing)h(w)o(a)o(y:)-12 1432 y Fs(T)24 b Fu(is)18 b Fs(C)123 1415 y Fp(2)142 1432 y Fu(,)g(monotone)f(and)h (increasing)g(on)g(eac)o(h)f Fs(I)880 1439 y Fr(n)904 1432 y Fu(,)h(it)f(ma)o(y)g(b)q(e)i(con)o(tin)o(ued)f(to)f(a)g(con)o (tin)o(uous)h(function)g(on)-12 1488 y(the)d(closure)h(of)f Fs(I)290 1495 y Fr(n)329 1488 y Fu(and)g(there)g(exists)h(some)f(in)o (teger)g Fs(K)j Fu(suc)o(h)e(that)741 1588 y Fs(T)6 b(I)794 1595 y Fr(n)830 1588 y Fu(=)970 1545 y Fi([)878 1643 y Fr(n)p Fq(\000)p Fr(K)r Fq(\024)p Fr(p)p Fq(\024)p Fr(n)p Fp(+)p Fr(K)1120 1588 y Fs(I)1140 1595 y Fr(p)-12 1732 y Fu(\(with)16 b(the)f(con)o(v)o(en)o(tion)h(that)f Fs(I)535 1739 y Fr(n)572 1732 y Fu(=)e Fn(;)i Fu(if)h Fs(n)e(<)g Fu(0\).)20 b(Moreo)o(v)o(er,)14 b Fs(T)21 b Fu(is)16 b(uniformly)h(expanding)g(and)e(there)h(exists)-12 1788 y Fs(R)d(>)g Fu(0)i(suc)o(h)g(that)g Fs(T)356 1772 y Fq(00)377 1788 y Fu(\()p Fs(x)p Fu(\))d Fn(\024)h Fs(R)i Fu(for)f(an)o(y)h Fs(x)e Fn(2)786 1754 y Fi(S)831 1788 y Fs(I)851 1795 y Fr(n)874 1788 y Fu(.)-12 1876 y(These)j(dynamics)g (are)f(ap)q(erio)q(dic)i(and)e(with)h(b)q(ounded)g(jumps.)k(W)l(e)c (will)g(denote)g(b)o(y)628 1991 y Fs(\032)652 1998 y Fr(n)688 1991 y Fu(=)806 1960 y Fs(\025)833 1967 y Fr(n)p 741 1980 181 2 v 764 2013 a(n)p Fp(+)p Fr(K)771 2027 y Fi(X)741 2125 y Fr(p)p Fp(=)p Fr(n)p Fq(\000)p Fr(K)875 2070 y Fs(\025)902 2077 y Fr(p)941 1991 y Fu(with)g Fs(\025)1072 1998 y Fr(n)1107 1991 y Fu(=)d Fs(\025)p Fu(\()p Fs(I)1220 1998 y Fr(n)1243 1991 y Fu(\))p Fs(:)-12 2211 y Fu(Let)j(us)f(remark)g (that)755 2277 y Fs(\032)779 2258 y Fq(\000)p Fp(1)779 2288 y Fr(n)838 2277 y Fu(=)e Fs(\025)913 2258 y Fq(\000)p Fp(1)913 2288 y Fr(n)968 2215 y Fi(Z)993 2318 y Fr(I)1008 2322 y Fl(n)1039 2277 y Fs(T)1072 2258 y Fq(0)1084 2277 y Fs(d\025:)-12 2407 y Fo(Lemma)18 b(4.3)k Fg(If)16 b(the)f(sequence)h Fu(\()p Fs(\025)620 2414 y Fr(n)643 2407 y Fu(\))661 2414 y Fr(n)p Fq(2)p Fm(N)747 2407 y Fg(satis\014es)f(one)g(of)g(the)g (t)o(w)o(o)f(follo)o(wing)i(prop)q(erties:)44 2517 y(1.)36 b Fu(lim)102 2544 y Fr(n)p Fq(!1)206 2486 y Fs(\025)233 2493 y Fr(n)p Fp(+1)p 206 2507 96 2 v 229 2548 a Fs(\025)256 2555 y Fr(n)319 2517 y Fs(<)13 b Fu(1)p Fg(,)i(in)h(this)f(case,)g(w)o (e)g(w)o(ould)h(sa)o(y)e(that)h(the)g(sequence)h Fs(\025)1338 2524 y Fr(n)1376 2517 y Fg(is)g(of)f(exp)q(onen)o(tial)i(t)o(yp)q(e,)44 2657 y(2.)107 2626 y Fs(\025)134 2633 y Fr(n)p Fp(+1)p 107 2646 V 129 2688 a Fs(\025)156 2695 y Fr(n)222 2657 y Fg(is)f(increasing)g(to)f Fu(1)g Fg(for)f Fs(n)f Fn(\025)g Fs(n)758 2664 y Fp(0)778 2657 y Fg(,)i(and)g Fs(\025)921 2664 y Fr(n)957 2657 y Fu(=)e Fs(o)p Fu(\()p Fs(n)1072 2640 y Fq(\000)p Fp(2)1119 2657 y Fu(\))p Fg(.)928 2796 y Fu(25)p eop %%Page: 26 26 26 25 bop -12 1 a Fg(then)16 b Fn(L)f Fg(satis\014es)h(\()p Fo(H)p Fg(\).)-12 91 y Fo(Pro)q(of)5 b Fu(:)40 b(W)l(e)15 b(ha)o(v)o(e)g(for)g(an)o(y)g Fs(x)g Fu(and)g Fs(y)i Fu(in)f Fs(I)753 98 y Fr(n)777 91 y Fu(,)605 193 y Fn(j)p Fs(T)651 174 y Fq(0)662 193 y Fu(\()p Fs(x)p Fu(\))9 b Fn(\000)i Fs(T)812 174 y Fq(0)823 193 y Fu(\()p Fs(y)r Fu(\))p Fn(j)h(\024)17 b Fu(sup)956 232 y Fr(z)q Fq(2)p Fr(I)1012 236 y Fl(n)1041 193 y Fs(T)1074 174 y Fq(00)1095 193 y Fu(\()p Fs(z)r Fu(\))p Fn(j)p Fs(x)9 b Fn(\000)i Fs(y)r Fn(j)p Fs(:)-12 321 y Fu(By)16 b(in)o(tegrating)f(on)g Fs(I)374 328 y Fr(n)397 321 y Fu(,)g(w)o(e)g(get:)561 440 y Fn(\000)p Fs(R\025)p Fu(\()p Fs(I)696 447 y Fr(n)718 440 y Fu(\))10 b(+)809 409 y(1)p 797 430 47 2 v 797 471 a Fs(\032)821 478 y Fr(n)861 440 y Fn(\024)j Fs(T)942 421 y Fq(0)954 440 y Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fs(R\025)p Fu(\()p Fs(I)1176 447 y Fr(n)1198 440 y Fu(\))d(+)1289 409 y(1)p 1276 430 V 1276 471 a Fs(\032)1300 478 y Fr(n)1328 440 y Fs(:)479 b Fu(\(4.2\))-12 592 y(Let)17 b(us)f(assume)g(that)30 b(lim)388 620 y Fr(n)p Fq(!1)492 562 y Fs(\025)519 569 y Fr(n)p Fp(+1)p 492 582 96 2 v 515 623 a Fs(\025)542 630 y Fr(n)606 592 y Fu(=)15 b Fs(\022)h(<)f Fu(1,)g(this)i(implies)h (that)e Fs(\032)1163 599 y Fr(n)1202 592 y Fu(go)q(es)g(to)1423 574 y Fr(\022)1440 563 y Fl(K)p 1364 581 164 2 v 1364 610 a Fp(1+)p Fq(\001\001\001)p Fp(+)p Fr(\022)1483 601 y Fk(2)p Fl(K)1548 592 y Fs(<)e Fu(\(2)p Fs(K)f Fu(+)e(1\))1777 576 y Fq(\000)p Fp(1)1840 592 y Fu(and)-12 681 y(sup)57 692 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])151 681 y Fn(L)p Fo(1)p Fu(\()p Fs(x)p Fu(\))k(to)f(\(2)p Fs(K)f Fu(+)d(1\))582 663 y Fr(\022)599 651 y Fl(K)p 523 670 V 523 699 a Fp(1+)p Fq(\001\001\001)p Fp(+)p Fr(\022)642 689 y Fk(2)p Fl(K)705 681 y Fs(<)j Fu(1)i(\(it)g(uses)g(\(4.2\)\))f(so)g(that)h(\()p Fo(H)p Fu(\))f(is)i(satis\014ed.)-12 769 y(Let)d(us)f(assume)g(that)g (it)g(exists)h Fs(n)564 776 y Fp(0)596 769 y Fu(suc)o(h)g(that)e(from)h Fs(n)923 776 y Fp(0)943 769 y Fu(,)g(the)h(sequence)1234 738 y Fs(\025)1261 745 y Fr(n)p Fp(+1)p 1234 758 96 2 v 1257 800 a Fs(\025)1284 807 y Fr(n)1347 769 y Fu(increases)g(to)e(1,) h(let)h(0)g Fs(<)g(u)1807 776 y Fr(n)1843 769 y Fs(<)g Fu(1)-12 843 y(b)q(e)j(suc)o(h)g(that)e(for)h Fs(n)e Fn(\025)g Fs(n)436 850 y Fp(0)456 843 y Fu(,)h Fs(u)509 850 y Fr(n)p Fp(+1)591 843 y Fn(\024)f Fs(u)665 850 y Fr(n)688 843 y Fu(,)i(the)g(sequence)h Fs(u)1008 850 y Fr(n)1047 843 y Fu(go)q(es)f(to)g(zero)g(and)785 934 y Fs(\025)812 941 y Fr(n)p Fp(+1)p 785 955 V 808 996 a Fs(\025)835 1003 y Fr(n)898 965 y Fu(=)e(\(1)c Fn(\000)i Fs(u)1068 972 y Fr(n)1091 965 y Fu(\))p Fs(;)-12 1084 y Fu(for)16 b(an)o(y)g Fs(j)g Fu(=)f(1)p Fs(;)8 b Fn(\001)g(\001)g (\001)t Fs(;)g(K)19 b Fu(and)d Fs(n)e Fn(\025)h Fs(n)620 1091 y Fp(0)651 1084 y Fu(+)c Fs(K)s Fu(,)16 b(it)g(exists)h(a)f (sequence)h Fs(u)1195 1091 y Fr(n;j)1261 1084 y Fu(suc)o(h)f(that)g Fs(u)1490 1091 y Fr(n;j)1555 1084 y Fu(go)q(es)g(to)g(zero)g(when)-12 1141 y Fs(n)g Fu(go)q(es)f(to)f(in\014nit)o(y)l(,)j(0)12 b Fs(<)h(u)461 1148 y Fr(n)p Fp(+1)p Fr(;j)568 1141 y Fn(\024)g Fs(u)642 1148 y Fr(n;j)707 1141 y Fu(and:)773 1235 y Fs(\025)800 1242 y Fr(n)p Fp(+)p Fr(j)p 773 1256 94 2 v 795 1298 a Fs(\025)822 1305 y Fr(n)884 1266 y Fu(=)g(\(1)d Fn(\000)g Fs(u)1054 1273 y Fr(n;j)1103 1266 y Fu(\))p Fs(:)-12 1389 y Fu(W)l(e)15 b(get)g(for)g(some)g Fs(n)353 1373 y Fq(0)353 1401 y Fp(0)385 1389 y Fn(\025)e Fs(n)460 1396 y Fp(0)495 1389 y Fu(and)j(for)e(an)o(y)h Fs(n)e Fn(\025)g Fs(n)854 1373 y Fq(0)854 1401 y Fp(0)884 1389 y Fu(+)e Fs(K)18 b Fu(:)292 1535 y Fs(\032)316 1516 y Fq(\000)p Fp(1)316 1546 y Fr(n)376 1535 y Fu(=)13 b(\(2)p Fs(K)f Fu(+)f(1\))e Fn(\000)674 1478 y Fr(K)658 1492 y Fi(X)660 1590 y Fr(j)r Fp(=1)731 1535 y Fs(u)757 1542 y Fr(n;j)816 1535 y Fu(+)879 1478 y Fr(K)862 1492 y Fi(X)864 1590 y Fr(j)r Fp(=1)927 1535 y Fu([)p Fs(u)966 1542 y Fr(n)p Fq(\000)p Fr(j;j)1067 1535 y Fu(+)1128 1478 y Fq(1)1113 1492 y Fi(X)1117 1590 y Fr(i)p Fp(=2)1178 1535 y Fu(\()p Fs(u)1222 1542 y Fr(n)p Fq(\000)p Fr(j;j)1313 1535 y Fu(\))1331 1516 y Fr(i)1345 1535 y Fu(])j Fn(\025)h Fu(\(2)p Fs(K)g Fu(+)d(1\))p Fs(:)-12 1692 y Fu(Remark)17 b(that)261 1658 y Fi(P)309 1705 y Fr(n)340 1692 y Fs(u)366 1699 y Fr(n;j)431 1692 y Fu(=)e Fn(1)i Fu(for)g(all)g Fs(j)s Fu(,)f(since)i Fs(\025)870 1699 y Fr(n)908 1692 y Fu(=)e Fs(o)p Fu(\()p Fs(n)1026 1675 y Fq(\000)p Fp(2)1073 1692 y Fu(\),)h(w)o(e)f(ha)o(v)o(e)g(that)1412 1674 y Fr(\025)1433 1678 y Fl(n)p 1400 1681 66 2 v 1400 1711 a Fr(u)1420 1699 y Fk(2)1420 1722 y Fl(n;j)1487 1692 y Fu(go)q(es)h(to)f(zero)h(for)f(all)h Fs(j)s Fu(.)-12 1765 y(So,)e(using)h(\(4.2\),)d(w)o(e)i(get)g Fn(L)p Fo(1)504 1772 y Fe(n)542 1765 y Fn(\024)e Fu(1)h(if)i Fs(n)d Fn(\025)g Fs(n)784 1772 y Fp(0)814 1765 y Fu(+)e Fs(K)18 b Fu(and)d(\()p Fo(H)p Fu(\))f(is)i(v)o(eri\014ed.)575 b Fn(4)3 1821 y Fu(As)14 b(w)o(e)g(already)h(noticed,)g(\()p Fo(H)p Fu(\))e(and)i(the)f(fact)g(that)g Fs(\025)g Fu(is)g(a)g (conformal)h(measure)f(for)g Fn(L)g Fu(imply)i(\()p Fo(K)p Fu(\),)d(let)i Fs(h)g Fu(b)q(e)-12 1878 y(the)g(in)o(v)m(arian)o(t)f (densit)o(y)h(and)g Fs(\026)e Fu(=)g Fs(h\025)p Fu(.)19 b(The)14 b(pro)q(of)g(of)g(lemma)h(4.3)e(sho)o(ws)g(that)h(if)h(\()p Fs(\025)1464 1885 y Fr(n)1486 1878 y Fu(\))1504 1885 y Fr(n)p Fq(2)p Fm(N)1589 1878 y Fu(is)g(of)f(exp)q(onen)o(tial)-12 1934 y(t)o(yp)q(e,)k(then)f Fn(L)h Fu(satis\014es)g(\()p Fo(Exp2)p Fu(\).)25 b(Since)19 b Fs(T)k Fu(has)17 b(b)q(ounded)i (jumps,)f(example)g(4.1)e(pro)o(v)o(es)h(that)f Fn(L)i Fu(satis\014es)-12 1991 y(\()p Fo(Exp1)p Fu(\),)c(so)h(that)g(the)g (deca)o(y)g(of)g(correlations)g(is)h(exp)q(onen)o(tial)h(on)e(the)g (space)h Fs(B)r Fu(.)-12 2047 y(Moreo)o(v)o(er,)11 b(if)h Fs(T)18 b Fu(is)12 b(a\016ne)g(on)f(eac)o(h)h Fs(I)619 2054 y Fr(n)654 2047 y Fu(then)g(it)g(is)g(easy)g(to)f(see)h(that)f Fs(')g Fu(de\014ned)j(b)o(y)d Fs(')p Fu(\()p Fs(x)p Fu(\))h(=)h Fs(\025)1585 2054 y Fr(n)p Fp(+)p Fr(K)1671 2047 y Fu(+)s Fn(\001)8 b(\001)g(\001)q Fu(+)s Fs(\025)1830 2054 y Fr(n)p Fq(\000)p Fr(K)-12 2104 y Fu(if)17 b Fs(x)d Fn(2)h Fs(I)136 2111 y Fr(n)176 2104 y Fu(is)h(a)g(\014xed)h(p)q(oin)o(t)g (for)e Fn(L)i Fu(\(so)e(it)i(is)g(the)f(only)h(one)f(up)h(to)e(a)h (normalization)h(b)o(y)f(theorem)g(1.1\).)22 b(Let)-12 2160 y(us)16 b(consider)g Fs(J)249 2167 y Fr(N)295 2160 y Fu(=)343 2126 y Fi(S)381 2174 y Fr(n>N)t Fp(+)p Fr(K)531 2160 y Fs(I)551 2167 y Fr(n)574 2160 y Fu(.)k(W)l(e)15 b(ha)o(v)o(e)247 2268 y Fn(j)p Fs(\026)p Fu(\()p Fs(T)338 2249 y Fq(\000)p Fr(N)399 2268 y Fs(J)424 2275 y Fr(N)468 2268 y Fn(\\)10 b Fs(I)528 2275 y Fp(0)548 2268 y Fu(\))g Fn(\000)g Fs(\026)p Fu(\()p Fs(J)691 2275 y Fr(N)725 2268 y Fu(\))p Fs(\026)p Fu(\()p Fs(I)808 2275 y Fp(0)827 2268 y Fu(\))p Fn(j)i Fu(=)h Fs(\026)p Fu(\()p Fs(J)988 2275 y Fr(N)1022 2268 y Fu(\))p Fs(\026)p Fu(\()p Fs(I)1105 2275 y Fp(0)1125 2268 y Fu(\))h(b)q(ecause)j Fs(T)1359 2249 y Fq(\000)p Fr(n)1409 2268 y Fs(J)1434 2275 y Fr(N)1478 2268 y Fn(\\)11 b Fs(I)1539 2275 y Fp(0)1571 2268 y Fu(=)i Fn(;)p Fs(:)-12 2370 y Fu(In)24 b(particular,)h(it)e(is)g(not)f(p)q (ossible)j(to)d(ha)o(v)o(e)g(an)h(exp)q(onen)o(tial)h(deca)o(y)f(of)f (correlations)i(of)e(t)o(yp)q(e)h(\(1.3\))e(if)-12 2427 y(the)h(sequence)h Fs(\025)295 2434 y Fr(n)339 2427 y Fu(is)g(not)e(ma)s(jored)g(b)o(y)g(an)h(exp)q(onen)o(tial)h(sequence.) 40 b(The)22 b(deca)o(y)g(of)f(correlations)h(ma)o(y)l(,)-12 2483 y(nev)o(ertheless,)g(b)q(e)f(estimated)g(in)g(some)f(cases.)35 b(F)l(or)20 b Fs(P)28 b Fn(2)21 b Fj(N)p Fu(,)f(let)h Fn(k)f(k)1263 2490 y Fr(P)1313 2483 y Fu(denotes)g(the)h(uniform)f (norm)g(on)-12 2506 y Fi(S)26 2553 y Fr(j)r Fq(\024)p Fr(P)107 2540 y Fs(I)127 2547 y Fr(j)145 2540 y Fu(.)928 2796 y(26)p eop %%Page: 27 27 27 26 bop -12 1 a Fo(Prop)q(osition)19 b(4.4)j Fg(If)15 b Fs(\025)432 8 y Fr(n)468 1 y Fu(=)e Fs(K)s(\015)585 -16 y Fr(n)606 -27 y Fl(\013)639 1 y Fu(+)d Fs(o)p Fu(\()p Fs(\015)751 -16 y Fr(n)772 -27 y Fl(\013)795 1 y Fu(\))p Fg(,)k Fu(1)p Fs(=)p Fu(2)e Fn(\024)h Fs(\013)g(<)g Fu(1)i Fg(then,)g(for)f(an)o(y)g Fs(r)g(>)f Fu(0)h Fg(and)h(an)o(y)g Fs(P)k Fn(2)13 b Fj(N)p Fg(,)g(there)-12 57 y(exist)j Fs(C)s Fu(\()p Fs(r)q Fu(\))11 b Fs(>)i Fu(0)i Fg(and)g Fs(C)s Fu(\()p Fs(r)o(;)8 b(P)e Fu(\))14 b Fg(suc)o(h)h(that:)k(for)c (an)o(y)g Fs(f)20 b Fg(and)15 b Fs(g)i Fg(suc)o(h)e(that)g Fs(f)5 b(h)13 b Fn(2)g Fs(B)18 b Fg(and)d Fs(g)f Fn(2)f Fs(L)1640 41 y Fq(1)1677 57 y Fg(,)501 161 y Fn(j)p Fs(\026)p Fu(\()p Fs(f)5 b(g)12 b Fn(\016)d Fs(T)685 142 y Fr(n)709 161 y Fu(\))h Fn(\000)g Fs(\026)p Fu(\()p Fs(f)5 b Fu(\))p Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fn(j)11 b(\024)i Fs(C)s Fu(\()p Fs(r)q Fu(\))1141 130 y(1)p 1130 151 47 2 v 1130 192 a Fs(n)1157 179 y Fr(r)1180 161 y Fn(k)p Fs(f)5 b(h)p Fn(k)15 b(k)p Fs(g)r Fn(k)1364 168 y Fq(1)417 285 y Fg(and)g Fn(kL)559 266 y Fr(n)583 285 y Fs(f)g Fn(\000)c Fs(hm)p Fu(\()p Fs(f)5 b Fu(\))p Fn(k)818 292 y Fr(P)859 285 y Fn(\024)13 b Fs(C)s Fu(\()p Fs(r)o(;)8 b(P)e Fu(\))1070 254 y(1)p 1060 274 V 1060 316 a Fs(n)1087 303 y Fr(r)1109 285 y Fn(k)p Fs(f)f Fn(k)15 b Fg(for)g(an)o(y)f Fs(f)k Fn(2)13 b Fs(B)r(:)-12 388 y Fo(Pro)q(of)5 b Fu(:)37 b(Let)13 b(us)g(assume)f(that)g Fs(\025)p Fu(\()p Fs(I)620 395 y Fr(n)642 388 y Fu(\))h(=)g Fs(K)s(\015)790 371 y Fr(n)811 359 y Fl(\013)839 388 y Fu(+)5 b Fs(o)p Fu(\()p Fs(\015)946 371 y Fr(n)967 359 y Fl(\013)990 388 y Fu(\),)12 b Fs(\013)h(<)g Fu(1.)19 b Fs(\032)1202 395 y Fr(n)1237 388 y Fu(satis\014es)13 b(for)f(some)g(p)q(ositiv)o(e)h(constan)o(t)-12 444 y Fs(C)s Fu(:)578 510 y Fs(\032)602 517 y Fr(n)637 510 y Fu(=)751 479 y(1)p 690 500 143 2 v 690 541 a(2)p Fs(K)g Fu(+)e(1)848 510 y Fn(\000)952 479 y Fs(C)p 899 500 V 899 543 a(n)926 530 y Fp(2\(1)p Fq(\000)p Fr(\013)p Fp(\))1056 510 y Fu(+)g Fs(o)p Fu(\()1206 479 y(1)p 1147 500 V 1147 543 a Fs(n)1174 530 y Fp(2\(1)p Fq(\000)p Fr(\013)p Fp(\))1293 510 y Fu(\))p Fs(:)-12 599 y Fu(So,)k(it)g(exists) h Fs(n)262 606 y Fp(0)297 599 y Fu(suc)o(h)g(that)e(for)h(some)g(p)q (ositiv)o(e)h(constan)o(t)e Fs(C)k Fu(and)d Fs(n)e Fn(\025)g Fs(n)1287 606 y Fp(0)1307 599 y Fu(,)710 702 y Fs(\032)734 709 y Fr(n)770 702 y Fn(\024)883 671 y Fu(1)p 823 691 V 823 733 a(2)p Fs(K)g Fu(+)d(1)981 702 y Fn(\000)1085 671 y Fs(C)p 1031 691 V 1031 735 a(n)1058 722 y Fp(2\(1)p Fq(\000)p Fr(\013)p Fp(\))1179 702 y Fs(:)-12 803 y Fu(So,)15 b(using)h(\(4.2\),)d(if)j Fs(n)d Fn(\025)g Fs(n)461 810 y Fp(0)481 803 y Fu(,)514 911 y(sup)506 953 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])598 911 y Fn(L)p Fo(1)p Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fu(1)d Fn(\000)860 881 y Fu(\(2)p Fs(K)j Fu(+)d(1\))p Fs(C)p 860 901 214 2 v 896 945 a(n)923 931 y Fp(2\(1)p Fq(\000)p Fr(\013)p Fp(\))1092 911 y Fu(:=)j(1)c Fn(\000)1297 881 y Fs(c)p 1236 901 143 2 v 1236 945 a(n)1263 931 y Fp(2\(1)p Fq(\000)p Fr(\013)p Fp(\))1383 911 y Fs(:)-12 1037 y Fu(Let)19 b Fs(\014)i Fu(=)e(2\(1)11 b Fn(\000)i Fs(\013)p Fu(\))19 b Fn(\024)g Fu(1)f(if)h Fs(\013)g Fn(\025)610 1019 y Fp(1)p 610 1026 18 2 v 610 1053 a(2)633 1037 y Fu(,)g(whic)o(h)h(w)o(e)e(will)j (assume.)30 b(This)19 b(means)g(that)f Fn(L)h Fu(v)o(eri\014es)h(\()p Fo(S-Exp2)p Fu(\).)-12 1094 y(Moreo)o(v)o(er,)f(for)g Fs(v)r Fu(\()p Fs(x)p Fu(\))g(=)i Fs(n)468 1077 y Fq(\000)p Fr(\013)520 1094 y Fu(,)f Fs(\013)h(>)f Fu(0)g(for)f Fs(x)h Fn(2)g Fs(I)891 1101 y Fr(n)914 1094 y Fu(,)h(if)f Fn(L)1025 1101 y Fr(v)1065 1094 y Fu(is)h(the)e(transfer)g(op)q(erator) g(asso)q(ciated)h(to)f(this)-12 1150 y(c)o(hange)14 b(of)g(p)q(oten)o (tial,)h(the)f(argumen)o(ts)f(of)h(lemma)h(4.3)e(pro)o(v)o(e)g(that)h Fn(L)1208 1157 y Fr(v)1243 1150 y Fu(satis\014es)g(\()p Fo(H)p Fu(\);)f(this)i(implies)h(that)d(the)-12 1206 y(in)o(v)m(arian)o(t)i(densit)o(y)h Fs(h)f Fu(of)f Fn(L)h Fu(go)q(es)f(to)h(zero)f(at)g(in\014nit)o(y)i(\(indeed,)g(w)o(e)f(ha)o (v)o(e)f(that)1382 1189 y Fr(h)p 1382 1196 21 2 v 1383 1222 a(v)1422 1206 y Fu(is)h(b)q(ounded\).)21 b(So)15 b(example)-12 1263 y(4.2,)f(sho)o(ws)h(the)g(follo)o(wing)h(estimate.) -12 1360 y Fo(Lemma)i(4.5)k Fg(It)15 b(exists)h Fs(n)465 1367 y Fp(1)500 1360 y Fg(suc)o(h)f(that)g(for)g Fs(N)i Fn(\025)c Fs(n)900 1367 y Fp(1)920 1360 y Fg(,)h(it)i(exists)f Fs(k)1142 1367 y Fp(0)1177 1360 y Fg(suc)o(h)g(that)g(for)f Fs(k)g Fn(\025)f Fs(k)1557 1367 y Fp(0)1577 1360 y Fg(,)184 1486 y Fs(\016)206 1467 y Fq(0)204 1497 y Fr(j;k)263 1486 y Fu(=)80 b(sup)311 1525 y Fr(N)t Fq(\024)p Fr(n)p Fq(\024)p Fr(N)t Fp(+)p Fr(k)q(j)520 1486 y Fn(L)551 1467 y Fr(k)573 1486 y Fo(1)599 1493 y Fe(n)637 1486 y Fn(\024)685 1429 y Fr(k)q Fq(\000)p Fp(1)688 1443 y Fi(Y)688 1540 y Fr(i)p Fp(=0)756 1486 y Fu(1)10 b Fn(\000)967 1455 y Fs(c)p 840 1475 274 2 v 840 1517 a Fu(\()p Fs(N)k Fu(+)c Fs(k)q(j)j Fu(+)d Fs(i)p Fu(\))1090 1504 y Fr(\014)1131 1486 y Fn(\024)1179 1429 y Fr(k)q Fq(\000)p Fp(1)1182 1443 y Fi(Y)1183 1540 y Fr(i)p Fp(=0)1251 1422 y Fi(\022)1284 1486 y Fu(1)g Fn(\000)1493 1455 y Fu(1)p 1368 1475 V 1368 1517 a(\()p Fs(N)k Fu(+)c Fs(k)q(j)j Fu(+)d Fs(i)p Fu(\))1618 1504 y Fr(\014)1646 1422 y Fi(\023)1680 1433 y Fr(c)1705 1486 y Fs(:)-12 1611 y Fu(So)19 b Fn(L)g Fu(satis\014es)g(\()p Fo(S-Exp1)p Fu(\).)30 b(Moreo)o(v)o(er,)18 b(\(3.6\))f(and)i(\(3.7\))e(giv)o(e)i(for)f Fs(f)24 b Fn(2)18 b Fs(L)p Fu(,)h Fs(q)r Fu(\()p Fs(j)s Fu(\))f(=)g Fs(j)1551 1595 y Fr(u)1573 1611 y Fu(,)h(0)g Fs(<)f(u)h(<)g Fu(1)f(and)-12 1668 y Fs(n)13 b Fu(=)g Fs(k)q Fu(\()p Fs(j)f Fu(+)f Fs(q)r Fu(\()p Fs(j)s Fu(\)\))d(+)j Fs(r)q Fu(,)j Fs(j)h Fu(=)e Fs(O)q Fu(\()538 1650 y Fr(n)p 538 1657 22 2 v 539 1683 a(k)564 1668 y Fu(\),)h(if)i Fs(\014)f(<)e Fu(1,)276 1763 y Fn(kL)330 1745 y Fr(n)354 1763 y Fs(f)i Fn(\000)10 b Fs(hm)p Fu(\()p Fs(f)5 b Fu(\))p Fn(k)588 1770 y Fr(P)630 1763 y Fn(\024)12 b Fu(Ct\()p Fs(P)q(;)c(N)d Fu(\))14 b([exp\()p Fn(\000)p Fs(c)p Fu(\()p Fs(N)g Fu(+)c Fs(k)q(j)s Fu(\))1205 1745 y Fp(1)p Fq(\000)p Fr(\014)1273 1763 y Fu(\))g(+)1407 1720 y Fi(X)1346 1821 y Fr(n)p Fq(\025)p Fr(N)t Fp(+)p Fr(k)q(q)q Fp(\()p Fr(j)r Fp(\))1541 1763 y Fs(\015)1568 1745 y Fr(n)1589 1733 y Fl(\013)1613 1763 y Fu(])-12 1897 y(and)16 b(for)e Fs(f)20 b Fu(suc)o(h)c(that)e Fs(f)5 b(h)13 b Fn(2)g Fs(B)18 b Fu(and)e Fs(g)e Fn(2)f Fs(L)749 1881 y Fq(1)786 1897 y Fu(,)237 1986 y Fn(j)p Fs(\026)p Fu(\()p Fs(f)5 b(g)11 b Fn(\016)f Fs(T)421 1967 y Fr(n)445 1986 y Fu(\))f Fn(\000)i Fs(\026)p Fu(\()p Fs(f)5 b Fu(\))p Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fn(j)11 b(\024)i Fu(Ct)o(\()p Fs(N)5 b Fu(\))14 b([exp)q(\()p Fn(\000)p Fs(c)p Fu(\()p Fs(N)f Fu(+)e Fs(k)q(j)s Fu(\))1244 1967 y Fp(1)p Fq(\000)p Fr(\014)1312 1986 y Fu(\))e(+)1446 1943 y Fi(X)1385 2044 y Fr(n)p Fq(\025)p Fr(N)t Fp(+)p Fr(k)q(q)q Fp(\()p Fr(j)r Fp(\))1580 1986 y Fs(\015)1607 1967 y Fr(n)1628 1956 y Fl(\013)1652 1986 y Fu(])155 b(\(4.3\))-12 2128 y(Since)102 2094 y Fi(P)150 2141 y Fr(p)p Fq(\025)p Fr(n)227 2128 y Fs(\015)254 2111 y Fr(p)272 2100 y Fl(\013)307 2128 y Fu(=)13 b Fs(O)q Fu(\()p Fs(n)436 2111 y Fp(1)p Fq(\000)p Fr(\013)506 2128 y Fs(\015)533 2111 y Fr(n)554 2100 y Fl(\013)578 2128 y Fu(\),)e(w)o(e)f(ha)o(v)o(e)h (the)f(announced)i(estimate)f(for)f Fs(\014)15 b(<)e Fu(1.)18 b(The)11 b(same)f(computation)-12 2184 y(leads)16 b(also)f(to)g(the)g(result)h(for)e Fs(\014)h Fu(=)e(1.)1225 b Fn(4)-12 2282 y Fo(Remark)17 b(4.1)22 b Fu(When)16 b(the)f(con)o(v)o(ergence)h(to)f(zero)g(of)g Fs(\025)p Fu(\()p Fs(I)1008 2289 y Fr(n)1030 2282 y Fu(\))g(is)h(slo)o(w)o(er)f (than)g Fs(\015)1379 2265 y Fr(n)1400 2253 y Fl(\013)1424 2282 y Fu(,)1457 2264 y Fp(1)p 1457 2271 18 2 v 1457 2297 a(2)1492 2282 y Fn(\024)e Fs(\013)g(<)g Fu(1,)i(for)g(example)-12 2338 y(if)h(it)f(is)h(p)q(olynomial,)g(w)o(e)f(ha)o(v)o(e)621 2441 y(sup)613 2482 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])705 2441 y Fn(L)p Fo(1)p Fu(\()p Fs(x)p Fu(\))d Fn(\024)h Fu(\(1)d Fn(\000)993 2410 y Fs(C)p 985 2431 51 2 v 985 2472 a(n)1012 2459 y Fr(\014)1041 2441 y Fu(\))15 b(with)g Fs(\014)g(>)e Fu(1)-12 2559 y(pro)o(vided)23 b Fs(n)f Fu(is)g(large)g(enough,)i(this)e(estimate)g(is)g(not)f(su\016cien)o(t)i (to)e(use)h(the)g(tec)o(hniques)h(of)f(section)g(3.)-12 2615 y(Ho)o(w)o(ev)o(er,)10 b(it)h(is)g(ma)o(yb)q(e)g(p)q(ossible)h(to) e(estimate)g(the)h(deca)o(y)f(of)g(correlations)h(b)o(y)g(impro)o(ving) g(the)f(ab)q(o)o(v)o(e)g(estimate)-12 2672 y(for)15 b(iterates)g(of)g Fn(L)p Fu(.)928 2796 y(27)p eop %%Page: 28 28 28 27 bop -12 1 a Fo(Birth-and-death)23 b(pro)q(cess.)j Fu(Using)18 b(the)g(same)f(metho)q(d,)h(w)o(e)f(obtain)h(the)g(follo)o (wing)g(results)g(for)f(birth-)-12 57 y(and-death)12 b(pro)q(cess)f(\(see)g([Se])g(for)f(a)h(review)g(on)g(non)h(negativ)o (e)f(matrices\).)18 b(W)l(e)11 b(consider)h(a)f(sto)q(c)o(hastic)g (matrix)-12 114 y Fs(P)20 b Fu(=)15 b(\()p Fs(p)128 121 y Fr(i;j)167 114 y Fu(\))185 121 y Fr(i;j)r Fq(2)p Fm(N)288 114 y Fu(\()306 80 y Fi(P)354 127 y Fr(i)376 114 y Fs(p)399 121 y Fr(i;j)452 114 y Fu(=)g(1)g(for)g(all)i Fs(j)s Fu(\).)k(W)l(e)16 b(assume)g(that)f(there)h(is)g(an)g(in)o(teger)g Fs(K)j Fu(suc)o(h)d(that)f Fs(p)1745 121 y Fr(i;j)1799 114 y Fu(=)f(0)i(if)-12 170 y Fn(j)p Fs(i)11 b Fn(\000)h Fs(j)s Fn(j)j Fs(>)h(K)k Fu(\(this)e(is)f(wh)o(y)g(w)o(e)g(call)i (these)e(pro)q(cess)h(birth-and-death)g(pro)q(cess\))f(and)h(w)o(e)f (assume)g(that)f(the)-12 227 y(matrix)i(is)g(ap)q(erio)q(dic.)29 b(In)19 b(this)f(situation,)h Fo(1)e Fu(is)i(a)e(\014xed)i(p)q(oin)o(t) f(for)f(the)h(Mark)o(o)o(v)e(op)q(erator)h Fs(P)24 b Fu(and)18 b(w)o(e)g(are)-12 283 y(lo)q(oking)e(for)e(an)g(stationary)g (measure,)g(i.e.)20 b(a)15 b(\014xed)g(p)q(oin)o(t)g(for)f(the)h(dual) 1273 267 y Fr(t)1288 283 y Fs(P)21 b Fu(of)14 b Fs(P)6 b Fu(.)20 b(Let)15 b(us)g(denote)1742 267 y Fr(t)1757 283 y Fs(P)21 b Fu(b)o(y)14 b Fn(L)p Fu(.)-12 340 y(The)i(measure)f (de\014ned)i(b)o(y)e Fs(m)519 347 y Fp(1)539 340 y Fu([)p Fs(i)p Fu(])d(=)h(1)i(and)g Fs(m)807 347 y Fp(1)827 340 y Fu([)p Fs(i)856 347 y Fp(1)875 340 y Fs(;)8 b Fn(\001)g(\001)g(\001)d Fs(;)j(i)993 347 y Fr(n)1015 340 y Fu(])13 b(=)g Fs(p)1112 347 y Fr(i)1124 352 y Fk(1)1141 347 y Fr(i)1153 352 y Fk(2)1180 340 y Fn(\001)8 b(\001)g(\001)e Fs(p)1264 347 y Fr(i)1276 352 y Fl(n)p Ff(\000)p Fk(1)1336 347 y Fr(i)1348 351 y Fl(n)1387 340 y Fu(is)15 b(a)h(conformal)f(measure)g(for)-12 396 y Fn(L)g Fu(but)f(it)h(is)g(not)f(\014nite.)20 b(T)l(o)14 b(an)o(y)g(function)i Fs(v)g Fu(constan)o(t)d(on)h(the)h(1-cylinders)h (and)e(suc)o(h)h(that)1623 362 y Fi(P)1671 410 y Fr(n)p Fq(2)p Fm(N)1749 396 y Fs(v)1771 403 y Fr(n)1807 396 y Fs(<)e Fn(1)p Fu(,)-12 453 y(w)o(e)j(asso)q(ciate)f(a)g(transfer)g (op)q(erator)g Fn(L)669 460 y Fr(v)705 453 y Fu(b)o(y)h(c)o(hange)f(of) h(p)q(oten)o(tial)g(\(see)g(page)f(6\).)20 b(The)c(measure)g Fs(m)1748 460 y Fr(v)1781 453 y Fu(=)e Fs(v)r(m)1894 460 y Fp(1)-12 509 y Fu(is)i(\014nite)g(and)f(conformal)g(for)g Fn(L)550 516 y Fr(v)571 509 y Fu(.)20 b(W)l(e)15 b(mak)o(e)g(the)g (follo)o(wing)h(assumptions)f(of)g(the)g(matrix.)-12 566 y Fn(\017)g Fu(F)l(or)g(an)o(y)f Fs(n)f Fn(2)g Fj(N)p Fu(,)h(let)i Fs(P)6 b(u)p Fu(\()p Fs(n)p Fu(\))12 b(=)587 531 y Fi(P)635 579 y Fr(i>n)706 566 y Fs(p)729 573 y Fr(i;n)789 566 y Fu(and)k Fs(P)6 b(d)p Fu(\()p Fs(n)p Fu(\))12 b(=)1060 531 y Fi(P)1108 579 y Fr(i)p Fq(\024)p Fr(n)1178 566 y Fs(p)1201 573 y Fr(i;n)1247 566 y Fu(.)-12 622 y Fn(\017)k Fu(W)l(e)g(assume)g(that)f(for)h Fs(n)g Fu(large)g(enough,)g Fs(P)6 b(u)p Fu(\()p Fs(n)p Fu(\))15 b(=)f Fs(a)p Fu(\(1)c Fn(\000)h Fs(w)1101 629 y Fr(n)1124 622 y Fu(\))16 b(and)g Fs(P)6 b(d)p Fu(\()p Fs(n)p Fu(\))14 b(=)g Fs(b)p Fu(\(1)c(+)h Fs(u)1575 629 y Fr(n)1599 622 y Fu(\))k(where)i Fs(a)f Fu(and)g Fs(b)-12 678 y Fu(are)f(p)q(ositiv)o (e)h(n)o(um)o(b)q(ers)g(and)f(\()p Fs(u)548 685 y Fr(n)571 678 y Fu(\))g(and)g(\()p Fs(w)743 685 y Fr(n)766 678 y Fu(\))g(are)g(p)q(ositiv)o(e)h(sequences)g(that)f(go)f(to)h(zero.)-12 735 y(W)l(e)g(ha)o(v)o(e)g(the)h(follo)o(wing)f(results:)-12 791 y Fn(\017)20 b Fu(If)h Fs(a)13 b Fu(+)h Fs(b)20 b(<)i Fu(1)d(then)i(there)f(exists)h(a)f(c)o(hange)g(of)g(p)q(oten)o(tial)g Fs(v)i Fu(suc)o(h)f(that)e Fn(L)1405 798 y Fr(v)1446 791 y Fu(satis\014es)i(\()p Fo(S-Exp2)p Fu(\).)34 b(So)-12 848 y(the)17 b(matrix)f(is)i(p)q(ositiv)o(e)g(recurren)o(t)e(and)h (geometrically)h(ergo)q(dic)f(in)h(the)f(sense)g(of)g(D.)f(V)l (ere-Jones)i(\([V-J1)o(],)-12 904 y([V-J2]\):)h(w)o(e)c(ha)o(v)o(e)g (the)g(follo)o(wing)h(exp)q(onen)o(tial)h(con)o(v)o(ergence)721 1003 y(sup)712 1042 y Fr(i;j)r Fq(2)p Fm(N)805 1003 y Fn(j)p Fs(p)841 979 y Fp(\()p Fr(n)p Fp(\))841 1017 y Fr(i;j)902 1003 y Fn(\000)10 b Fs(\027)969 1010 y Fr(j)988 1003 y Fn(j)i(\024)h Fu(Ct)i Fs(\015)1154 984 y Fr(n)1177 1003 y Fs(;)-12 1124 y Fu(where)h Fs(\027)i Fu(is)e(the)f(stationary)f (measure)i(and)f(0)d Fs(<)h(\015)i(<)e Fu(1.)-12 1180 y Fn(\017)18 b Fu(If)g Fs(a)11 b Fu(+)h Fs(b)17 b Fu(=)g(1,)h Fs(a)e(<)i(b)p Fu(,)f(for)g Fs(n)h Fu(large)g(enough)g Fs(w)868 1187 y Fr(n)908 1180 y Fs(<)f(u)986 1187 y Fr(n)1027 1180 y Fu(and)1118 1146 y Fi(P)1166 1194 y Fr(n)1197 1180 y Fs(w)1230 1187 y Fr(n)1270 1180 y Fu(=)g Fn(1)h Fu(then)g(there)g(exists)g(a)f(c)o(hange)-12 1237 y(of)g(p)q(oten)o (tial)h Fs(v)h Fu(suc)o(h)f(that)e Fn(L)513 1244 y Fr(v)551 1237 y Fu(satis\014es)i(\()p Fo(S-Exp2)p Fu(\).)26 b(So)17 b(that)g(the)g(matrix)g(is)h(p)q(ositiv)o(e)g(recurren)o(t)f(and)h(w)o (e)-12 1293 y(ha)o(v)o(e)e(the)h(follo)o(wing)g(estimate:)22 b(for)16 b(an)o(y)g Fs(N)j Fn(2)c Fj(N)g Fu(and)h(an)o(y)g Fs(r)g Fn(2)f Fj(N)g Fu(there)h(exists)h Fs(C)s Fu(\()p 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y(p)q(ositiv)o(e)h(recurren)o(t)e (and)g(w)o(e)h(ha)o(v)o(e)e(the)i(follo)o(wing)g(estimate:)k(for)15 b(an)o(y)g Fs(N)i Fn(2)d Fj(N)g Fu(and)h(an)o(y)g Fs(r)f Fn(2)f Fj(N)i Fu(there)g(exists)-12 1659 y Fs(C)s Fu(\()p Fs(N)r(;)8 b(r)q Fu(\))j Fs(>)i Fu(0)i(suc)o(h)g(that)g(if)g Fs(i;)8 b(j)14 b Fn(\024)f Fs(N)20 b Fu(then)713 1757 y Fn(j)p Fs(p)749 1734 y Fp(\()p Fr(n)p Fp(\))749 1771 y Fr(i;j)810 1757 y Fn(\000)10 b Fs(\027)877 1764 y Fr(j)896 1757 y Fn(j)i(\024)h Fu(C\(N,r\))o Fs(n)1129 1739 y Fq(\000)p Fr(r)1176 1757 y Fs(:)-12 1931 y Fh(4.2)56 b(Non)18 b(uniformly)f (expanding)h(maps)g(of)h(the)f(in)n(terv)m(al.)-12 2016 y Fu(W)l(e)e(conclude)h(this)f(article)h(with)f(the)g(estimation)g(of)f (the)h(deca)o(y)f(of)h(correlations)f(for)g(Gaspard-W)l(ang)g(t)o(yp)q (e)-12 2073 y(applications.)39 b(Let)22 b(\()p Fs(I)402 2080 y Fr(n)425 2073 y Fu(\))443 2080 y Fr(n)p Fq(2)p Fm(N)535 2073 y Fu(b)q(e)g(a)e(partition)i(\(mo)q(d)f(0\))f(of)h Fs(I)k Fu(with)c Fs(\025)p Fu(\()p Fs(I)1306 2080 y Fr(n)1329 2073 y Fu(\))h(=)1474 2055 y Fr(K)p 1432 2062 117 2 v 1432 2089 a Fp(\()p Fr(n)p Fp(+1\))1526 2080 y Fl(\013)1553 2073 y Fu(,)g Fs(K)k(>)c Fu(0,)g Fs(\013)h(>)g Fu(1.)-12 2129 y(Let)f(us)g(consider)h(the)e(follo)o(wing)i(piecewise)h(a\016ne)d (application.)41 b Fs(T)28 b Fu(is)22 b(increasing,)i(a\016ne)e(on)g (eac)o(h)f Fs(I)1877 2136 y Fr(n)1901 2129 y Fu(,)-12 2186 y Fs(T)6 b(I)41 2193 y Fr(n)81 2186 y Fu(=)16 b Fs(I)152 2193 y Fr(n)p Fq(\000)p Fp(1)238 2186 y Fu(for)h Fs(n)f Fn(\025)g Fu(1)h(and)h Fs(T)6 b(I)588 2193 y Fp(0)624 2186 y Fu(=)16 b Fs(I)t Fu(.)26 b(This)18 b(is)f(a)g(linearization)j (of)d(smo)q(oth)f(non)i(uniformly)g(expanding)-12 2242 y(maps)i(of)g(the)g(in)o(terv)m(al)h(considered)g(for)f(example)h(b)o (y)f(M.)f(Thaler)h(\([T]\),)g(C.)f(Liv)o(erani,)j(B.)e(Saussol)h(et)f (S.)-12 2299 y(V)l(aien)o(ti)i(\([L,)14 b(S,)h(V2]\))20 b(and)g(in)o(tro)q(duced)i(b)o(y)e(P)l(.)h(Gaspard)e(et)i(X.-J.)f(W)l (ang)g(\([G,)14 b(W)o(],)21 b([W)l(an]\))f(in)h(order)f(to)-12 2355 y(mo)q(del)15 b(in)o(termitency)g(phenomenons.)20 b(It)14 b(is)g(w)o(ell)h(kno)o(wn)f(that)f Fs(T)19 b Fu(admits)14 b(a)g(unique)h(absolutely)g(con)o(tin)o(uous)-12 2412 y(in)o(v)m(arian)o(t)j(measure)g(whose)f(densit)o(y)h Fs(h)g Fu(v)o(eri\014es)g Fs(cn)e Fn(\024)h Fs(h)p Fu(\()p Fs(x)p Fu(\))f Fn(\024)g Fs(C)s(n)i Fu(if)g Fs(x)e Fn(2)h Fs(I)1358 2419 y Fr(n)1398 2412 y Fu(\([La,Si,V]\).)26 b(In)18 b(particular,)-12 2468 y Fs(\026)d Fu(=)f Fs(h\025)i Fu(is)h(a)e(\014nite)i(measure)f(if)h(and)f(only)h(if)g Fs(\013)d(>)g Fu(2.)23 b(Moreo)o(v)o(er,)14 b(this)j(measure)f(is)g (mixing.)24 b(Let)16 b(us)h(notice)-12 2524 y(that)e(the)g(dynamic)h (is)g(without)f(big)h(branc)o(hes)f(at)g(in\014nit)o(y)h(and)g(ap)q (erio)q(dic.)-12 2581 y(If)g Fs(d)c(>)h Fu(0)i(w)o(e)g(denote)g(b)o(y)g Fs(v)454 2588 y Fr(d)487 2581 y Fu(:)d Fs(I)17 b Fn(!)c Fj(R)643 2564 y Fp(+)684 2581 y Fu(the)i(lo)q(cally)i(constan)o(t)e (function:)695 2672 y Fs(v)717 2679 y Fr(d)737 2672 y Fu(\()p Fs(x)p Fu(\))d(=)h Fs(v)881 2679 y Fr(n)917 2672 y Fu(=)g Fs(n)992 2653 y Fr(d)1028 2672 y Fu(if)i Fs(x)e Fn(2)g Fs(I)1171 2679 y Fr(n)1194 2672 y Fs(;)928 2796 y Fu(28)p eop %%Page: 29 29 29 28 bop -12 1 a Fu(let)16 b Fs(E)h Fu(b)q(e)f(the)f(space)h(of)f (functions)h Fs(f)k Fu(suc)o(h)15 b(that)463 94 y Fs(f)5 b(h)p 463 114 54 2 v 469 156 a(v)491 163 y Fr(d)534 125 y Fn(2)13 b Fs(B)18 b Fu(for)c(an)o(y)h Fs(d)d(>)h Fu(1)i(and)23 b(sup)1005 164 y Fr(d>)p Fp(1)1078 59 y Fi(\015)1078 86 y(\015)1078 113 y(\015)1078 141 y(\015)1108 94 y Fs(f)5 b(h)p 1108 114 V 1114 156 a(v)1136 163 y Fr(d)1167 59 y Fi(\015)1167 86 y(\015)1167 113 y(\015)1167 141 y(\015)1205 125 y Fu(:=)12 b Fn(j)-8 b(j)g(j)p Fs(f)5 b Fn(j)-8 b(j)g(j)12 b Fs(<)h Fn(1)-12 251 y Fu(W)l(e)i(are)g(going)h(to)e(pro)o(v)o(e)h (the)g(follo)o(wing)h(result.)-12 357 y Fo(Prop)q(osition)j(4.6)j Fg(F)l(or)15 b(an)o(y)g Fs(")e(>)g Fu(0)p Fg(,)h(there)h(exists)h Fs(C)s Fu(\()p Fs(")p Fu(\))e Fg(suc)o(h)i(that)e(for)h(all)h Fs(f)i Fn(2)13 b Fs(E)k Fg(and)e Fs(g)f Fn(2)f Fs(L)1675 341 y Fq(1)1713 357 y Fg(,)463 457 y Fn(j)p Fs(\026)p Fu(\()p Fs(f)5 b(g)11 b Fn(\016)f Fs(T)647 439 y Fr(n)670 457 y Fu(\))g Fn(\000)h Fs(\026)p Fu(\()p Fs(f)5 b Fu(\))p Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fn(j)11 b(\024)i Fs(C)s Fu(\()p Fs(")p Fu(\))p Fn(j)-8 b(j)g(j)p Fs(f)5 b Fn(j)-8 b(j)g(j)14 b(k)p Fs(g)r Fn(k)1243 464 y Fq(1)1343 427 y Fu(1)p 1284 447 141 2 v 1284 489 a Fs(n)1311 476 y Fr(\013)p Fq(\000)p Fp(2)p Fq(\000)p Fr(")1430 457 y Fs(:)403 b Fg(\(I\))-12 628 y Fu(Let)15 b(us)g(remark)f(that)g(since)h Fs(cn)e Fn(\024)g Fs(h)p Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fs(C)s(n)i Fu(for)f Fs(x)e Fn(2)h Fs(I)994 635 y Fr(n)1018 628 y Fu(,)h(the)h(space)g Fs(B)i Fu(is)e(included)i(in)f Fs(E)h Fu(and)d(for)g Fs(f)k Fn(2)13 b Fs(B)r Fu(,)-12 684 y(w)o(e)i(ha)o(v)o(e)g Fn(j)-8 b(j)g(j)p Fs(f)5 b Fn(j)-8 b(j)g(j)12 b(\024)h Fu(Ct)p Fn(k)p Fs(f)5 b Fn(k)p Fu(.)-12 790 y Fo(Remark)17 b(4.2)22 b Fu(There)c(exists)f(man)o(y)g (results)g(on)g(the)g(deca)o(y)h(of)e(correlations)i(for)e(this)i(map)f (\(linearized)i(or)-12 847 y(not\).)25 b(The)17 b(oldest)h(are)f(from)f (A.)h(Lam)o(b)q(ert,)g(S.)g(Sib)q(oni,)i(S.)e(V)l(aien)o(ti)h (\([La,Si,V]\),)e(M.)g(Mori)h(\([Mo)o(]\))f(and)i(N.)-12 903 y(Cherno)o(v)13 b(\([Ch]\).)18 b(A.M.)13 b(Fisher)h(and)g(A.)f(Lop) q(es)h(\([F,)g(L]\))f(and)h(S.)f(Isola)h(\([I]\))f(get)g(a)g(sp)q(eed)i (of)e(con)o(v)o(ergence)h(in)26 942 y Fp(1)p -7 949 83 2 v -7 977 a Fr(n)14 967 y Fl(\013)p Ff(\000)p Fk(2)97 960 y Fu(for)i(observ)m(ables)i(whic)o(h)f(are)f(\014nite)i(linear)g (com)o(binations)f(of)f(c)o(haracteristic)h(functions)g(of)f (cylinders.)-12 1016 y(Concerning)22 b(the)g(smo)q(oth)f(mo)q(del,)j (using)e(appro)o(ximation)f(tec)o(hniques,)j(C.)d(Liv)o(erani,)j(B.)d (Saussol)h(et)g(S.)-12 1073 y(V)l(aien)o(ti)16 b(\([L,)e(S,)h(V2]\))f (obtain)h(a)f(rate)g(of)g(con)o(v)o(ergence)h(in)1019 1052 y Fp(log)5 b Fr(n)p 1014 1062 V 1014 1090 a(n)1035 1080 y Fl(\013)p Ff(\000)p Fk(2)1116 1073 y Fu(for)14 b(Lipsc)o(hitz)i(functions)g(on)e(the)h(in)o(terv)m(al)g Fs(I)t Fu(,)-12 1129 y(this)h(space)f(is)h(included)h(in)f Fs(E)s Fu(.)j(Using)d(a)f(coupling)h(metho)q(d,)f(L.-S.)g(Y)l(oung)h (\([Y)l(o)o(]\))e(get,)h(on)g(the)g(same)f(space,)-12 1186 y(a)j(rate)g(of)g(con)o(v)o(ergence)h(of)f(order)644 1168 y Fp(1)p 612 1175 V 612 1202 a Fr(n)633 1193 y Fl(\013)p Ff(\000)p Fk(2)700 1186 y Fu(;)h(in)g([H])f(H.)g(Hu)g(pro)o(v)o(es)g (the)h(same)f(result)h(and)f(that)g(this)h(result)g(is)-12 1242 y(optimal)i(for)f(Lipsc)o(hitz)i(functions)f(on)f Fs(I)t Fu(.)32 b(More)18 b(recen)o(tly)l(,)j(M.)e(P)o(ollicott)g(and)h (M.)e(Y)l(uri)j(\([P)o(o,Y)n(]\))e(get)f(an)-12 1298 y(estimation)e(for)e(observ)m(ables)j(in)f(a)e(space)i(con)o(taining)g Fn(f)992 1281 y Fp(1)p 981 1288 40 2 v 981 1314 a Fr(x)1001 1305 y Fl(\015)1025 1298 y Fs(;)23 b Fu(0)12 b Fs(<)h(\015)i(<)1261 1281 y Fp(1)p 1236 1288 68 2 v 1236 1314 a Fr(\013)p Fp(+1)1308 1298 y Fn(g)p Fu(.)-12 1388 y Fo(Pro)q(of)20 b(of)f(of)g(the)h(prop)q(osition)h(4.6)p Fu(:)47 b(The)17 b(transfer)f(op)q(erator)g Fn(L)1236 1395 y Fp(\010)1281 1388 y Fu(asso)q(ciated)h(to)f(the)h(p)q(oten)o(tial)h(\010)d(=)-12 1444 y Fn(\000)8 b Fu(log)g Fs(T)130 1428 y Fq(0)157 1444 y Fu(satis\014es)677 1524 y Fn(L)708 1531 y Fp(\010)736 1524 y Fo(1)762 1531 y Fe(n)800 1524 y Fu(=)847 1460 y Fi(\022)886 1494 y Fs(n)i Fu(+)h(1)p 886 1514 106 2 v 886 1556 a Fs(n)f Fu(+)h(2)996 1460 y Fi(\023)1030 1472 y Fr(\013)1065 1524 y Fu(+)f Fs(\025)p Fu(\()p Fs(I)1175 1531 y Fp(0)1194 1524 y Fu(\))p Fs(:)-12 1632 y Fu(Since)17 b Fs(h)e Fu(is)h(not)f(b)q(ounded,)h Fn(L)504 1639 y Fp(\010)547 1632 y Fu(cannot)f(v)o(eri\014es)h(\()p Fo(K)p Fu(\).)e(This)i(is)g(wh)o(y)e(w)o(e)h(use)h(a)f(cohomologous)f(p)q (oten)o(tial.)-12 1689 y(F)l(or)22 b Fs(d)i(>)g Fu(1,)g(let)e Fn(L)347 1696 y Fr(d)390 1689 y Fu(b)q(e)h(the)f(transfer)g(op)q (erator)f(asso)q(ciated)h(to)g(the)g(c)o(hange)h(of)e(p)q(oten)o(tial)i Fs(v)1718 1696 y Fr(d)1738 1689 y Fu(,)h Fn(L)1806 1696 y Fr(d)1827 1689 y Fs(f)29 b Fu(=)2 1727 y Fp(1)p -7 1734 36 2 v -7 1761 a Fr(v)10 1767 y Fl(d)33 1745 y Fn(L)64 1752 y Fp(\010)92 1745 y Fu(\()p Fs(f)5 b(v)159 1752 y Fr(d)179 1745 y Fu(\).)597 1847 y(sup)589 1888 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])681 1847 y Fn(L)712 1854 y Fr(d)732 1847 y Fo(1)p Fu(\()p Fs(x)p Fu(\))12 b(=)880 1783 y Fi(\022)919 1816 y Fs(n)e Fu(+)h(1)p 919 1836 106 2 v 919 1878 a Fs(n)f Fu(+)h(2)1029 1783 y Fi(\023)1063 1794 y Fr(\013)p Fq(\000)p Fr(d)1143 1847 y Fu(+)1194 1816 y Fs(\025)p Fu(\()p Fs(I)1259 1823 y Fp(0)1278 1816 y Fu(\))p 1194 1836 102 2 v 1221 1878 a Fs(n)1248 1865 y Fr(d)1300 1847 y Fs(:)-12 1966 y Fu(So,)k(if)h Fs(n)f Fu(is)h(large)f(enough,)g Fn(L)505 1973 y Fr(d)526 1966 y Fo(1)552 1973 y Fe(n)589 1966 y Fn(\024)e Fu(1,)i(whic)o(h)h(means)f (that)g Fn(L)1087 1973 y Fr(d)1122 1966 y Fu(satis\014es)h(\()p Fo(H)p Fu(\))e(so)h(it)h(satis\014es)f(\()p Fo(K)p Fu(\))g(pro)o(vided) -12 2022 y(that)i(the)g(conformal)h(measure)f Fs(m)602 2029 y Fr(d)638 2022 y Fu(=)g Fs(v)712 2029 y Fr(d)732 2022 y Fs(\025)g Fu(remains)h(\014nite:)25 b(the)18 b(p)q(oten)o(tial)g (is)g(constan)o(t)e(on)i(eac)o(h)f Fs(I)1784 2029 y Fr(n)1825 2022 y Fu(so)g(it)-12 2079 y(is)i(uniformly)g(lo)q(cally)g(Lipsc)o (hitz)h(on)e(the)g(partition)h(\()p Fs(I)968 2086 y Fr(n)991 2079 y Fu(\))1009 2086 y Fr(n)p Fq(2)p Fm(N)1079 2079 y Fu(.)29 b(W)l(e)18 b(ha)o(v)o(e)f Fs(m)1349 2086 y Fr(d)1369 2079 y Fu(\()p Fs(I)t Fu(\))f(=)1498 2045 y Fi(P)1554 2079 y Fs(v)1576 2086 y Fr(n)1599 2079 y Fs(\025)p Fu(\()p Fs(I)1664 2086 y Fr(n)1687 2079 y Fu(\),)i(so)f Fs(m)1834 2086 y Fr(d)1854 2079 y Fu(\()p Fs(I)t Fu(\))-12 2135 y(is)i(\014nite)h(if)f(and)g(only)g(if)g Fs(\013)13 b Fn(\000)g Fs(d)18 b(>)g Fu(1,)h(since)h Fs(d)e(>)h Fu(1,)g(w)o(e)f(reco)o(v)o(er)g(the)h(condition)h Fs(\013)e(>)h Fu(2)f(whic)o(h)i(guaran)o(t)o(y)-12 2192 y(the)d(existence)h(of)f(an)f (absolutely)i(con)o(tin)o(uous)f(in)o(v)m(arian)o(t)h(measure)f Fs(\026)p Fu(.)25 b(In)17 b(what)g(follo)o(ws,)g(w)o(e)f(assume)h(that) -12 2248 y Fs(\013)e(>)g Fu(2,)g Fs(\013)c Fn(\000)h Fs(d)i(>)g Fu(1)i(and)h(the)f(measure)g Fs(m)733 2255 y Fr(d)770 2248 y Fu(is)g(normalized)i(\(i.e.)23 b Fs(m)1189 2255 y Fr(d)1209 2248 y Fu(\()p Fs(I)t Fu(\))14 b(=)g(1\).)23 b(Let)16 b Fs(h)1517 2255 y Fr(d)1554 2248 y Fu(b)q(e)h(the)f (normalized)-12 2305 y(\014xed)g(p)q(oin)o(t)g(of)e Fn(L)299 2312 y Fr(d)320 2305 y Fu(,)h(w)o(e)g(ha)o(v)o(e)g Fs(\026)e Fu(=)g Fs(h)634 2312 y Fr(d)654 2305 y Fs(m)694 2312 y Fr(d)727 2305 y Fu(=)g Fs(h\025)p Fu(.)-12 2361 y(Let)j(us)f(pro)o(v) o(e)g(that)f Fn(L)380 2368 y Fr(d)416 2361 y Fu(v)o(eri\014es)i(\()p Fo(S-Exp1)p Fu(\).)j(Let)d(us)f(\014x)h(0)c Fs(<)h(\021)h(<)f(\013)d Fn(\000)h Fs(d)p Fu(,)j(let)i Fs(\014)e Fu(=)f Fs(\013)e Fn(\000)f Fs(d)g Fn(\000)g Fs(\021)r Fu(.)-12 2467 y Fo(Lemma)18 b(4.7)k Fg(It)15 b(exists)h Fs(n)465 2474 y Fp(1)497 2467 y Fu(=)d Fs(n)572 2474 y Fp(1)592 2467 y Fu(\()p Fs(d;)8 b(\021)r Fu(\))p Fg(,)13 b(it)i(exists)h Fs(k)919 2474 y Fp(0)953 2467 y Fg(suc)o(h)g(that)e(for)h Fs(N)i Fn(\025)c Fs(n)1353 2474 y Fp(1)1388 2467 y Fg(and)i Fs(k)f Fn(\025)f Fs(k)1586 2474 y Fp(0)1605 2467 y Fg(,)362 2612 y Fs(\016)384 2593 y Fq(0)382 2623 y Fr(k)q(;j)442 2612 y Fu(=)80 b(sup)490 2651 y Fr(N)t Fq(\024)p Fr(n)p Fq(\024)p Fr(N)t Fp(+)p Fr(k)q(j)708 2612 y Fu(sup)700 2653 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])792 2612 y Fn(L)823 2593 y Fr(k)823 2623 y(d)845 2612 y Fo(1)p Fu(\()p Fs(x)p Fu(\))12 b Fn(\024)993 2555 y Fr(k)q Fq(\000)p Fp(1)996 2568 y Fi(Y)995 2667 y Fr(`)p Fp(=0)1065 2548 y Fi(\022)1098 2612 y Fu(1)e Fn(\000)1318 2581 y Fu(1)p 1181 2601 297 2 v 1181 2643 a Fs(N)15 b Fu(+)10 b Fs(k)q(j)j Fu(+)d Fs(`)g Fu(+)h(2)1483 2548 y Fi(\023)1516 2559 y Fr(\014)928 2796 y Fu(29)p eop %%Page: 30 30 30 29 bop -12 1 a Fo(Pro)q(of)5 b Fu(:)57 b Fn(L)221 8 y Fr(d)261 1 y Fu(satis\014es)19 b(\()p Fo(K)p Fu(\),)f(so)h(there)h (exists)f Fs(M)25 b(>)20 b Fu(0)f(suc)o(h)g(that)g Fn(kL)1279 -16 y Fr(n)1279 15 y(d)1303 1 y Fo(1)p Fn(k)1352 8 y Fq(1)1408 1 y Fn(\024)h Fs(M)k Fu(for)19 b(all)h Fs(n)g Fn(2)g Fj(N)p Fu(.)31 b(Let)-12 70 y(us)21 b(\014x)g Fs(n)149 77 y Fp(0)189 70 y Fu(suc)o(h)g(that)f(if)h Fs(n)h Fn(\025)g Fs(n)581 77 y Fp(0)621 70 y Fu(then)f(\(1)14 b Fn(\000)863 52 y Fp(1)p 839 59 67 2 v 839 86 a Fr(n)p Fp(+2)910 70 y Fu(\))928 54 y Fr(\021)962 70 y Fu(+)1016 48 y Fp(2)1034 36 y Fl(\014)1055 48 y Fr(\025)p Fp(\()p Fr(I)1105 53 y Fk(0)1122 48 y Fp(\))p Fr(M)p 1016 60 157 2 v 1075 89 a(n)1096 79 y Fl(d)1199 70 y Fn(\024)22 b Fu(1.)36 b(This)21 b(implies)i(that)d(if)h Fs(n)h Fn(\025)g Fs(n)1881 77 y Fp(0)1901 70 y Fu(,)-12 133 y Fn(L)19 140 y Fr(d)40 133 y Fo(1)66 140 y Fe(n)103 133 y Fn(\024)13 b Fu(\(1)d Fn(\000)277 115 y Fp(1)p 252 122 67 2 v 252 149 a Fr(n)p Fp(+2)324 133 y Fu(\))342 117 y Fr(\014)365 133 y Fu(.)20 b(No)o(w,)14 b(the)h(follo)o(wing)h(estimate)f(can)g(b)q (e)h(pro)o(v)o(ed)f(b)o(y)g(induction.)434 287 y(F)l(or)g(an)o(y)g Fs(n)d Fn(\025)h Fs(n)716 294 y Fp(0)747 287 y Fu(+)d Fs(k)q(;)22 b Fn(L)883 268 y Fr(k)883 298 y(d)905 287 y Fo(1)931 294 y Fe(n)969 287 y Fn(\024)1017 230 y Fr(k)q Fq(\000)p Fp(1)1020 243 y Fi(Y)1019 342 y Fr(`)p Fp(=0)1088 223 y Fi(\022)1122 287 y Fu(1)10 b Fn(\000)1284 256 y Fu(1)p 1205 276 181 2 v 1205 318 a Fs(n)h Fu(+)f Fs(`)g Fu(+)h(2)1390 223 y Fi(\023)1424 234 y Fr(\014)1455 287 y Fs(:)-12 436 y Fu(It)k(remains)h(to)f(estimate)g(sup)513 447 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])607 436 y Fn(L)638 420 y Fr(k)638 450 y(d)660 436 y Fo(1)p Fu(\()p Fs(x)p Fu(\))f(for)h Fs(N)i Fn(\024)c Fs(n)g(<)g(n)1049 443 y Fp(0)1079 436 y Fu(+)e Fs(k)q Fu(.)19 b(Let)d Fs(N)h Fn(\024)c Fs(n)g(<)g(n)1481 443 y Fp(0)1511 436 y Fu(+)d Fs(k)q Fu(,)368 581 y(sup)360 622 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])452 581 y Fn(L)483 562 y Fr(k)q Fp(+1)483 596 y Fr(d)549 581 y Fo(1)p Fu(\()p Fs(x)p Fu(\))i Fn(\024)697 517 y Fi(\022)731 581 y Fu(1)e Fn(\000)856 550 y Fu(1)p 814 571 106 2 v 814 612 a Fs(n)g Fu(+)h(2)925 517 y Fi(\023)958 528 y Fr(\013)p Fq(\000)p Fr(d)1067 581 y Fu(sup)1036 622 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(+1])1173 581 y Fn(L)1204 562 y Fr(k)1204 593 y(d)1226 581 y Fo(1)p Fu(\()p Fs(x)p Fu(\))e(+)1374 550 y Fs(\025)p Fu(\()p Fs(I)1439 557 y Fp(0)1458 550 y Fu(\))p Fs(M)p 1374 571 151 2 v 1425 612 a(n)1452 599 y Fr(d)1529 581 y Fs(;)-12 717 y Fu(b)o(y)15 b(induction,)i(w)o(e)e(pro)o(v)o(e)173 860 y(sup)165 901 y Fr(x)p Fq(2)p Fp([)p Fr(n)p Fp(])257 860 y Fn(L)288 841 y Fr(k)288 871 y(d)309 860 y Fo(1)p Fu(\()p Fs(x)p Fu(\))d Fn(\024)457 803 y Fr(k)q Fq(\000)p Fp(1)460 816 y Fi(Y)460 915 y Fr(`)p Fp(=0)522 860 y Fu(\(1)d Fn(\000)702 829 y Fu(1)p 623 849 183 2 v 623 891 a Fs(n)h Fu(+)g Fs(j)j Fu(+)d(2)810 860 y(\))828 841 y Fr(\013)p Fq(\000)p Fr(d)908 860 y Fu(+)959 829 y Fs(\025)p Fu(\()p Fs(I)1024 836 y Fp(0)1042 829 y Fu(\))p Fs(M)p 959 849 151 2 v 1010 891 a(n)1037 878 y Fr(d)1114 860 y Fu([1)g(+)1206 803 y Fr(k)q Fq(\000)p Fp(1)1205 816 y Fi(X)1206 914 y Fr(p)p Fp(=2)1299 800 y Fr(p)1279 816 y Fi(Y)1278 915 y Fr(`)p Fp(=1)1338 860 y Fu(\(1)f Fn(\000)1517 829 y Fu(1)p 1439 849 181 2 v 1439 891 a Fs(n)h Fu(+)h Fs(`)f Fu(+)g(2)1624 860 y(\))1642 841 y Fr(\013)p Fq(\000)p Fr(d)1712 860 y Fu(])p Fs(:)-12 1006 y Fu(This)16 b(leads)g(to)-3 1132 y(sup)-3 1171 y Fr(x)p Fq(2)p Fm(N)73 1132 y Fn(L)104 1113 y Fr(k)104 1143 y(d)126 1132 y Fo(1)p Fu(\()p Fs(x)p Fu(\))c Fn(\024)274 1068 y Fi(\022)313 1101 y Fs(n)f Fu(+)f(1)p 312 1121 108 2 v 312 1163 a Fs(n)g Fu(+)h Fs(k)425 1068 y Fi(\023)458 1079 y Fr(\013)p Fq(\000)p Fr(d)539 1132 y Fu(+)615 1101 y Fs(K)p 589 1121 93 2 v 589 1163 a(n)616 1150 y Fr(d)p Fq(\000)p Fp(1)702 1132 y Fu(where)k Fs(K)j Fu(is)e(a)f(constan)o(t)f(whic)o(h)i(dep)q(ends)h (neither)f(on)g Fs(n)f Fu(nor)g(on)g Fs(k)q(:)-12 1277 y Fu(Let)72 1240 y Fi(\000)98 1259 y Fp(1)p 98 1266 18 2 v 98 1292 a(2)121 1240 y Fi(\001)141 1250 y Fr(\013)p Fq(\000)p Fr(d)229 1277 y Fs(<)i(\015)308 1260 y Fq(0)335 1277 y Fs(<)h(\015)h(<)e Fu(1,)h(and)g Fs(n)655 1284 y Fp(1)692 1277 y Fu(b)q(e)h(suc)o(h)f(that)f Fs(n)g Fn(\025)g Fs(n)1086 1284 y Fp(1)1106 1277 y Fu(,)1165 1259 y Fr(K)p 1142 1266 79 2 v 1142 1295 a(n)1163 1285 y Fl(d)p Ff(\000)p Fk(1)1243 1277 y Fs(<)g(\015)d Fn(\000)e Fs(\015)1407 1260 y Fq(0)1435 1277 y Fu(and)18 b Fs(N)k Fn(\025)17 b Fs(n)1664 1284 y Fp(1)1684 1277 y Fu(,)h(w)o(e)f(c)o(ho)q (ose)-12 1340 y Fs(k)12 1347 y Fp(0)48 1340 y Fu(suc)o(h)g(that)f Fs(k)g Fn(\025)f Fs(k)366 1347 y Fp(0)385 1340 y Fu(,)i(\(1)10 b Fn(\000)560 1322 y Fr(k)p 518 1329 104 2 v 518 1355 a(n)539 1360 y Fk(0)556 1355 y Fp(+2)p Fr(k)626 1340 y Fu(\))644 1323 y Fr(\013)p Fq(\000)p Fr(d)729 1340 y Fs(<)15 b(\015)806 1323 y Fq(0)817 1340 y Fu(.)24 b(Since)18 b Fs(N)h Fn(\024)c Fs(n)g(<)h(n)1200 1347 y Fp(0)1231 1340 y Fu(+)11 b Fs(k)q Fu(,)1337 1322 y Fr(n)p Fp(+1)p 1337 1329 69 2 v 1337 1355 a Fr(n)p Fp(+)p Fr(k)1425 1340 y Fn(\024)k Fu(1)c Fn(\000)1602 1322 y Fr(k)p 1560 1329 104 2 v 1560 1355 a(n)1581 1360 y Fk(0)1599 1355 y Fp(+2)p Fr(k)1685 1340 y Fu(w)o(e)16 b(ha)o(v)o(e)g(for)-12 1404 y Fs(x)d Fn(2)g Fu([)p Fs(n)p Fu(],)h Fn(L)181 1387 y Fr(k)181 1418 y(d)203 1404 y Fo(1)p Fu(\()p Fs(x)p Fu(\))e Fn(\024)h Fs(\015)s Fu(.)19 b(This)c(is)h(su\016cien)o(t)g(to)f (get)f(the)i(lemma.)769 b Fn(4)3 1460 y Fu(So,)15 b(w)o(e)g(ha)o(v)o (e,)610 1549 y Fs(\016)632 1530 y Fq(0)630 1561 y Fr(k)q(;j)690 1549 y Fn(\024)738 1485 y Fi(\022)772 1549 y Fu(1)9 b Fn(\000)1011 1518 y Fu(1)p 855 1539 336 2 v 855 1580 a Fs(N)14 b Fu(+)d Fs(k)q Fu(\()p Fs(j)h Fu(+)e(1\))g(+)g(2)1195 1485 y Fi(\023)1229 1496 y Fr(\014)r(k)1279 1549 y Fs(;)-12 1657 y Fu(and)21 b Fn(L)113 1664 y Fr(d)153 1657 y Fu(satis\014es)g(\() p Fo(S-Exp1)p Fu(\).)34 b(In)21 b(order)f(to)f(adapt)h(the)g(metho)q(d) h(of)f(section)g(3,)h(it)f(su\016ces)h(to)e(estimate)-12 1714 y Fs(K)27 1721 y Fr(j)45 1714 y Fu(\()p Fn(L)94 1697 y Fr(k)94 1728 y(d)115 1714 y Fs(f)5 b Fu(\))16 b(for)g Fs(f)j Fn(2)c Fs(C)369 1692 y Fr(j)366 1728 y(N)399 1714 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\).)20 b(Let)c(us)h(note)f Fs(\032)p Fu(\()p Fs(x)p Fu(\))d(=)i(\()p Fs(T)1018 1697 y Fq(0)1029 1714 y Fu(\()p Fs(x)p Fu(\)\))1109 1697 y Fq(\000)p Fp(1)1171 1714 y Fu(and)i Fs(\032)1285 1721 y Fr(k)1305 1714 y Fu(\()p Fs(x)p Fu(\))d(=)1431 1680 y Fi(Q)1474 1693 y Fr(k)q Fq(\000)p Fp(1)1474 1727 y Fr(i)p Fp(=0)1548 1714 y Fs(\032)p Fu(\()p Fs(T)1623 1697 y Fr(i)1636 1714 y Fs(x)p Fu(\).)22 b(F)l(or)16 b Fs(x)g Fu(and)-12 1770 y Fs(y)f Fn(2)e Fs(I)88 1777 y Fr(n)111 1770 y Fu(,)h Fs(n)f Fn(\024)g Fs(N)e Fu(+)c Fs(k)q Fu(\()p Fs(j)j Fn(\000)d Fu(1\))13 b(and)h Fs(f)k Fn(2)13 b Fs(B)r Fu(,)h(since)h Fs(T)k Fu(is)14 b(without)g(big)g (branc)o(hes)g(at)f(in\014nit)o(y)i(and)f(a\016ne,)g(w)o(e)f(get:)408 1879 y Fn(jL)452 1861 y Fr(k)452 1891 y(d)474 1879 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))k Fn(\000)i(L)649 1861 y Fr(k)649 1891 y(d)670 1879 y Fs(f)5 b Fu(\()p Fs(y)r Fu(\))p Fn(j)12 b(\024)h Fs(K)869 1886 y Fr(j)887 1879 y Fu(\()p Fs(f)5 b Fu(\))15 b Fs(d)p Fu(\()p Fs(x;)8 b(y)r Fu(\))1145 1836 y Fi(X)1116 1939 y Fr(T)1142 1929 y Fl(k)1161 1939 y Fr(x)1181 1929 y Ff(0)1192 1939 y Fp(=)p Fr(x)1247 1879 y Fs(g)1269 1886 y Fr(k)1290 1879 y Fu(\()p Fs(x)1334 1861 y Fq(0)1345 1879 y Fu(\))p Fs(\032)1387 1886 y Fr(k)1408 1879 y Fu(\()p Fs(x)1452 1861 y Fq(0)1463 1879 y Fu(\))p Fs(;)-12 2025 y Fn(\017)15 b Fs(\032)50 2032 y Fr(k)84 2025 y Fn(\024)e Fu(1,)h(so)h(that)f(for)h Fs(n)e Fn(\025)g Fs(N)567 2127 y Fn(jL)611 2108 y Fr(k)611 2138 y(d)632 2127 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))10 b Fn(\000)g(L)807 2108 y Fr(k)807 2138 y(d)829 2127 y Fs(f)5 b Fu(\()p Fs(y)r Fu(\))p Fn(j)11 b(\024)i Fs(K)1027 2134 y Fr(j)1045 2127 y Fu(\()p Fs(f)5 b Fu(\))15 b Fs(d)p Fu(\()p Fs(x;)8 b(y)r Fu(\))13 b Fs(\016)1289 2108 y Fq(0)1287 2138 y Fr(k)q(;j)-12 2229 y Fn(\017)i Fu(Let)h(0)c Fn(\024)h Fs(n)g(<)g(N)5 b Fu(.)19 b(F)l(or)c(an)o(y)f Fs(p)f Fn(\024)g Fs(k)j Fu(and)g Fs(z)e Fn(2)f Fu(\006,)i(w)o(e)g(ha)o (v)o(e)g Fs(\032)1068 2236 y Fr(k)1088 2229 y Fu(\()p Fs(z)r Fu(\))e Fn(\024)g Fs(\032)1232 2236 y Fr(p)1251 2229 y Fu(\()p Fs(z)r Fu(\).)19 b(So,)470 2296 y Fi(X)442 2399 y Fr(T)468 2389 y Fl(k)486 2399 y Fr(x)506 2389 y Ff(0)517 2399 y Fp(=)p Fr(x)572 2340 y Fs(g)594 2347 y Fr(k)615 2340 y Fu(\()p Fs(x)659 2321 y Fq(0)670 2340 y Fu(\))p Fs(\032)712 2347 y Fr(k)733 2340 y Fu(\()p Fs(x)777 2321 y Fq(0)788 2340 y Fu(\))13 b Fn(\024)895 2296 y Fi(X)867 2399 y Fr(T)893 2389 y Fl(k)911 2399 y Fr(x)931 2389 y Ff(0)942 2399 y Fp(=)p Fr(x)997 2340 y Fs(g)1019 2347 y Fr(k)1040 2340 y Fu(\()p Fs(x)1084 2321 y Fq(0)1095 2340 y Fu(\))p Fs(\032)1137 2347 y Fr(p)1156 2340 y Fu(\()p Fs(x)1200 2321 y Fq(0)1212 2340 y Fu(\))f(=)h Fn(L)1321 2321 y Fr(k)1321 2351 y(d)1343 2340 y Fs(\032)1367 2347 y Fr(p)1386 2340 y Fu(\()p Fs(x)p Fu(\))p Fs(;)-12 2493 y Fu(if)i Fs(x)d Fn(2)110 2459 y Fi(S)148 2506 y Fr(n)p Fq(\024)p Fr(N)238 2493 y Fs(I)258 2500 y Fr(n)282 2493 y Fu(,)h(theorem)h(1.1)f(applied)j(to)d Fn(L)799 2500 y Fr(d)834 2493 y Fu(implies)j(that)d Fn(L)1115 2476 y Fr(k)1115 2507 y(d)1137 2493 y Fs(\032)1161 2500 y Fr(p)1180 2493 y Fu(\()p Fs(x)p Fu(\))h(go)q(es)g(to)f Fs(h)1435 2500 y Fr(d)1455 2493 y Fu(\()p Fs(x)p Fu(\))p Fs(m)1557 2500 y Fr(d)1577 2493 y Fu(\()p Fs(\032)1619 2500 y Fr(p)1638 2493 y Fu(\))g(uniformly)j(in)-12 2549 y Fs(x)p Fu(.)j(Birkho\013)t('s)12 b(ergo)q(dic)i(theorem)f(applied)i (to)d(log)c Fs(\032)13 b Fu(and)g(the)g(fact)f(that)g Fs(\026)p Fu(\(log)d Fs(\032)p Fu(\))j Fs(<)h Fu(0)f(imply)j(that)d Fs(\032)1738 2556 y Fr(p)1757 2549 y Fu(\()p Fs(z)r Fu(\))h(go)q(es)-12 2606 y(to)i(zero)h(when)g Fs(p)g Fu(go)q(es)f(to)g(in\014nit)o(y)j(for) d Fs(\026)p Fu(-almost)g(all)i Fs(z)r Fu(.)22 b(W)l(e)16 b(ha)o(v)o(e)f Fs(\026)f Fu(=)g Fs(h)1294 2613 y Fr(d)1314 2606 y Fs(m)1354 2613 y Fr(d)1374 2606 y Fu(,)i(so)f(that)g Fs(\032)1582 2613 y Fr(p)1602 2606 y Fu(\()p Fs(z)r Fu(\))g(go)q(es)g (to)g(zero)928 2796 y(30)p eop %%Page: 31 31 31 30 bop -12 1 a Fu(when)18 b Fs(p)e Fu(go)q(es)h(to)f(in\014nit)o(y)j (for)d Fs(m)577 8 y Fr(d)597 1 y Fu(-almost)h(all)h Fs(z)r Fu(.)25 b(Leb)q(esgue's)18 b(dominated)f(con)o(v)o(ergence)h(theorem)e (implies)-12 57 y(that)f Fs(m)127 64 y Fr(d)147 57 y Fu(\()p Fs(\032)189 64 y Fr(p)208 57 y Fu(\))g(go)q(es)g(to)f(zero)h (when)h Fs(p)f Fu(go)q(es)g(to)g(in\014nit)o(y)l(.)21 b(So,)15 b(there)g(exists)h Fs(k)q Fu(\()p Fs(N)5 b Fu(\))14 b(suc)o(h)h(that)g(for)f Fs(k)g(>)f(k)q Fu(\()p Fs(N)5 b Fu(\),)618 125 y Fi(X)589 228 y Fr(T)615 218 y Fl(k)634 228 y Fr(x)654 218 y Ff(0)665 228 y Fp(=)p Fr(x)720 168 y Fs(g)742 175 y Fr(k)763 168 y Fu(\()p Fs(x)807 149 y Fq(0)818 168 y Fu(\))p Fs(\032)860 175 y Fr(k)881 168 y Fu(\()p Fs(x)925 149 y Fq(0)936 168 y Fu(\))12 b Fn(\024)h(L)1045 149 y Fr(k)1045 180 y(d)1067 168 y Fs(\032)1091 175 y Fr(p)1110 168 y Fu(\()p Fs(x)p Fu(\))f Fn(\024)h Fs(\016)1254 149 y Fq(0)1252 180 y Fr(k)q(;j)1300 168 y Fs(:)-12 322 y Fu(Finally)l(,)k(for)e Fs(k)g Fn(\025)f Fs(k)q Fu(\()p Fs(N)5 b Fu(\),)14 b(w)o(e)i(ha)o(v)o(e)f Fs(K)652 329 y Fr(j)r Fq(\000)p Fp(1)715 322 y Fu(\()p Fn(L)764 305 y Fr(k)764 335 y Fp(\010)791 322 y Fs(f)5 b Fu(\))14 b Fn(\024)g Fs(K)938 329 y Fr(j)955 322 y Fu(\()p Fs(f)5 b Fu(\))p Fs(\016)1040 305 y Fq(0)1038 335 y Fr(k)q(;j)1086 322 y Fu(.)21 b(With)16 b(the)g(notations)f(of)g(section)h(3,)g(for)f (an)o(y)-12 378 y Fs(k)f Fn(\025)f Fu(max)o(\()p Fs(k)q Fu(\()p Fs(N)5 b Fu(\))p Fs(;)j(k)324 385 y Fp(0)342 378 y Fu(\))14 b(and)i(0)c Fs(<)h(\015)i(<)e Fu(1)621 480 y Fn(L)652 461 y Fr(k)652 492 y(d)674 480 y Fs(C)710 458 y Fr(j)707 494 y(N)740 480 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))h Fn(\032)k Fs(C)975 458 y Fr(j)r Fq(\000)p Fp(1)972 494 y Fr(N)1039 480 y Fu(\()p Fs(\015)s(a;)8 b(\016)1149 487 y Fr(j)1165 480 y Fs(b;)g(\016)1226 487 y Fr(j)1243 480 y Fs(c)p Fu(\))-12 591 y(and)18 b(the)f(h)o(yp)q(erb)q(olic)j (diameter)e(of)f Fn(L)661 575 y Fr(k)661 605 y(d)682 591 y Fs(C)718 569 y Fr(j)715 606 y(N)748 591 y Fu(\()p Fs(a;)8 b(b;)g(c)p Fu(\))15 b(in)j Fs(C)996 569 y Fr(j)r Fq(\000)p Fp(1)993 606 y Fr(N)1059 591 y Fu(\()p Fs(\015)s(a;)8 b(\016)1169 598 y Fr(j)1186 591 y Fs(b;)g(\016)1247 598 y Fr(j)1264 591 y Fs(c)p Fu(\))16 b(is)i(ma)s(jored)f(b)o(y)g(2)8 b(log)1714 567 y Fp(1+)p Fr(\016)1775 572 y Fl(j)p 1714 581 77 2 v 1714 607 a Fp(1)p Fq(\000)p Fr(\016)1775 612 y Fl(j)1796 591 y Fu(.)26 b(The)-12 665 y(\014xed)16 b(p)q(oin)o(t)f(of)g Fn(L)299 672 y Fr(d)334 665 y Fu(v)o(eri\014es)h Fs(h)512 672 y Fr(d)545 665 y Fu(=)606 647 y Fr(h)p 598 654 36 2 v 598 680 a(v)615 686 y Fl(d)638 665 y Fu(.)k(If)15 b Fs(f)21 b Fu(is)15 b(suc)o(h)g(that)g Fs(f)5 b(h)1058 672 y Fr(d)1091 665 y Fu(=)1144 644 y Fr(f)t(h)p 1144 654 41 2 v 1147 680 a(v)1164 686 y Fl(d)1202 665 y Fn(2)13 b Fs(B)18 b Fu(and)d Fs(g)f Fn(2)f Fs(L)1495 648 y Fq(1)1532 665 y Fu(,)i(the)g(estimate)g(\(3.7\))-12 721 y(giv)o(es)h(with)f Fs(q)r Fu(\()p Fs(j)s Fu(\))c(=)i Fs(j)364 705 y Fr(u)386 721 y Fu(,)i(0)d Fs(<)h(u)g(<)g Fu(1)i(and)g Fs(n)e Fu(=)g Fs(k)q Fu(\()p Fs(j)f Fu(+)f Fs(q)r Fu(\()p Fs(j)s Fu(\)\))d(+)i Fs(r)q Fu(,)15 b Fs(j)g Fu(=)e Fs(O)q Fu(\()1260 703 y Fr(n)p 1260 710 22 2 v 1261 737 a(k)1286 721 y Fu(\),)155 823 y Fn(j)p Fs(\026)p Fu(\()p Fs(f)5 b(g)11 b Fn(\016)f Fs(T)339 804 y Fr(n)362 823 y Fu(\))g Fn(\000)h Fs(\026)p Fu(\()p Fs(f)5 b Fu(\))p Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fn(j)238 931 y(\024)28 b Fu(Ct)359 853 y Fi(")386 931 y Fu(log\()p Fs(N)15 b Fu(+)10 b Fs(k)q Fu(\()p Fs(j)i Fu(+)f Fs(q)r Fu(\()p Fs(j)s Fu(\)\)\))803 867 y Fi(\022)839 900 y Fs(N)j Fu(+)d Fs(k)q(q)r Fu(\()p Fs(j)s Fu(\))p 839 920 201 2 v 867 962 a Fs(N)k Fu(+)10 b Fs(k)q(j)1045 867 y Fi(\023)1078 878 y Fr(\014)r(k)1131 931 y Fu(+)g Fs(\026)p Fu(\([)p Fs(N)15 b Fu(+)10 b Fs(k)q(q)r Fu(\()p Fs(j)s Fu(\)])1449 912 y Fr(c)1465 931 y Fu(\))1483 853 y Fi(#)1517 931 y Fn(k)1545 900 y Fs(f)5 b(h)p 1545 920 54 2 v 1551 962 a(v)1573 969 y Fr(d)1603 931 y Fn(k)14 b(k)p Fs(g)r Fn(k)1710 938 y Fq(1)238 1070 y Fn(\024)28 b Fu(Ct\()p Fs(d;)8 b(u)p Fu(\))467 1006 y Fi(\024)561 1039 y Fu(1)p 493 1060 158 2 v 493 1103 a Fs(j)515 1090 y Fp(\(1)p Fq(\000)p Fr(u)p Fp(\))p Fr(\014)r(k)666 1070 y Fu(+)774 1039 y(1)p 716 1060 140 2 v 716 1103 a Fs(j)738 1090 y Fr(u)p Fp(\()p Fr(\013)p Fq(\000)p Fp(2\))860 1006 y Fi(\025)892 1070 y Fn(k)920 1039 y Fs(f)d(h)p 920 1060 54 2 v 926 1101 a(v)948 1108 y Fr(d)978 1070 y Fn(k)15 b(k)p Fs(g)r Fn(k)1086 1077 y Fq(1)1122 1070 y Fs(:)-12 1202 y Fu(Let)h Fs(f)k Fu(b)q(elongs)c(to)f Fs(E)s Fu(.)k(W)l(e)c(\014x)g Fs(")e(>)g Fu(0)i(and)g(1)e Fs(<)g(d)f(<)h Fu(1)d(+)g Fs(")p Fu(,)15 b(let)h Fs(u)d Fu(=)1203 1184 y Fr(\013)p Fq(\000)p Fp(2)p Fq(\000)p Fr(")p 1202 1191 114 2 v 1202 1218 a(\013)p Fq(\000)p Fr(d)p Fq(\000)p Fp(1)1333 1202 y Fs(<)g Fu(1)i(and)h Fs(k)d(>)1600 1184 y Fr(\013)p Fq(\000)p Fp(2)p Fq(\000)p Fr(")p 1598 1191 115 2 v 1598 1219 a(\014)r Fp(\(1)p Fq(\000)p Fr(u)p Fp(\))1718 1202 y Fu(,)i(then)457 1337 y Fn(j)p Fs(\026)p Fu(\()p Fs(f)5 b(g)11 b Fn(\016)f Fs(T)641 1318 y Fr(n)664 1337 y Fu(\))g Fn(\000)g Fs(\026)p Fu(\()p Fs(f)5 b Fu(\))p Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fn(j)12 b(\024)h Fu(Ct)o(\()p Fs(")p Fu(\))1155 1306 y(1)p 1099 1326 135 2 v 1099 1368 a Fs(j)1121 1355 y Fr(\013)p Fq(\000)p Fp(2)p Fq(\000)p Fr(")1238 1337 y Fn(j)-8 b(j)g(j)p Fs(f)5 b Fn(j)-8 b(j)g(j)15 b(k)p Fs(g)r Fn(k)1396 1344 y Fq(1)1432 1337 y Fs(:)-12 1461 y Fu(Since)i Fs(j)e Fu(=)e Fs(O)q Fu(\()248 1443 y Fr(n)p 248 1450 22 2 v 249 1476 a(k)274 1461 y Fu(\))i(and)g Fs(k)f Fu(=)f Fs(k)q Fu(\()p Fs(")p Fu(\))h(w)o(e)h(deduce:)454 1586 y Fn(j)p Fs(\026)p Fu(\()p Fs(f)5 b(g)11 b Fn(\016)f Fs(T)638 1567 y Fr(n)661 1586 y Fu(\))g Fn(\000)g Fs(\026)p Fu(\()p Fs(f)5 b Fu(\))p Fs(\026)p Fu(\()p Fs(g)r Fu(\))p Fn(j)12 b(\024)h Fu(Ct)o(\()p Fs(")p Fu(\))1155 1555 y(1)p 1096 1576 141 2 v 1096 1617 a Fs(n)1123 1604 y Fr(\013)p Fq(\000)p Fp(2)p Fq(\000)p Fr(")1241 1586 y Fn(j)-8 b(j)g(j)p Fs(f)5 b Fn(j)-8 b(j)g(j)15 b(k)p Fs(g)r Fn(k)1399 1593 y Fq(1)1435 1586 y Fs(:)-12 1699 y Fu(This)h(conclude)h (the)e(pro)o(v)o(e)g(of)f(the)i(prop)q(osition.)1873 1755 y Fn(4)-12 1918 y Fo(Remark)h(4.3)22 b Fu(W)l(e)17 b(can)f(also)g(apply)h(the)f(same)g(tec)o(hniques)h(to)f(a\016ne)g(non) g(uniformly)h(expanding)h(Mark)o(o-)-12 1974 y(vian)h(maps)f(of)g(the)h (in)o(terv)m(al)g(with)g(b)q(ounded)h(jumps)f(pro)o(vided)g(they)f (satisfy)g(\()p Fo(S-Exp2)p Fu(\).)30 b(Moreo)o(v)o(er,)17 b(the)-12 2031 y(tec)o(hniques)k(ma)o(y)d(b)q(e)i(impro)o(v)o(ed)g(to)f (consider)h(dynamics)g(whic)o(h)g(do)f(not)g(v)o(erify)h(the)f(b)q (ounded)i(distortion)-12 2087 y(prop)q(ert)o(y)16 b(\(but)h(a)f(b)q (ounded)i(distortion)f(on)g(eac)o(h)f Fs(I)901 2094 y Fr(n)925 2087 y Fu(\))g(and)h(then)g(obtain)g(estimates)f(for)g (piecewise)j(smo)q(oth)-12 2144 y(non)d(uniformly)g(expanding)g(maps.) -12 2287 y Fv(References)-12 2388 y Fu([A,D,U])59 b(J.)19 b(AAR)o(ONSON,)h(M.)f(DENKER)g(&)h(M.)e(URBANSKI)j Ft(Er)n(go)n(dic)e (the)n(ory)h(for)h(Markov)f(\014b)n(er)n(e)n(d)202 2445 y(systems)e(and)i(p)n(ar)n(ab)n(olic)e(r)n(ational)h(maps.)g Fu(T)l(rans.)f(Amer.)g(Math.)f(So)q(c.)h(\(1993\),)f Fo(337)i Fu(\(2\),)f(495-)202 2501 y(548.)-12 2595 y([B,K,S])70 b(V.)13 b(BALADI,)g(A.)g(K)o(OND)o(AH,)g(B.)g(SCHMITT)f Ft(R)n(andom)j(c)n(orr)n(elations)e(for)i(smal)r(l)f(p)n(erturb)n (ations)202 2652 y(of)j(exp)n(anding)e(maps.)h Fu(Random)f(&)g(Comput.) g(Dyn.,)f(\(1996\))g Fo(4)p Fu(,)h(179-204.)928 2796 y(31)p eop %%Page: 32 32 32 31 bop -12 1 a Fu([B,Y])109 b(M.)10 b(BENEDICKS)h(&)g(L.-S.)f(YOUNG) h Ft(De)n(c)n(ay)g(of)h(c)n(orr)n(elations)f(for)h(c)n(ertain)g(Henon)f (maps.)f Fu(\(1996\))202 57 y(preprin)o(t.)-12 150 y([Bi1])121 b(G.)15 b(BIRKHOFF)h Ft(Extensions)f(of)h(Jentzch's)f(the)n(or)n(em.)g Fu(T.A.M.S.)f(\(1957\),)f Fo(85)p Fu(,)i(219-227.)-12 244 y([Bi2])121 b(G.)15 b(BIRKHOFF)h Ft(L)n(attic)n(e)f(the)n(ory)i (\(3r)n(d)f(e)n(dition\).)e Fu(Amer.)h(Math.)f(So)q(c.)h(\(1967\).)-12 337 y([Bog])110 b(T.)15 b(BOGENSCH)529 325 y(\177)523 337 y(UTZ.)g Ft(Equilibrium)i(states)f(for)h(R)n(andom)f(Dynamic)n(al)h (systems.)d Fu(\(1991\))f(PhD)202 393 y(thesis)j(Univ)o(ersit)o(y)g(of) f(Brenen.)-12 486 y([Bre])118 b(X.)16 b(BRESSA)o(UD)h Ft(Op)o(\023)-22 b(er)n(ateurs)18 b(de)f(tr)n(ansfert)g(sur)g(le)g(d)o (\023)-22 b(e)n(c)n(alage)17 b(\022)-23 b(a)17 b(alphab)n(et)h(d)o (\023)-22 b(enombr)n(able)16 b(et)i(ap-)202 542 y(plic)n(ations.)c Fu(\(1996\))g(Pr)o(\023)-21 b(epublication.)-12 636 y([Buz])111 b(J.)15 b(BUZZI)h Ft(Exp)n(onential)f(de)n(c)n(ay)h(of)g(c)n(orr)n (elations)g(for)g(r)n(andom)h(L)n(asota-Y)m(orke)e(maps.)h Fu(\(1998\))-12 729 y([Ch])130 b(N.)22 b(CHERNO)o(V)g Ft(Markov)h(appr)n(oximations)g(and)f(de)n(c)n(ay)g(of)h(c)n(orr)n (elations)e(for)i(A)o(nosov)f(\015ows.)202 785 y Fu(Pr)o(\023)-21 b(epublication.)-12 878 y([F,S)15 b(1])82 b(P)l(.)12 b(FERRER)o(O,)h(B.)f(SCHMITT)g Ft(R)o(uel)r(le)i(Perr)n(on)f(F)m(r)n (ob)n(enius)f(the)n(or)n(ems)h(and)g(pr)n(oje)n(ctive)h(metrics.)202 934 y Fu(Collo)q(que)j(Math.)d(So)q(c.)h(J.)g(Boly)o(ai)h(Random)f (Fields.)h(Estergom)e(\(Hungary\))h(\(1979\).)-12 1028 y([F,S)g(2])82 b(P)l(.)14 b(FERRER)o(O,)h(B.)e(SCHMITT)h Ft(On)h(the)g(r)n(ate)g(of)h(c)n(onver)n(genc)n(e)d(for)i(some)g(limit) g(r)n(atio)h(the)n(or)n(ems)202 1084 y(r)n(elate)n(d)g(to)g (endomorphisms)g(with)h(a)f(non)f(r)n(e)n(gular)h(invariant)f(density.) g Fu(Pr)o(\023)-21 b(ebuplication)15 b(\(1994\).)-12 1177 y([F,)f(L])103 b(A.M.)14 b(FISHER,)h(A.)f(LOPES)h Ft(Polynomial)g(de)n(c)n(ay)h(of)g(c)n(orr)n(elations)e(and)i(the)g(c)n (entr)n(al)f(limit)g(the)n(o-)202 1233 y(r)n(em)h(for)h(the)f(e)n (quilibrium)h(state)f(of)g(a)h(non-H\177)-23 b(older)16 b(p)n(otential.)e Fu(\(1997\).)-12 1326 y([G,)g(S])100 b(A.)12 b(GAL)-5 b(VES)12 b(&)h(B.)f(SCHMITT)g Ft(Ine)n(qualities)f (for)j(hitting)f(time)h(in)e(mixing)h(dynamic)n(al)g(systems.)202 1383 y Fu(Random)j(and)f(Computational)g(Dynamics)h(\(1997\).)-12 1476 y([G,)e(W])78 b(P)l(.)15 b(GASP)l(ARD)h(&)f(X.J.)g(W)-5 b(ANG.)14 b(Pro)q(c.)h(Math.)f(Acad.)h(Sci.)h(USA)g(\(1988\))d Fo(85)j Fu(4591.)-12 1569 y([H])154 b(H.)15 b(HU)g Ft(De)n(c)n(ay)h(of) h(Corr)n(elations)e(for)i(Maps)f(with)h(Indi\013er)n(ent)d(Fixe)n(d)i (Points)e Fu(\(1998\))g(preprin)o(t.)-12 1662 y([IT,M])84 b(C.T.)17 b(IONESCU)i(&)f(G.)g(MARINESCU)g Ft(Th)o(\023)-22 b(eorie)19 b(er)n(go)n(dique)g(p)n(our)g(des)g(classes)e(d'op)o(\023) -22 b(er)n(ations)202 1718 y(non)16 b(c)n(ompl)o(\022)-22 b(etement)16 b(c)n(ontinues.)e Fu(Annals)i(of)e(Math.)g(\(1950\),)f Fo(52)p Fu(,)j(140-147.)-12 1812 y([I])172 b(S.)23 b(ISOLA)i Ft(On)f(the)g(r)n(ate)g(of)g(c)n(onver)n(genc)n(e)d(to)j(e)n (quilibrium)g(for)g(c)n(ountable)g(er)n(go)n(dic)f(Markov)202 1868 y(chains.)15 b Fu(Pr)o(\023)-21 b(epublication)16 b(\(1997\).)-12 1961 y([K,M,S])60 b(A.)11 b(K)o(OND)o(AH,)f(V.)h(MA)o (UME)f(&)h(B.)g(SCHMITT)f Ft(Vitesse)h(de)i(c)n(onver)n(genc)n(e)d (vers)i(l')o(\023)-22 b(etat)12 b(d')o(\023)-22 b(equlibr)n(e)202 2017 y(p)n(our)18 b(des)e(dynamiques)h(markoviennes)f(non)g(h\177)-23 b(old)o(\023)h(eriennes.)14 b Fu(Ann.)i(Inst.)f(P)o(oincarr)o(\023)-21 b(e)15 b(Sec.)h(Prob.)202 2074 y(Stat.)e(\(1997\))g Fo(33)h Fu(\(6\))f(675-695.)-12 2167 y([La,Si,V])40 b(A.)20 b(LAMBER)l(T,)h(S.) f(SIBONI)h(&)g(S.)f(V)-5 b(AIENTI)21 b Ft(Statistic)n(al)e(pr)n(op)n (erties)i(of)g(a)g(non)f(uniformly)202 2223 y(hyp)n(erb)n(olic)c(map)h (of)f(the)h(interval.)d Fu(Journ.)i(of)e(Stat.)g(Ph)o(ysics.)i (\(1993\),)d Fo(72)p Fu(,)i(1305-1330.)-12 2316 y([Li1])125 b(C.)15 b(LIVERANI)j Ft(De)n(c)n(ay)e(of)h(Corr)n(elations)e(in)h(Pie)n (c)n(ewise)f(Exp)n(anding)h(maps.)g Fu(Journal)g(of)f(Statis-)202 2373 y(tical)h(Ph)o(ysics,)g(\(1995\),)d Fo(78)p Fu(,)i(3/4,)f (1111-1129.)-12 2466 y([Li2])125 b(C.)15 b(LIVERANI)i Ft(De)n(c)n(ay)f(of)g(c)n(orr)n(elations.)e Fu(Ann.)i(of)f(Math.)f (\(1995\),)f Fo(142)i Fu(\(2\),)f(239-301)-12 2559 y([Li3])125 b(C.)15 b(LIVERANI)i Ft(Centr)n(al)e(limit)h(the)n(or)n(em)g(for)h (deterministic)e(systems.)f Fu(Pro)q(ceedings)j(of)d(the)i(In-)202 2615 y(ternational)f(Congress)f(on)g(Dynamical)h(Systems,)f(Mon)o (tevideo)g(95,)g(Researc)o(h)h(Notes)f(in)h(Math-)202 2672 y(ematics)h(series,)f(Pittman,)g(\(1997\).)928 2796 y(32)p eop %%Page: 33 33 33 32 bop -12 1 a Fu([L,)15 b(S,)g(V1])22 b(C.)17 b(LIVERANI,)h(B.)f (SA)o(USSOL)h(&)f(S.)g(V)-5 b(AIENTI)18 b Ft(Conformal)f(me)n(asur)n(e) h(and)g(de)n(c)n(ay)f(of)h(c)n(orr)n(e-)202 57 y(lations)e(for)g(c)n (overing)g(weighte)n(d)g(systems.)e Fu(\(1996\))g(to)g(app)q(ear)i(in)g (Erg.)e(Th.)h(and)g(Dyn.)g(Syst.)-12 151 y([L,)g(S,)g(V2])22 b(C.)14 b(LIVERANI,)i(B.)e(SA)o(USSOL)i(&)e(S.)g(V)-5 b(AIENTI)15 b Ft(A)h(pr)n(ob)n(abilistic)e(appr)n(o)n(ach)i(to)g (intermitency.)202 208 y Fu(\(1997\))e(to)g(app)q(ear)i(in)g(Erg.)e (Th.)h(and)g(Dyn.)g(Syst.)-12 302 y([Ma])123 b(V.)50 b(MA)o(UME-DESCHAMPS)e Ft(Pr)n(opri)o(\023)-22 b(et)o(\023)g(es)48 b(de)g(m)o(\023)-22 b(elange)47 b(p)n(our)i(des)f(syst)o(\022)-22 b(emes)47 b(dy-)202 358 y(namiques)62 b(markoviens.)i Fu(PhD)g(Thesis,)78 b(Univ)o(ersit)o(\023)-21 b(e)64 b(de)h(Bourgogne)f(\(1998\),)202 414 y Fa(http://www.u-bourgogne.fr/mo) o(nge/v.m)o(aume/ac)o(cueil.ht)o(ml)p Fu(.)-12 508 y([Mo])123 b(M.)14 b(MORI)i Ft(On)g(the)g(intermitency)f(of)i(a)f(pie)n(c)n(ewise) f(line)n(ar)g(map)i(\(T)m(akahashi)e(mo)n(del\).)f Fu(T)l(oky)o(o)g(J.) 202 565 y(Math.)g(\(1993\))g Fo(16)p Fu(,)h(2,)f(411-428.)-12 659 y([P)o(o,Y])88 b(M.)18 b(POLLICOTT)i(&)f(M.)f(YURI)h Ft(Statistic)n(al)g(pr)n(op)n(erties)g(of)h(maps)f(with)h(indi\013er)n (ent)e(p)n(erio)n(dic)202 715 y(p)n(oints.)d Fu(Pr)o(\023)-21 b(epublication)16 b(\(1998\).)-12 809 y([Sa])140 b(O.)15 b(SARIG)h Ft(Thermo)n(dynamic)h(F)m(ormalism)e(for)i(Countable)f (Markov)g(Shifts.)e Fu(\(1997\).)-12 903 y([Sau])115 b(B.)18 b(SA)o(USSOL)497 891 y Ft(\023)490 903 y(Etude)h(statistique)g (de)g(syst)o(\022)-22 b(emes)18 b(dynamiques)h(dilatants.)e Fu(PhD.)h(Thesis,)h(Uni-)202 959 y(v)o(ersit)o(\023)-21 b(e)15 b(de)g(T)l(oulon.)-12 1053 y([Se])143 b(E.)15 b(SENET)l(A)g Ft(Non-ne)n(gative)h(matric)n(es)g(and)g(Markov)g (chains.)f Fu(Springer)h(\(1981\).)-12 1147 y([T])155 b(M.)18 b(THALER)h Ft(Estimates)g(of)g(the)h(invariant)f(densities)f (of)h(endomorphisms)h(with)g(indi\013er)n(ent)202 1203 y(\014xe)n(d)c(p)n(oint.)f Fu(Israel)h(Journal)g(of)e(Math.)g(\(1980\)) g Fo(37)h Fu(303-314.)-12 1297 y([V-J1])93 b(D.)11 b(VERE-JONES)j Ft(Ge)n(ometric)f(er)n(go)n(dicity)g(in)g(denumer)n(able)f(Markov)i (Chains.)d Fu(Quartely)h(Jour-)202 1353 y(nal)k(of)f(Math.)f(\(1962\))f Fo(13)j Fu(7-28.)-12 1447 y([V-J2])93 b(D.)12 b(VERE-JONES)h Ft(Er)n(go)n(dic)g(pr)n(op)n(erties)g(of)h(non-ne)n(gative)e(matric)n (es.)f Fu(P)o(aci\014c)i(Journal)g(of)e(Math.)202 1504 y(\(1990\))j Fo(22)h Fu(361-386.)-12 1598 y([V])154 b(M.)15 b(VIANA)h Ft(Sto)n(chastic)f(dynamics)h(of)h(deterministic)e(systems.)g Fu(\(1997\).)-12 1691 y([W)l(an])97 b(X.J.)14 b(W)-5 b(ANG)14 b Ft(statistic)n(al)h(physics)g(of)h(temp)n(or)n(al)g (intermitency.)d Fu(Ph)o(y)l(.)i(Rev.)f(A.)g(\(1989\))f Fo(40)i Fu(6647.)-12 1785 y([Y)l(o])135 b(L.-S.)15 b(YOUNG)h Ft(R)n(e)n(curr)n(enc)n(e)f(times)h(and)g(r)n(ates)g(of)g(mixing.)f Fu(Pr)o(\023)-21 b(epublication)16 b(\(1997\).)-12 1891 y(V)o(\023)-21 b(eronique)12 b(Maume-Desc)o(hamps,)f(Univ)o(ersit)o (\023)-21 b(e)12 b(de)f(Bourgogne,)h(Lab)q(oratoire)f(de)h(T)l(op)q (ologie,)g(BP)g(400,)f(21011)-12 1948 y(Dijon,)k(F)l(rance.)-12 2004 y(Curren)o(t)d(address:)19 b(Univ)o(ersit)o(\023)-21 b(e)12 b(de)h(Gen)o(\022)-21 b(ev)o(e,)12 b(Section)i(de)f(Math)o(\023) -21 b(ematiques)11 b(2)h(-)h(4)f(rue)h(du)g(li)o(\022)-21 b(evre)13 b(1211)e(Gen)o(\022)-21 b(ev)o(e)-12 2061 y(24,)14 b(Suisse.)-12 2117 y(e-mail:)21 b(V)l(eronique.Maume@math.unige.c)o(h) 928 2796 y(33)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9811161127406--