Content-Type: multipart/mixed; boundary="-------------0103281123837" This is a multi-part message in MIME format. ---------------0103281123837 Content-Type: text/plain; name="01-114.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-114.keywords" transfer operator, dimension 1, zeta function, sharp determinant ---------------0103281123837 Content-Type: application/postscript; name="stage.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="stage.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: stage.dvi %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o stage.ps stage.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2001.03.27:1708 %%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 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn 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 TeXDict begin 39158280 55380996 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8003F890397F0001FC01FE6D7E4848147F485A0007ED3F805B000FED1FC05B001F16E05B 003F150FA2007F16F0A25BA212FFA290B7FCA20180C9FCA8127FA27FA2123F17706C7E17 F0000F16E07F0007ED01C06C6C14036C6C158000001507017EEC0F006D141E6D6C5BD907 E013F0903901F807C09026007FFFC7FCEC0FF82C4E7DCC33>235 D E /FM 19 234 df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end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 130 375 a FM(Sp)t(ectre)51 b(de)i(l'op)t(\351rateur)e(de)i (transfert)e(en)h(dimension)e(1)1431 706 y FL(S\351bastien)37 b(Gou\353zel)947 855 y(e-mail)e(:)j FK(sebastien.gouezel@e)o(ns.)o(fr) 800 1005 y FL(adresse)f(:)h(45)f(rue)h(d'Ulm,)e(75005)i(P)m(aris,)f(F) -9 b(rance)1551 1267 y(19)38 b(mars)f(2001)1701 1570 y FJ(R\351sum\351)134 1705 y FI(On)d(\351tudie)g(les)f (propri\351t\351s)h(sp)s(ectrales)g(d'un)g(op)s(\351rateur)g(de)g (transfert)g FH(M)p FG(\010\()p FF(x)p FG(\))c(=)3087 1637 y FE(P)3182 1732 y FD(!)3248 1705 y FF(g)3291 1719 y FD(!)3342 1705 y FG(\()p FF(x)p FG(\)\010\()p FF( )3624 1719 y FD(!)3675 1705 y FF(x)p FG(\))-2 1818 y FI(agissan)m(t)23 b(sur)f(les)g(fonctions)h(\340)f(v)-5 b(ariation)23 b(b)s(orn\351e.)g (Apr\350s)f(des)g(estimations)h(\025)f(utilisan)m(t)i(une)e(in)m (t\351grale)i(sym\351-)-2 1931 y(trique)j(\025)g(du)f(ra)m(y)m(on)i(sp) s(ectral)f(et)g(du)g(ra)m(y)m(on)g(sp)s(ectral)g(essen)m(tiel,)h(on)e 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Fz(B)k FA(l'ensem)m(ble)33 b(des)i(p)s(oin)m(ts)f Fz(x)h FA(tels)e(qu'il)g(existe)i Fz(!)i FA(a)m(v)m(ec)f Fz( )2282 1976 y FD(!)2367 1961 y FA(croissan)m(te,)e(et)h Fz(!)3024 1925 y Fv(0)3080 1961 y FA(a)m(v)m(ec)h Fz( )3359 1976 y FD(!)3405 1957 y Fl(0)3466 1961 y FA(d\351croissan)m(te,)-246 2082 y(tels)j(que)h Fz( )191 2097 y FD(!)242 2082 y Fx(\()p Fz(x)p Fx(\))f(=)f Fz( )589 2097 y FD(!)635 2078 y Fl(0)662 2082 y Fx(\()p Fz(x)p Fx(\))p FA(.)i(Comme,)e(p)s(our)h(c)m(haque)h(couple)f Fx(\()p Fz(!)t(;)17 b(!)2341 2046 y Fv(0)2363 2082 y Fx(\))p FA(,)39 b(il)e(y)i(a)g(un)g(seul)g Fz(x)g FA(qui)g(v)m (\351ri\034e)h Fz( )3710 2097 y FD(!)3761 2082 y Fx(\()p Fz(x)p Fx(\))e(=)-246 2202 y Fz( )-183 2217 y FD(!)-137 2198 y Fl(0)-109 2202 y Fx(\()p Fz(x)p Fx(\))p FA(,)e Fz(B)41 b FA(est)36 b(\034ni.)g(Soit)f Fz(C)42 b FA(l'ensem)m(ble)36 b(des)h(discon)m(tin)m(uit\351s)e(des)i Fz(g)2350 2217 y FD(!)2417 2202 y FA(;)f(il)e(est)i(au)g(plus)g(d\351nom)m(brable.)f (Comme)-246 2322 y Fy(kM)-76 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Fv(1)101 588 y FA(.)p 3936 588 4 66 v 3940 525 59 4 v 3940 588 V 3998 588 4 66 v -246 734 a Fu(Remarque)38 b(2.4.4.)49 b FA(En)29 b(d\351comp)s(osan)m(t)f(une)h(fonction)f Fx(\010)g Fy(2)g(F)38 b FA(comme)27 b(somme)h(d'une)h(fonction)e(n)m(ulle)h(hors)h(de)-246 855 y Fz(A)k FA(\(ce)h(qui)e(corresp)s(ond)i(\340)e(un)i(op)s (\351rateur)e(de)i(rang)e(\034ni\))g(et)h(d'une)h(fonction)e(n)m(ulle)g (sur)h Fz(A)p FA(,)g(on)g(en)g(d\351duit)g(par)g(le)-246 975 y(th\351or\350me)g(de)g(Nussbaum)g(que,)h(p)s(our)e(l'action)f(de)i Fy(M)f FA(sur)h Fx(\()p Fy(F)10 b Fz(;)17 b Fy(k)31 b(k)2308 1004 y Fv(1)2383 975 y Fx(\))p FA(,)i Fz(r)2525 990 y FD(ess)2627 975 y Fx(\()p Fy(M)p Fx(\))27 b Fs(6)h Fz(R)3029 990 y FD(c)3064 975 y FA(.)-246 1120 y Fu(Prop)s(osition)36 b(2.4.5.)49 b Ft(Si)34 b(les)h(p)-5 b(oids)33 b Fz(g)1229 1135 y FD(!)1314 1120 y Ft(sont)i(c)-5 b(ontinus,)34 b(on)g(a)h Fz(R)29 b Fx(=)e Fz(R)2448 1135 y FD(c)2518 1120 y Ft(\(et)35 b(les)f(tr)-5 b(ois)35 b(\351)-5 b(galit\351s)34 b(analo)-5 b(gues)33 b(p)-5 b(our)-246 1253 y Fz(R)-171 1217 y FD(")-134 1253 y Fz(;)-72 1228 y Fw(b)-90 1253 y Fz(R)q(;)35 1228 y Fw(c)29 1253 y Fz(R)104 1225 y FD(")140 1253 y Ft(\).)35 b(En)g(p)-5 b(articulier,)34 b Fz(R)29 b Fx(=)f Fz(R)1183 1217 y FD(")1254 1253 y Ft(et)1384 1228 y Fw(b)1366 1253 y Fz(R)h Fx(=)1579 1228 y Fw(c)1573 1253 y Fz(R)1648 1225 y FD(")1684 1253 y Ft(.)-246 1446 y(D\351monstr)-5 b(ation.)48 b FA(Mon)m(trons)43 b(que)g Fz(R)j Fx(=)e Fz(R)1415 1461 y FD(c)1450 1446 y FA(.)f(On)f(sait)g (d\351j\340)g(\(prop)s(osition)f(2.4.1\))g(que)j Fz(R)h Fs(>)g Fz(R)3453 1461 y FD(c)3488 1446 y FA(,)e(il)e(su\036t)i(de)-246 1567 y(prouv)m(er)34 b(que)f Fz(R)c Fs(6)f Fz(R)575 1582 y FD(c)610 1567 y FA(.)-246 1723 y(Il)42 b(d\351coule)g(de)h(la)e (preuv)m(e)j(du)f(lemme)e(pr\351c\351den)m(t)j(que,)f(quand)g(les)g(p)s (oids)e(son)m(t)i(con)m(tin)m(us,)g(on)g(p)s(eut)f(prendre)-246 1843 y Fz(A)28 b Fx(=)g Fy(;)p FA(.)k(Dans)h(ce)h(cas,)f(on)f(obtien)m (t)h Fy(k)o(M)1264 1807 y FD(m)1331 1843 y Fy(k)1380 1872 y Fv(1)1483 1843 y Fs(6)28 b Fy(kM)1758 1807 y FD(m)1758 1868 y(c)1824 1843 y Fy(k)1874 1872 y Fv(1)p FD(;c)1999 1843 y FA(,)k(ce)h(qui)g(donne)g(en)g(passan)m(t)g(\340)g(la)e(limite)e (sur)k Fz(m)g FA(que)-246 1963 y Fz(R)c Fs(6)f Fz(R)36 1978 y FD(c)71 1963 y FA(.)-246 2119 y(Les)43 b(trois)d(autres)i (\351galit\351s)f Fz(R)904 2083 y FD(")983 2119 y Fx(=)i Fz(R)1177 2083 y FD(")1176 2144 y(c)1214 2119 y FA(,)1301 2094 y Fw(b)1283 2119 y Fz(R)h Fx(=)1537 2094 y Fw(b)1520 2119 y Fz(R)1594 2134 y FD(c)1670 2119 y FA(et)1799 2094 y Fw(c)1793 2119 y Fz(R)1868 2090 y FD(")1947 2119 y Fx(=)2072 2094 y Fw(c)2066 2119 y Fz(R)2141 2090 y FD(")2140 2144 y(c)2220 2119 y FA(son)m(t)e(iden)m(tiques.)g(Comme)f(on)g(sait)g (d\351j\340)g(que)-246 2252 y Fz(R)-172 2267 y FD(c)-109 2252 y Fx(=)28 b Fz(R)70 2216 y FD(")69 2277 y(c)139 2252 y FA(\(prop)s(osition)i(2.4.2\),)i(on)h(en)g(d\351duit)f Fz(R)d Fx(=)f Fz(R)1838 2216 y FD(")1874 2252 y FA(,)33 b(et)g(de)g(la)e(m\352me)h(fa\347on)2843 2227 y Fw(b)2825 2252 y Fz(R)d Fx(=)3038 2227 y Fw(c)3032 2252 y Fz(R)3107 2223 y FD(")3143 2252 y FA(.)p 3936 2252 V 3940 2189 59 4 v 3940 2252 V 3998 2252 4 66 v -246 2399 a Fu(Remarque)38 b(2.4.6.)49 b FA(On)23 b(p)s(eut)g(prouv)m(er)h(cette)f(prop)s(osition) e(directemen)m(t)i(sans)h(utiliser)d(le)h(lemme)16 b(;)21 b(cela)i(revien)m(t)-246 2519 y(\340)33 b(refaire)f(la)g(preuv)m(e)j (du)f(lemme,)d(sans)j(les)f(probl\350mes)g(tec)m(hniques)i(dus)f(aux)f (discon)m(tin)m(uit\351s.)g(Simplemen)m(t,)e(on)-246 2639 y(p)s(ose)i Fx(\011)g FA(\351gale)f(\340)g Fx(\010\()p Fz( )579 2654 y FD(!)630 2639 y Fz(x)p Fx(\))h FA(au)f(v)m(oisinage)g (de)h(c)m(haque)h Fz( )1827 2654 y FD(!)1878 2639 y Fz(x)f FA(\(apr\350s)g(s'\352tre)h(ramen\351)e(au)h(cas)g(o\371)f Fx(\012)h FA(est)g(\034ni\).)-246 2784 y(La)40 b(prop)s(osition)e(qui)h (pr\351c\350de)j(a)d(\351t\351)h(\351nonc\351e)i(p)s(our)d(des)i(p)s 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FA(agit)f(sur)i Fy(B)i FA(l'ensem)m(ble)d (des)i(fonctions)d(\340)h(v)-5 b(ariation)33 b(b)s(orn\351e.)j(Cette)g (action)e(est)i(con)m(tin)m(ue,)g(et)g(on)f(a)g(m\352me)-246 609 y Fy(kMk)-26 638 y FC(V)-6 b(ar)115 609 y Fs(6)28 b Fz(V)22 b FA(,)j(o\371)f Fz(V)49 b Fx(=)688 535 y Fw(P)810 609 y Fy(k)p Fz(g)907 624 y FD(!)957 609 y Fy(k)1007 638 y Fv(1)1081 609 y FA(.)25 b(De)h(la)d(m\352me)h(fa\347on,)h(les)f (trois)g(autres)h(op)s(\351rateurs)h Fy(M)3153 573 y FD(")3189 609 y FA(,)3265 584 y Fw(c)3241 609 y Fy(M)e FA(et)3497 584 y Fw(d)3491 609 y Fy(M)3611 581 y FD(")3672 609 y FA(agissen)m(t)-246 730 y(\351galemen)m(t)32 b(sur)h Fy(B)s FA(.)-246 886 y(On)38 b(notera)g Fz(r)s Fx(\()p Fy(M)p Fx(\))474 901 y FC(V)-6 b(ar)625 886 y FA(le)38 b(ra)m(y)m(on)g(sp)s(ectral)g(de)g Fy(M)g FA(p)s(our)g(son)g(action)f (sur)i Fx(\()p Fy(B)s Fz(;)17 b Fx(V)-8 b(ar\))p FA(.)38 b(Dans)g(cette)h(partie,)f(on)f(v)-5 b(a)-246 1016 y(\351tudier)40 b(le)f(lien)g(en)m(tre)h(ce)h(ra)m(y)m(on)f(sp)s(ectral)g(et)g Fz(R)h FA(et)1798 991 y Fw(b)1780 1016 y Fz(R)q FA(,)f(en)g(utilisan)m (t)e(essen)m(tiellemen)m(t)h(l'in)m(t\351grale)f(construite)-246 1136 y(dans)33 b(l'app)s(endice)f(A.)-246 1461 y Fr(3.1)131 b(Lien)44 b(en)l(tre)g Fk(M)g Fr(et)1054 1430 y Fj(c)1025 1461 y Fk(M)-246 1681 y FA(Le)33 b(lien)e(en)m(tre)j Fy(M)e FA(et)612 1656 y Fw(c)588 1681 y Fy(M)g FA(s'exprime)h(bien)f (par)g(une)i(form)m(ule)d(in)m(t\351grale)g(:)-246 1835 y Fu(Th\351or\350me)37 b(3.1.1.)50 b Ft(Soient)34 b Fz( )t(;)17 b Fx(\010)28 b Fy(2)g(B)38 b Ft(et)d Fz(m)28 b Fy(2)g Fq(N)9 b Ft(.)41 b(A)n(lors)302 1990 y Fw(Z)418 2126 y Fz( )20 b Fy(r)p Fx(\()p Fy(M)742 2085 y FD(m)809 2126 y Fx(\010\))28 b(=)1089 2001 y FD(m)1048 2031 y Fw(X)1056 2243 y FD(k)r FC(=1)1209 2031 y Fw(X)1240 2240 y FD(!)1284 2252 y Fp(k)1369 1990 y Fw(Z)1509 2101 y(c)1485 2126 y Fy(M)1605 2085 y FD(k)r Fv(\000)p FC(1)1738 2126 y Fz( )20 b Fy(r)p Fx(\()p Fz(g)1989 2141 y FD(!)2033 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Fp(c)1290 5753 y Fs(6)28 b Fx(max\()p Fz(R)1689 5768 y FD(c)1724 5753 y Fz(;)1785 5728 y Fw(b)1768 5753 y Fz(R)1842 5768 y FD(c)1876 5753 y Fx(\))p Ft(.)1832 6027 y FA(10)p eop %%Page: 11 11 11 10 bop -246 -28 a Ft(D\351monstr)-5 b(ation.)48 b FA(La)43 b(preuv)m(e)i(est)f(analogue)e(\340)i(celle)e(du)i (th\351or\350me)g(3.2.1,)f(mais)f(en)i(rempla\347an)m(t)e(toutes)i(les) -246 92 y(in)m(t\351grales)191 12 y Fw(R)290 92 y FA(par)32 b(des)i(in)m(t\351grales)1069 12 y Fw(R)1116 127 y FD(c)1150 92 y FA(,)f(et)g(en)g(utilisan)m(t)d(le)i(th\351or\350me)h(A.5.1.)p 3936 92 4 66 v 3940 30 59 4 v 3940 92 V 3998 92 4 66 v -246 255 a Fu(Th\351or\350me)k(3.5.2.)50 b Ft(On)34 b(a)h Fz(r)s Fx(\()p Fy(M)1045 270 y FD(c)1079 255 y Fx(\))1117 270 y FC(V)-6 b(ar)1226 278 y Fp(c)1290 255 y Fs(>)1401 230 y Fw(c)1395 255 y Fz(R)1470 226 y FD(")1469 280 y(c)1507 255 y Fx(\(=)1667 230 y Fw(b)1649 255 y Fz(R)1723 270 y FD(c)1758 255 y Fx(\))p Ft(.)-246 457 y(D\351monstr)h(ation.)48 b FA(Comme)27 b(dans)h(la)f(preuv)m(e)i(du)g (th\351or\350me)f(3.3.1,)f(on)h(mon)m(tre)g(que)h Fy(k)o(M)3066 472 y FD(c)3101 457 y Fy(k)3151 486 y FC(V)-6 b(ar)3260 494 y Fp(c)3324 457 y Fs(>)3429 372 y Fw(\015)3429 432 y(\015)3491 431 y(d)3485 457 y Fy(M)3605 428 y FD(")3605 481 y(c)3641 372 y Fw(\015)3641 432 y(\015)3696 496 y Fv(1)p FD(;c)3821 457 y FA(.)28 b(On)-246 586 y(suit)37 b(le)f(m\352me)h(sc)m(h\351ma)g(de)h(preuv)m(e,)g(le)f(seul)g (probl\350me)f(\351tan)m(t)h(que)h(dans)g(le)e(cas)i(non)f(quotien)m (t\351)g(on)g(prend)h(un)-246 706 y(Dirac,)33 b(tu\351)g(dans)g(le)f (quotien)m(t,)g(ce)i(qu'il)d(faut)h(\351viter.)h(On)f(p)s(eut)h(supp)s (oser)h Fx(\012)f FA(\034ni.)-246 862 y(On)28 b(prend)h Fz( )j FA(a)m(v)m(ec)e Fy(k)p Fz( )t Fy(k)656 891 y Fv(1)758 862 y Fx(=)e(1)f FA(et)1048 777 y Fw(\015)1048 837 y(\015)6 b(d)1103 862 y Fy(M)1223 833 y FD(")1223 886 y(c)1260 862 y Fz( )1323 877 y FD(c)1358 777 y Fw(\015)1358 837 y(\015)1413 901 y Fv(1)p FD(;c)1566 862 y FA(grand.)28 b(Soit)f Fz(x)i FA(un)f(p)s(oin)m(t)f(de)i(con)m(tin)m(uit\351)f(des)h Fz(g)3319 877 y FD(!)3382 862 y Fy(\016)13 b Fz( )3512 826 y Fv(\000)p FC(1)3508 886 y FD(!)3634 862 y FA(en)29 b(lequel)-239 997 y Fw(d)-246 1022 y Fy(M)-126 993 y FD(")-89 1022 y Fz( )k FA(est)d(grand.)g(Si)f Fz(')g FA(est)h(\351gale)f(\340)g Fx(1)h FA(sur)g(un)g(p)s(etit)e(v)m (oisinage)h(de)h Fz(x)g FA(\(c'est)h(l'analogue)c(de)j Fz(\016)3250 1037 y FD(x)3294 1022 y FA(\),)g(on)f(v)m(\351ri\034e)h (alors)-246 1142 y(que)37 b Fy(M)p Fz(')e FA(tend)h(v)m(ers)h Fy(M)p Fz(\016)750 1157 y FD(x)829 1142 y FA(en)f(c)m(hacun)h(des)f Fz( )1531 1106 y Fv(\000)p FC(1)1527 1167 y FD(!)1626 1142 y Fz(x)g FA(\(par)f(con)m(tin)m(uit\351\),)g(et)h(qu'elle)f(s'ann) m(ule)h(en)m(tre)g(deux)h(de)f(ces)-246 1263 y(p)s(oin)m(ts)i(\(si)f (on)h(a)g(pris)g(le)f(v)m(oisinage)g(de)i Fz(x)f FA(o\371)g Fz(')g FA(v)-5 b(aut)38 b Fx(1)g FA(su\036sammen)m(t)g(p)s(etit,)g(et)g (en)h(utilisan)m(t)d Fx(\012)i FA(\034ni\).)g(Donc)-246 1396 y Fx(V)-8 b(ar)-94 1411 y FD(c)-59 1396 y Fx(\()p Fy(M)99 1411 y FD(c)133 1396 y Fz(')197 1411 y FD(c)232 1396 y Fx(\))27 b Fs(>)h Fx(V)-8 b(ar\()p Fy(M)p Fz(\016)755 1411 y FD(x)799 1396 y Fx(\))27 b Fs(>)970 1311 y Fw(\014)970 1371 y(\014)6 b(d)1003 1396 y Fy(M)1123 1367 y FD(")1159 1396 y Fz( )t Fx(\()p Fz(x)p Fx(\))1357 1311 y Fw(\014)1357 1371 y(\014)1391 1396 y FA(,)32 b(et)h(on)g(conclut)f(comme)f (pr\351c\351demmen)m(t.)p 3936 1396 V 3940 1333 59 4 v 3940 1396 V 3998 1396 4 66 v -246 1556 a Fu(Th\351or\350me)37 b(3.5.3.)50 b Ft(On)34 b(a)h Fz(r)884 1571 y FD(ess)986 1556 y Fx(\()p Fy(M)1144 1571 y FD(c)1178 1556 y Fx(\))1216 1571 y FC(V)-6 b(ar)1326 1579 y Fp(c)1390 1556 y Fs(6)1513 1531 y Fw(b)1495 1556 y Fz(R)1569 1571 y FD(c)1604 1556 y Ft(.)-246 1745 y(D\351monstr)h(ation.)48 b FA(On)39 b(p)s(ourrait)f(r\351p)s(\351ter)i(la)e(preuv)m(e)j(du)f(cas)g(non)g (quotien)m(t\351,)g(en)g(utilisan)m(t)d(les)i(propri\351t\351s)h(de) -246 1865 y(compacit\351)29 b(de)338 1785 y Fw(R)385 1900 y FD(c)420 1865 y FA(,)g(mais)f(il)f(n'y)j(en)g(pas)f(b)s(esoin.)g (En)h(e\033et,)g(on)f(sait)f(d\351j\340)h(que)i Fz(r)2697 1880 y FD(ess)2799 1865 y Fx(\()p Fy(M)p Fx(\))2995 1880 y FC(V)-6 b(ar)3136 1865 y Fs(6)3259 1840 y Fw(b)3241 1865 y Fz(R)3315 1880 y FD(c)3350 1865 y FA(,)29 b(et)h(le)e (th\351or\350me)-246 1985 y(B.0.2)33 b(assure)g(que)h Fz(r)527 2000 y FD(ess)629 1985 y Fx(\()p Fy(M)787 2000 y FD(c)821 1985 y Fx(\))859 2000 y FC(V)-6 b(ar)969 2009 y Fp(c)1033 1985 y Fs(6)28 b Fz(r)1182 2000 y FD(ess)1284 1985 y Fx(\()p Fy(M)p Fx(\))1480 2000 y FC(V)-6 b(ar)1593 1985 y FA(.)p 3936 1985 V 3940 1923 59 4 v 3940 1985 V 3998 1985 4 66 v -246 2304 a Fr(3.6)131 b(Lien)44 b(en)l(tre)g Fk(B)j Fr(et)d Fk(B)1043 2322 y Fh(c)-246 2524 y FA(La)31 b(pro)5 b(jection)31 b Fz(\031)g Fx(:)d Fy(B)j(!)c(B)786 2539 y FD(c)852 2524 y FA(v)-5 b(a)31 b(en)m(v)m(o)m(y)m(er)j(les)c (sous-espaces)k(caract\351ristiques)d(de)g Fy(M)g FA(dans)g(ceux)i Fy(M)3561 2539 y FD(c)3595 2524 y FA(.)e(On)g(p)s(eut)-246 2644 y(pr\351ciser)39 b(son)g(comp)s(ortemen)m(t.)f(En)h(particulier,)d (dans)j(le)f(cas)h(des)g(p)s(oids)f(con)m(tin)m(us,)h(on)f(v)-5 b(a)38 b(v)m(oir)g(que)h(c'est)h(un)-246 2765 y(isomorphisme.)-246 2907 y Fu(Lemme)d(3.6.1.)49 b Ft(Si)31 b Fx(\010)d Fy(2)g(B)831 2922 y Fv(1)937 2907 y Ft(\(les)i(fonctions)f(\340)i(variation)f(b)-5 b(orn\351)g(e)29 b(nul)5 b(les)31 b(hors)f(d'un)g(ensemble)f (d\351nombr)-5 b(able\))-246 3040 y(v\351ri\034e)34 b Fy(M)p Fx(\010)28 b(=)g Fz(\025)p Fx(\010)35 b Ft(ave)-5 b(c)34 b Fy(j)p Fz(\025)p Fy(j)27 b Fz(>)997 3015 y Fw(c)991 3040 y Fz(R)1066 3011 y FD(")1102 3040 y Ft(,)35 b(alors)f Fx(\010)29 b(=)e(0)p Ft(.)-246 3229 y(D\351monstr)-5 b(ation.)48 b FA(On)33 b(utilise)f(la)h(notation)f Fy(h)p Fz( )t(;)17 b Fx(\010)p Fy(i)29 b Fx(=)1808 3154 y Fw(P)1930 3229 y Fz( )t Fx(\()p Fz(x)p Fx(\)\010\()p Fz(x)p Fx(\))35 b FA(in)m(tro)s(duite)d(dans)i(la)f(preuv)m(e)i(du)f(th\351or\350me) -246 3362 y(3.3.1)16 b(;)33 b(elle)e(v)m(\351ri\034e)i Fy(h)p Fz( )t(;)17 b Fy(M)p Fx(\010)p Fy(i)27 b Fx(=)h Fy(h)1046 3337 y Fw(d)1040 3362 y Fy(M)1160 3333 y FD(")1196 3362 y Fz( )t(;)17 b Fx(\010)p Fy(i)p FA(,)32 b(donc)h(ici)e Fy(h)p Fz( )t(;)17 b Fx(\010)p Fy(i)28 b Fx(=)f Fz(\025)2285 3326 y Fv(\000)p FC(1)2379 3362 y Fy(h)2424 3337 y Fw(d)2418 3362 y Fy(M)2538 3333 y FD(")2575 3362 y Fz( )t(;)17 b Fx(\010)p Fy(i)p FA(.)-246 3533 y(Soit)-39 3508 y Fw(c)-45 3533 y Fz(R)30 3504 y FD(")94 3533 y Fz(<)28 b(C)34 b(<)28 b Fy(j)p Fz(\025)p Fy(j)p FA(,)k(et)h Fz(K)39 b FA(tel)32 b(que)1137 3448 y Fw(\015)1137 3508 y(\015)6 b(d)1192 3533 y Fy(M)1312 3504 y FD(")1348 3459 y(m)1415 3448 y Fw(\015)1415 3508 y(\015)1498 3533 y Fs(6)28 b Fz(K)7 b(C)1770 3497 y FD(m)1837 3533 y FA(.)32 b(Alors)69 3737 y Fy(jh)p Fz( )t(;)17 b Fx(\010)p Fy(ij)27 b Fx(=)h Fy(j)p Fz(\025)p Fy(j)628 3695 y Fv(\000)p FD(m)748 3737 y Fy(jh)821 3711 y Fw(d)815 3737 y Fy(M)935 3708 y FD(")971 3663 y(m)1038 3737 y Fz( )t(;)17 b Fx(\010)p Fy(ij)27 b Fs(6)h Fy(j)p Fz(\025)p Fy(j)1531 3695 y Fv(\000)p FD(m)1652 3652 y Fw(\015)1652 3712 y(\015)1713 3711 y(d)1707 3737 y Fy(M)1827 3708 y FD(")1864 3663 y(m)1930 3737 y Fz( )1997 3652 y Fw(\015)1997 3712 y(\015)2052 3776 y Fv(1)2144 3737 y Fx(V)-8 b(ar)16 b(\010)p Fz(=)p Fx(2)28 b Fs(6)g Fy(j)p Fz(\025)p Fy(j)2726 3695 y Fv(\000)p FD(m)2847 3737 y Fz(K)7 b(C)3014 3695 y FD(m)3097 3737 y Fy(k)p Fz( )t Fy(k)3264 3766 y Fv(1)3355 3737 y Fx(V)-8 b(ar)16 b(\010)p Fz(=)p Fx(2)-246 3915 y FA(En)37 b(passan)m(t)g(\340)f(la)f (limite,)e Fy(h)p Fz( )t(;)17 b Fx(\010)p Fy(i)33 b Fx(=)h(0)p FA(.)i(En)g(prenan)m(t)h(p)s(our)f Fz( )k FA(la)35 b(fonction)g(qui)h (v)-5 b(aut)36 b Fx(1)g FA(en)h Fz(x)f FA(et)g Fx(0)g FA(ailleurs,)e(on)-246 4036 y(obtien)m(t)f Fx(\010\()p Fz(x)p Fx(\))28 b(=)g(0)p FA(.)p 3936 4036 V 3940 3973 59 4 v 3940 4036 V 3998 4036 4 66 v -246 4188 a Fu(Prop)s(osition)36 b(3.6.2.)49 b Ft(Pour)29 b Fy(j)p Fz(\025)p Fy(j)e Fz(>)1158 4163 y Fw(c)1152 4188 y Fz(R)1227 4160 y FD(")1263 4188 y Ft(,)i Fz(\031)j Fx(:)c Fy(B)j(!)c(B)1752 4203 y FD(c)1816 4188 y Ft(r)-5 b(\351)g(alise)29 b(un)g(isomorphisme)e(de)h Fz(E)3055 4203 y FD(\025)3101 4188 y Fx(\()p Fy(M)p Fx(\))g Ft(sur)i Fz(E)3560 4203 y FD(\025)3605 4188 y Fx(\()p Fy(M)3763 4203 y FD(c)3797 4188 y Fx(\))f Ft(\(o\371)-246 4309 y Fz(E)-174 4324 y FD(\025)-128 4309 y Fx(\()p Fy(M)p Fx(\))39 b Ft(d\351signe)g(le)g(sous-esp)-5 b(ac)g(e)38 b(c)-5 b(ar)g(act\351ristique)40 b(asso)-5 b(ci\351)38 b(\340)i(l'action)f(de)g Fy(M)g Ft(sur)h Fy(B)s Ft(,)h(et)e(\340)h(la)f (valeur)h(pr)-5 b(opr)g(e)-246 4429 y Fz(\025)p Ft(\).)-246 4631 y(D\351monstr)g(ation.)48 b FA(On)37 b(a)g(vu)h(dans)g(les)f(prop) s(ositions)f(2.4.1)h(et)g(2.4.2)g(que)2578 4606 y Fw(c)2572 4631 y Fz(R)2647 4602 y FD(")2720 4631 y Fs(>)2839 4606 y Fw(c)2833 4631 y Fz(R)2908 4602 y FD(")2907 4655 y(c)2981 4631 y Fx(=)3111 4606 y Fw(b)3093 4631 y Fz(R)3167 4646 y FD(c)3202 4631 y FA(.)g(Donc)h Fy(j)p Fz(\025)p Fy(j)d Fz(>)3802 4606 y Fw(b)3784 4631 y Fz(R)3858 4646 y FD(c)3929 4631 y Fs(>)-246 4751 y Fz(r)-202 4766 y FD(ess)-100 4751 y Fx(\()p Fy(M)p Fx(\))96 4766 y FC(V)-6 b(ar)210 4751 y FA(,)30 b(ce)h(qui)g(assure)g(que)g Fz(E)1090 4766 y FD(\025)1136 4751 y Fx(\()p Fy(M)p Fx(\))f FA(est)h(bien)f (d\351\034ni,)h(de)g(m\352me)e(que)j Fz(E)2671 4766 y FD(\025)2716 4751 y Fx(\()p Fy(M)2874 4766 y FD(c)2908 4751 y Fx(\))p FA(.)f(On)f(sait)g(alors)f(\(th\351or\350me)-246 4872 y(B.0.2\))k(que)g Fz(\031)j FA(est)e(une)f(surjection)f(de)i Fz(E)1305 4887 y FD(\025)1350 4872 y Fx(\()p Fy(M)p Fx(\))e FA(sur)h Fz(E)1813 4887 y FD(\025)1859 4872 y Fx(\()p Fy(M)2017 4887 y FD(c)2051 4872 y Fx(\))p FA(.)g(Il)e(reste)j(\340)e(v) m(\351ri\034er)h(son)g(injectivit\351.)-246 5027 y(Soit)e Fx(\010)d Fy(2)g Fz(E)218 5042 y FD(\025)264 5027 y Fx(\()p Fy(M)p Fx(\))j FA(telle)f(que)j Fx(\010)952 5042 y FD(c)1015 5027 y Fx(=)27 b(0)p FA(.)32 b(Autremen)m(t)g(dit,)f Fx(\010)d Fy(2)g(B)2153 5042 y Fv(1)2228 5027 y FA(.)k(Soit)e Fz(n)i FA(tel)f(que)h Fx(\()p Fz(\025)17 b Fx(Id)g Fy(\000M)p Fx(\))3349 4991 y FD(n)3396 5027 y Fx(\010)28 b(=)g(0)p FA(.)j(Notons)-246 5161 y Fx(\011)h(=)g(\()p Fz(\025)17 b Fx(Id)f Fy(\000M)p Fx(\))422 5124 y FD(n)p Fv(\000)p FC(1)559 5161 y Fx(\010)36 b FA(:)f Fx(\011)d Fy(2)g(B)998 5176 y Fv(1)1108 5161 y FA(et)j Fx(\011)g FA(v)m(\351ri\034e)g Fy(M)p Fx(\011)d(=)g Fz(\025)p Fx(\011)i FA(a)m(v)m(ec)j Fy(j)p Fz(\025)p Fy(j)31 b Fz(>)2602 5135 y Fw(c)2595 5161 y Fz(R)2670 5132 y FD(")2707 5161 y FA(,)k(donc)h(par)f(le)f (lemme)f(pr\351c\351den)m(t)-246 5281 y Fx(\011)28 b(=)f(0)p FA(,)33 b(i.e.)f Fx(\()p Fz(\025)17 b Fx(Id)f Fy(\000M)p Fx(\))679 5245 y FD(n)p Fv(\000)p FC(1)817 5281 y Fx(\010)28 b(=)f(0)p FA(.)33 b(Ainsi,)e(on)i(fait)e(dimin)m(uer)g(le)h Fz(n)p FA(,)h(jusqu'\340)g Fx(0)p FA(,)f(ce)i(qui)e(donne)h Fx(\010)28 b(=)g(0)p FA(.)p 3936 5281 V 3940 5218 59 4 v 3940 5281 V 3998 5281 4 66 v -246 5431 a Fu(Corollaire)35 b(3.6.3.)49 b Ft(On)30 b(supp)-5 b(ose)29 b(les)g(p)-5 b(oids)29 b Fz(g)1526 5446 y FD(!)1606 5431 y Ft(c)-5 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(d\351terminan)m(t)-246 354 y(p)s(our)48 b(les)f(op)s(\351rateurs)h(de) g(transfert)g(:)f(la)f(trace)i(et)g(le)f(d\351terminan)m(t)g(sharp)h (\(qu'on)g(p)s(ourrait)e(traduire)g(par)-246 474 y(d\351terminan)m(t)i (di\350se)g(en)g(fran\347ais,)f(mais)g(je)h(pr\351f\350re)g(m'en)g (tenir)f(\340)h(la)e(terminologie)f(\020sharp\021)55 b(p)s(our)48 b(ne)g(pas)-246 595 y(in)m(tro)s(duire)29 b(de)i(confusion\).)e(Dans)i(cet)g(article)d([BR96])i(\(et)g(donc)h (dans)f(cette)h(partie\),)f(les)f(p)s(oids)h(son)m(t)h(supp)s(os\351s) -246 715 y(con)m(tin)m(us.)j(On)e(g\351n\351ralisera)g(les)g (r\351sultats)h(\340)f(des)i(p)s(oids)d(discon)m(tin)m(us)i(dans)h(la)d (partie)h(suiv)-5 b(an)m(te.)-246 1039 y Fr(4.1)131 b(D\351\034nition) 44 b(de)g(la)g(trace)g(sharp)-246 1259 y FA(En)30 b(g\351n\351ral,)g (les)f(op)s(\351rateurs)h(de)g(transfert)g(consid\351r\351s)h(dans)f (cet)g(article)e(ne)i(son)m(t)h(pas)f(compacts)f(\(sauf)h(lorsque)-246 1380 y(tous)36 b(les)f(p)s(oids)f(on)m(t)h(des)h(supp)s(orts)g (\034nis\).)f(Il)f(n'y)i(a)f(donc)h(pas)f(d'esp)s(oir)g(de)g (d\351\034nir)g(leur)g(trace)g(\(et)g(leur)g(d\351ter-)-246 1500 y(minan)m(t\))f(de)h(fa\347on)g(canonique)g(\(i.e.)f(comme)g (trace)h(et)g(d\351terminan)m(t)g(d'op)s(\351rateurs)g(n)m (ucl\351aires\).)g(Il)f(faut)h(donc)-246 1620 y(p)s(oser)e(une)g (d\351\034nition)f Ft(ad)i(ho)-5 b(c)p FA(,)32 b(en)h(esp)s(\351ran)m (t)h(qu'elle)e(v)m(\351ri\034era)h(de)g(b)s(onnes)g(propri\351t\351s.) -246 1776 y(On)g(commence)f(par)h(d\351\034nir)f(une)h(\020trace\021,)f (not\351e)h Fx(T)-8 b(r)1746 1734 y FC(#)1808 1776 y FA(,)33 b(p)s(our)f(les)h(op)s(\351rateurs)g(\351l\351men)m(taires)f(:) -246 1931 y Fu(D\351\034nition)k(4.1.1.)50 b Ft(Si)34 b Fy(L)p Fx(\010\()p Fz(x)p Fx(\))28 b(=)g Fz(g)t Fx(\()p Fz(x)p Fx(\)\010\()p Fz( )t(x)p Fx(\))p Ft(,)35 b(on)f(p)-5 b(ose)34 b Fx(T)-8 b(r)2078 1889 y FC(#)2141 1931 y Fx(\()p Fy(L)p Fx(\))27 b(=)2416 1851 y Fw(R)2509 1892 y FC(1)p 2509 1908 36 4 v 2509 1965 a(2)2571 1931 y Fx(sgn)q(\()p Fz( )t Fx(\()p Fz(x)p Fx(\))22 b Fy(\000)h Fz(x)p Fx(\))17 b(d)p Fz(g)t Fx(\()p Fz(x)p Fx(\))p Ft(.)-246 2086 y FA(\(la)45 b(notation)f Fx(sgn)i FA(d\351signe)g(la)f(fonction)f(qui)i (v)-5 b(aut)45 b Fx(1)h FA(sur)g Fx(\(0)p Fz(;)17 b Fx(+)p Fy(1)p Fx(\))p FA(,)45 b Fy(\000)p Fx(1)h FA(sur)g Fx(\()p Fy(\0001)p Fz(;)17 b Fx(0\))45 b FA(et)h(est)g(n)m(ulle)f(en)i Fx(0)p FA(\).)-246 2206 y(L'in)m(t\351grale)32 b(utilis\351e)f(ici)g (est)i(une)g(in)m(t\351grale)e(de)i(Stieltjes)f(classique.)-246 2362 y(On)h(d\351\034nit)f(ensuite)h(la)f(trace)h(d'un)g(op)s (\351rateur)f(g\351n\351ral)g(:)-246 2517 y Fu(D\351\034nition)k (4.1.2.)50 b Ft(Si)34 b Fy(M)p Fx(\010\()p Fz(x)p Fx(\))28 b(=)1160 2442 y Fw(P)1281 2517 y Fz(g)1328 2532 y FD(!)1379 2517 y Fx(\()p Fz(x)p Fx(\)\010\()p Fz( )1681 2532 y FD(!)1732 2517 y Fz(x)p Fx(\))g(=)1957 2442 y Fw(P)2078 2517 y Fy(L)2147 2532 y FD(!)2198 2517 y Fx(\010\()p Fz(x)p Fx(\))p Ft(,)35 b(on)g(p)-5 b(ose)34 b Fx(T)-8 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b(si)e(l'une)g(est)h(croissan)m(te) g(et)g(l'autre)f(d\351croissan)m(te,)h Fz( )2053 -13 y FC(2)2093 -28 y Fz( )2156 -13 y FC(1)2231 -28 y FA(a)f(un)h(unique)g (p)s(oin)m(t)e(\034xe)j Fz(a)p FA(,)e(et)h Fx(T)-8 b(r)3544 -70 y FC(#)3607 -28 y Fx(\()p Fy(L)3714 -13 y FC(2)3753 -28 y Fy(L)3822 -13 y FC(1)3862 -28 y Fx(\))30 b(=)-246 92 y Fz(g)-199 107 y FC(1)-160 92 y Fx(\()p Fz(a)p Fx(\))p Fz(g)14 107 y FC(2)54 92 y Fx(\()p Fz( )155 107 y FC(1)194 92 y Fz(a)p Fx(\))p FA(.)k(De)h(m\352me,)e Fz( )869 107 y FC(1)909 92 y Fz( )972 107 y FC(2)1045 92 y FA(a)h(un)g(unique)g(p)s (oin)m(t)f(\034xe)h Fz(b)p FA(\()p Fx(=)c Fz( )2274 107 y FC(1)2314 92 y Fx(\()p Fz(a)p Fx(\))p FA(\))k(et)g Fx(T)-8 b(r)2728 51 y FC(#)2791 92 y Fx(\()p Fy(L)2898 107 y FC(1)2937 92 y Fy(L)3006 107 y FC(2)3045 92 y Fx(\))30 b(=)f Fz(g)3265 107 y FC(1)3305 92 y Fx(\()p Fz( )3406 107 y FC(2)3445 92 y Fz(b)p Fx(\))p Fz(g)3571 107 y FC(2)3611 92 y Fx(\()p Fz(b)p Fx(\))p FA(.)34 b(C'est)-246 213 y(la)e(m\352me)f(v)-5 b(aleur.)p 3936 213 4 66 v 3940 150 59 4 v 3940 213 V 3998 213 4 66 v -246 411 a(On)33 b(p)s(eut)g(\351tablir)e(un)h(lien)g(en)m(tre)h(les)g(traces)g(de)g Fy(M)f FA(et)h(de)1996 386 y Fw(c)1972 411 y Fy(M)p FA(,)f(qui)g(nous)h (sera)h(utile)d(par)h(la)f(suite.)-246 559 y Fu(Prop)s(osition)36 b(4.2.4.)49 b Ft(On)34 b(a)h Fx(T)-8 b(r)1019 517 y FC(#)1082 559 y Fx(\()1144 534 y Fw(c)1120 559 y Fy(M)p Fx(\))27 b(=)h Fy(\000)17 b Fx(T)-8 b(r)1603 517 y FC(#)1666 559 y Fx(\()p Fy(M)p Fx(\))p Ft(.)-246 757 y(D\351monstr)j(ation.)48 b FA(Il)35 b(su\036t)h(de)g(le)g(prouv)m(er)g(p)s(our)g(un)g(op)s (\351rateur)g(\351l\351men)m(taire)e Fy(L)p FA(.)i(Notons)g Fz(")d Fx(=)g(1)i FA(si)h Fz( )j FA(cro\356t)c(et)-246 878 y Fy(\000)p Fx(1)e FA(si)f(elle)g(d\351cro\356t.)g(On)g(a)h(alors) 357 1108 y Fx(T)-8 b(r)457 1066 y FC(#)520 1108 y Fx(\()579 1083 y Fw(b)558 1108 y Fy(L)p Fx(\))28 b(=)796 973 y Fw(Z)912 1108 y Fz(")968 1041 y Fx(1)p 968 1085 49 4 v 968 1177 a(2)1043 1108 y(sgn)q(\()p Fz( )1290 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Fd(/)p Fc(/)p 1775 2200 200 4 v 1999 2225 a Fy(F)2071 2240 y FD(c)3882 2058 y FA(\(1\))-246 2424 y(Sur)e(c)m(hacun)g(de)f(ces)h(espaces)h (agit)d(un)h(op)s(\351rateur)g(\(resp)s(ectiv)m(emen)m(t)2347 2399 y Fw(f)2324 2424 y Fy(M)o FA(,)2521 2399 y Fw(f)2497 2424 y Fy(M)2617 2439 y FD(c)2652 2424 y FA(,)g Fy(M)f FA(et)i Fy(M)3081 2439 y FD(c)3115 2424 y FA(\),)f(et)g(tout)g(est)h (comm)m(u-)-246 2555 y(tatif.)i(Ces)j(op)s(\351rateurs)f(on)m(t)f(des)i (ra)m(y)m(ons)f(sp)s(ectraux)h(resp)s(ectifs)2189 2530 y Fw(e)2171 2555 y Fz(R)q FA(,)2322 2530 y Fw(e)2304 2555 y Fz(R)2378 2570 y FD(c)2413 2555 y FA(,)f Fz(R)g FA(et)g Fz(R)2765 2570 y FD(c)2800 2555 y FA(.)f(Le)h(th\351or\350me)g (B.0.2)f(garan)m(tit)-246 2675 y(que)25 b(ces)f(ra)m(y)m(ons)h(sp)s (ectraux)g(d\351croissen)m(t)g(le)d(long)h(des)h(\035\350c)m(hes.)i(On) d(sait)g(en)h(plus)f(que)2915 2650 y Fw(e)2897 2675 y Fz(R)29 b Fx(=)3121 2650 y Fw(e)3103 2675 y Fz(R)3177 2690 y FD(c)3212 2675 y FA(,)24 b(par)f(la)f(prop)s(osition)-246 2795 y(2.4.5)32 b(\(car)h(les)f Fw(e)-55 b Fz(g)376 2810 y FD(!)459 2795 y FA(son)m(t)33 b(con)m(tin)m(us\).)-246 3113 y Fr(5.2)131 b(Injectivit\351)46 b(de)e Fa(\013)-246 3370 y Fu(Lemme)37 b(5.2.1.)49 b Ft(Soit)35 b Fx(\010)28 b Fy(2)873 3345 y Fw(e)852 3370 y Fy(B)38 b Ft(nul)5 b(le)35 b(hors)f(des)h Fz(I)1625 3385 y FD(s)1662 3370 y Ft(,)f(et)h(v\351ri\034ant)2249 3345 y Fw(f)2226 3370 y Fy(M)o Fx(\010)28 b(=)g Fz(\025)p Fx(\010)35 b Ft(ave)-5 b(c)34 b Fy(j)p Fz(\025)p Fy(j)27 b Fz(>)3177 3316 y Fw(e)3185 3345 y(b)3167 3370 y Fz(R)q Ft(.)34 b(A)n(lors)h Fx(\010)28 b(=)g(0)p Ft(.)-246 3556 y(D\351monstr)-5 b(ation.)48 b FA(P)m(our)d Fz( )j FA(b)s(orn\351e)d(qui,)g(sur)g(c)m (haque)h Fz(I)1910 3571 y FD(s)1947 3556 y FA(,)f(est)g(non)g(n)m(ulle) f(en)h(au)f(plus)h(un)g(p)s(oin)m(t,)f(on)g(d\351\034nit)-246 3676 y Fy(h)p Fz( )t(;)17 b Fx(\010)p Fy(i)52 b Fx(=)192 3602 y Fw(P)297 3705 y FD(x)358 3676 y Fz( )t Fx(\()p Fz(x)p Fx(\)\010\()p Fz(x)p Fx(\))p FA(.)47 b(C'est)h(bien)f 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b(o\371)h(on)f(fait)g(des)h(mo)s(di\034cations)e(juste)i(sur) g(les)g Fz(I)1934 5552 y FD(s)2005 5537 y FA(et)g(hors)g(des)h(cycles.) g(Lorsque)f(l'h)m(yp)s(oth\350se)h Fx(\()p Fz(G)p Fx(\))f FA(est)-246 5658 y(v)m(\351ri\034\351e,)c(on)f(p)s(eut)g(donc)h(mo)s (di\034er)d Fw(e)-55 b Fz(g)1121 5673 y FD(!)1201 5658 y FA(sur)31 b(les)f(in)m(terv)-5 b(alles)28 b Fz(I)2002 5673 y FD(s)2069 5658 y FA(tels)i(que)g Fz(g)2471 5673 y FD(!)2551 5658 y FA(ne)h(soit)e(pas)h(con)m(tin)m(u)h(en)f Fz(s)p FA(,)g(puisqu'ils)-246 5778 y(n'in)m(terviennen)m(t)k(pas)f (dans)g(des)g(cycles.)1832 6027 y(22)p eop %%Page: 23 23 23 22 bop -246 -28 a FA(On)40 b(d\351\034nit)f(en)g(particulier)872 -53 y Fw(f)848 -28 y Fy(M)968 -64 y Fv(0)991 -28 y Fx(\010\()p Fz(x)p Fx(\))h(=)1347 -102 y Fw(P)1469 -28 y(e)-55 b Fz(g)1520 -64 y Fv(0)1516 -3 y FD(!)1566 -28 y Fx(\010\()1691 -54 y Fw(e)1674 -28 y Fz( )1737 -13 y FD(!)1788 -28 y Fz(x)p Fx(\))p FA(,)39 b(o\371)h Fw(e)-55 b Fz(g)2141 -64 y Fv(0)2137 -3 y FD(!)2226 -28 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b(m\352me)e(en)i(utilisan)m(t)e(la)g(prop)s(osition)g(5.3.3) 16 b(!)1832 6027 y(23)p eop %%Page: 24 24 24 23 bop -246 -28 a Fr(5.4)131 b(Conclusion)-246 206 y Fu(Th\351or\350me)37 b(5.4.1.)50 b Ft(On)36 b(supp)-5 b(ose)35 b(que)i Fx(\()p Fz(G)p Fx(\))f Ft(est)h(v\351ri\034\351)-5 b(e.)35 b(A)n(lors)2282 181 y Fw(e)2264 206 y Fz(R)d Fx(=)2495 181 y Fw(e)2477 206 y Fz(R)2551 221 y FD(c)2617 206 y Fx(=)e Fz(R)i Fx(=)f Fz(R)3010 221 y FD(c)3081 206 y Ft(et)3205 152 y Fw(e)3213 181 y(b)3195 206 y Fz(R)h Fx(=)3418 152 y Fw(e)3426 181 y(b)3408 206 y Fz(R)3483 221 y FD(c)3548 206 y Fx(=)3673 181 y Fw(b)3655 206 y Fz(R)g Fx(=)3885 181 y Fw(b)3867 206 y Fz(R)3941 221 y FD(c)3976 206 y Ft(.)-246 327 y(De)j(plus,)f(sur)h(le)g(diagr)-5 b(amme)1684 558 y Fw(e)1663 584 y Fy(B)1627 723 y FD(\013)1698 823 y Fd(\017)p Fc(\017)p 1696 823 4 215 v 1843 510 a FD(\031)1956 535 y Fd(/)p Fc(/)p 1757 537 200 4 v 2001 551 a Fw(e)1981 576 y Fy(B)2046 591 y FD(c)2056 718 y(\013)2101 726 y Fp(c)2031 815 y Fd(\017)p 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5659 V 3998 5659 4 66 v 1832 6027 a(24)p eop %%Page: 25 25 25 24 bop -246 -28 a FA(Comme)35 b Fx(\()p Fz(G)230 -64 y Fv(0)253 -28 y Fx(\))g FA(implique)e Fx(\()p Fz(G)p Fx(\))p FA(,)i(les)g(conclusions)g(son)m(t)h(en)g(particulier)d(vraies) i(lorsque)h(les)f(p)s(oids)g(son)m(t)h(con)m(tin)m(us)-246 92 y(aux)d(p)s(oin)m(ts)f(p)s(\351rio)s(diques.)-246 248 y(Remarquons)38 b(que)h(le)f(th\351or\350me)g(5.4.1)f(p)s(ermet)h (de)g(v)m(oir)g(que)g Fx(Det)2301 212 y FC(#)2402 248 y FA(traduit)f(aussi)h(les)g(propri\351t\351s)g(sp)s(ectrales)-246 369 y(de)44 b(l'action)d(de)i Fy(M)g FA(sur)h(le)e(quotien)m(t)h Fy(B)1302 384 y FD(c)1337 369 y FA(.)g(Comme)f Fx(Det)1931 332 y FC(#)2037 369 y FA(ne)h(d\351p)s(end)h(des)g(v)-5 b(aleurs)43 b(des)h(p)s(oids)e(que)i(presque)-246 489 y(partout)32 b(\(car)f(c'est)i(le)e(cas)h(de)g(la)e(mesure)i(d)q Fz(g)1459 504 y FD(!)1509 489 y FA(\),)f(on)h(aurait)e(directemen)m(t)i (tout)f(pu)h(form)m(uler)e(dans)i(le)f(quotien)m(t.)-246 609 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FC(0)1178 1718 y Fs(6)h Fz(:)17 b(:)g(:)38 b Fs(6)h Fz(a)1614 1733 y FD(n)1700 1718 y FA(de)g Fz(X)8 b FA(.)39 b(On)g(note)g(alors)e Fx(v)-5 b(ar)17 b Fz(')39 b FA(la)e(b)s(orne)i(sup)s(\351rieure)h(de)f(ces)-246 1838 y(sommes.)e(On)f(note)h(aussi)g Fx(V)-8 b(ar)16 b Fz(')35 b Fx(=)g(sup)1305 1758 y Fw(\010)1364 1838 y Fy(j)p Fz(')p Fx(\()p Fz(a)1545 1853 y FC(0)1584 1838 y Fx(\))p Fy(j)24 b Fx(+)1775 1764 y Fw(P)1897 1838 y Fy(j)p Fz(')p Fx(\()p Fz(a)2078 1853 y FD(i)2106 1838 y Fx(\))h Fy(\000)g Fz(')p Fx(\()p Fz(a)2424 1853 y FD(i)p Fv(\000)p FC(1)2543 1838 y Fx(\))p Fy(j)f Fx(+)h Fy(j)p Fz(')p Fx(\()p Fz(a)2915 1853 y FD(n)2962 1838 y Fx(\))p Fy(j)3028 1758 y Fw(\011)3085 1838 y FA(.)37 b(Si)f(on)h(note)g Fy(B)s Fx(\()p Fz(X)8 b Fx(\))36 b FA(les)-246 1959 y(fonctions)29 b(\340)f(v)-5 b(ariation)26 b(b)s(orn\351e)j(sur)h Fz(X)8 b FA(,)28 b Fx(V)-8 b(ar)29 b FA(est)g(une)h(norme)e(sur)h Fy(B)s Fx(\()p Fz(X)8 b Fx(\))29 b FA(\(\351quiv)-5 b(alen)m(te)29 b(\340)f Fy(k)k(k)3240 1988 y Fv(1)3329 1959 y Fx(+)14 b(v)-5 b(ar)p FA(,)29 b(mais)e(plus)-246 2079 y(agr\351able)32 b(\340)g(manipuler\),)f(qui)h(en)h(fait)e(un)i(espace)h(de)f(Banac)m (h.)-246 2235 y(On)g(notera)f Fy(B)k FA(au)d(lieu)e(de)i Fy(B)s Fx(\()p Fq(R)5 b Fx(\))p FA(.)-246 2391 y(Si)43 b Fz(X)51 b FA(est)43 b(un)h(in)m(terv)-5 b(alle)41 b(b)s(orn\351)i(de) h Fq(R)5 b FA(,)49 b(on)43 b(a)g(un)g(plongemen)m(t)g(isom\351trique)f (de)h Fy(B)s Fx(\()p Fz(X)8 b Fx(\))43 b FA(dans)h Fy(B)s FA(,)g(obten)m(u)g(en)-246 2511 y(prolongean)m(t)37 b(les)h(fonctions)f (par)g Fx(0)h FA(en)g(dehors)h(de)f Fz(X)8 b FA(.)37 b(Dans)i(la)d(suite,)i(on)g(iden)m(ti\034era)f(donc)h(une)g(fonction)f (de)-246 2631 y Fy(B)s Fx(\()p Fz(X)8 b Fx(\))33 b FA(a)m(v)m(ec)h(son) f(prolongemen)m(t.)-246 2787 y(Les)h(propri\351t\351s)e(qui)h(suiv)m (en)m(t)g(d\351coulen)m(t)g(directemen)m(t)g(des)h(d\351\034nitions.) -246 2934 y Fu(Prop)s(osition)i(A.1.1.)49 b Ft(Soient)32 b Fz(X)41 b Ft(une)33 b(p)-5 b(artie)34 b(de)e Fq(R)45 b Ft(et)33 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)2518 384 y FC(1)2558 369 y Fz(;)17 b( )2665 384 y FC(2)2737 369 y FA(et)33 b Fz(')p FA(.)f(Elle)g(v)m(\351ri\034e)1129 479 y Fw(\014)1129 539 y(\014)1129 599 y(\014)1129 659 y(\014)1162 488 y(Z)1279 624 y Fz( )1342 639 y FC(1)1398 624 y Fy(r)p Fz(')17 b( )1625 639 y FC(2)1664 479 y Fw(\014)1664 539 y(\014)1664 599 y(\014)1664 659 y(\014)1725 624 y Fs(6)29 b Fy(k)o Fz( )1943 639 y FC(1)1983 624 y Fy(k)2033 653 y Fv(1)2124 624 y Fy(k)p Fz( )2237 639 y FC(2)2277 624 y Fy(k)2327 653 y Fv(1)2418 624 y Fx(v)-5 b(ar)17 b Fz(')-246 925 y FA(Il)27 b(faut)f(remarquer)h(que)h(la)e(place)h(des) h(fonctions)f(dans)g(l'in)m(t\351grale)e(est)j(imp)s(ortan)m(te)d(:)i (en)h(g\351n\351ral,)3450 845 y Fw(R)3534 925 y Fz( )3597 940 y FC(1)3636 925 y Fz( )3699 940 y FC(2)3756 925 y Fy(r)p Fz(')f Fy(6)p Fx(=)-246 965 y Fw(R)-163 1046 y Fz( )-100 1061 y FC(1)-43 1046 y Fy(r)p Fz(')17 b( )184 1061 y FC(2)223 1046 y FA(.)-246 1201 y(Cette)37 b(in)m(t\351grale)e (est)h(sym\351trique)h(en)f Fz( )1279 1216 y FC(1)1355 1201 y FA(et)g Fz( )1535 1216 y FC(2)1610 1201 y FA(\(d'o\371)g(son)h (nom\).)e(En)h(fait,)f(c'est)i(un)f(analogue)f(sym\351tris\351)h(de) -246 1322 y(l'in)m(t\351grale)31 b(a)m(v)m(ec)j(des)f(semi-mesures)f (d\351\034nie)h(dans)g([Rue91].)-246 1615 y Fu(A.2.3)112 b(Premi\350res)36 b(propri\351t\351s)g(de)i(l'in)m(t\351grale)d (sym\351trique)-246 1835 y FA(La)e(prop)s(osition)d(qui)i(suit)h (traduit)e(la)g(p)s(ossibilit\351)f(d'in)m(t\351grer)j(par)f(parties)g (:)-246 1984 y Fu(Prop)s(osition)k(A.2.3.)49 b Ft(Si)34 b Fz( )881 1999 y FC(1)921 1984 y Fz(;)17 b( )1028 1999 y FC(2)1102 1984 y Ft(et)35 b Fz(')1278 1999 y FC(1)1318 1984 y Fz(;)17 b(:)g(:)g(:)32 b(;)17 b(')1617 1999 y FD(m)1718 1984 y Ft(sont)35 b(\340)f(variation)h(b)-5 b(orn\351)g(e,)33 b(alors)583 2123 y Fw(Z)699 2258 y Fz( )762 2273 y FC(1)819 2258 y Fy(r)p Fx(\()p Fz(')1004 2273 y FC(1)1060 2258 y Fz(:)17 b(:)g(:)f(')1255 2273 y FD(m)1321 2258 y Fx(\))p Fz( )1422 2273 y FC(2)1490 2258 y Fx(=)1634 2134 y FD(m)1593 2163 y Fw(X)1608 2373 y FD(i)p FC(=1)1754 2123 y Fw(Z)1870 2258 y Fz( )1933 2273 y FC(1)1973 2258 y Fz(')2037 2273 y FC(1)2093 2258 y Fz(:)h(:)g(:)f(')2288 2273 y FD(i)p Fv(\000)p FC(1)2423 2258 y Fy(r)p Fz(')2570 2273 y FD(i)2614 2258 y Fz(')2678 2273 y FD(i)p FC(+1)2813 2258 y Fz(:)h(:)g(:)f(')3008 2273 y FD(m)3075 2258 y Fz( )3138 2273 y FC(2)-246 2537 y Ft(D\351monstr)-5 b(ation.)48 b FA(Il)31 b(su\036t)j(de)f(le)f(prouv) m(er)h(p)s(our)g Fz(m)28 b Fx(=)f(2)p FA(.)32 b(On)h(utilise)e(alors)g (la)h(factorisation)-100 2740 y Fz(')-36 2755 y FC(1)3 2740 y Fx(\()p Fz(a)92 2755 y FD(i)121 2740 y Fx(\))p Fz(')223 2755 y FC(2)262 2740 y Fx(\()p Fz(a)351 2755 y FD(i)379 2740 y Fx(\))22 b Fy(\000)h Fz(')603 2755 y FC(1)642 2740 y Fx(\()p Fz(a)731 2755 y FD(i)p Fv(\000)p FC(1)850 2740 y Fx(\))p Fz(')952 2755 y FC(2)991 2740 y Fx(\()p Fz(a)1080 2755 y FD(i)p Fv(\000)p FC(1)1198 2740 y Fx(\))28 b(=)1368 2660 y Fw(\000)1413 2740 y Fz(')1477 2755 y FC(1)1517 2740 y Fx(\()p Fz(a)1606 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b Fz(')p Fx(\()p Fz(x)p Fx(\))p 3936 876 4 66 v 3940 814 59 4 v 3940 876 V 3998 876 4 66 v -246 1082 a FA(En)h(fait,)d(on)i(a)g(c)m (hoisi)f(l'expression)h(de)41 b Fw(e)-68 b Fz(')28 b FA(sp)s(\351cialemen)m(t)f(p)s(our)h(que)h(\347a)f(marc)m(he,)g(en)g (s'arrangean)m(t)g(p)s(our)g(\351liminer)-246 1203 y(les)33 b Fz(')p Fx(\()p Fz(x)53 1218 y Fv(\000)112 1203 y Fx(\))p FA(.)-246 1356 y Fu(Th\351or\350me)k(A.3.2.)49 b Ft(Soit)35 b Fz(')27 b Fy(2)i(B)1072 1371 y FC(0)1146 1356 y Ft(tel)5 b(le)35 b(que)g Fy(8)p Fz( )d Fy(2)c(B)s Fz(;)1891 1272 y Fw(\014)1891 1332 y(\014)1924 1276 y(R)2007 1356 y Fz( )21 b Fy(r)p Fz(')2238 1272 y Fw(\014)2238 1332 y(\014)2299 1356 y Fs(6)28 b Fz(C)23 b Fy(k)p Fz( )t Fy(k)2664 1385 y Fv(1)2739 1356 y Ft(.)34 b(A)n(lors)h Fx(V)-8 b(ar)16 b Fz(')28 b Fs(6)g Fx(3)p Fz(C)7 b Ft(.)-246 1562 y(D\351monstr)-5 b(ation.)48 b FA(Comme)33 b Fz(')h FA(tend)h(v)m(ers)h Fx(0)e FA(en)h(l'in\034ni,)d Fx(v)-5 b(ar)16 b Fz(')31 b Fx(=)g(V)-8 b(ar)16 b Fz(')p FA(.)34 b(Il)g(su\036t)h(donc)g (d'\351tudier)f Fx(v)-5 b(ar)16 b Fz(')p FA(.)35 b(P)m(our)-246 1683 y(cela,)d(on)h(v)-5 b(a)32 b(utiliser)f(l'expression)i(de)g Fz(')f FA(donn\351e)i(par)e(la)g(prop)s(osition)e(pr\351c\351den)m(te.) -246 1839 y(Notons)j Fz(\021)t Fx(\()p Fz(x)p Fx(\))28 b(=)402 1758 y Fw(R)485 1839 y Fz(\037)546 1854 y FC(\()p Fv(\0001)p FD(;x)p FC(\))807 1839 y Fy(r)p Fz(')p FA(.)k(Mon)m(trons)i (que)f Fx(v)-5 b(ar)17 b Fz(\021)31 b Fs(6)d Fz(C)7 b FA(.)33 b(Si)f Fz(a)2270 1854 y FC(0)2337 1839 y Fz(<)27 b(:)17 b(:)g(:)28 b(<)f(a)2737 1854 y FD(n)2784 1839 y FA(,)33 b(on)f(a)-238 2017 y Fw(X)-78 2111 y Fy(j)p Fz(\021)t Fx(\()p Fz(a)91 2126 y FD(i)119 2111 y Fx(\))22 b Fy(\000)h Fz(\021)t Fx(\()p Fz(a)420 2126 y FD(i)p Fv(\000)p FC(1)538 2111 y Fx(\))p Fy(j)k Fx(=)735 2017 y Fw(X)895 2111 y Fz(")941 2126 y FD(i)969 2030 y Fw(\000)1015 2111 y Fz(\021)t Fx(\()p Fz(a)1156 2126 y FD(i)1184 2111 y Fx(\))22 b Fy(\000)g Fz(\021)t Fx(\()p Fz(a)1484 2126 y FD(i)p Fv(\000)p FC(1)1603 2111 y Fx(\))1641 2030 y Fw(\001)1714 2111 y Fx(=)1817 1976 y Fw(Z)1934 2001 y(\020)1993 2017 y(X)2154 2111 y Fz(")2200 2126 y FD(i)2227 2111 y Fz(\037)2288 2127 y FC([)p FD(a)2345 2137 y Fp(i)p Fl(\000)p Fm(1)2450 2127 y FD(;a)2507 2137 y Fp(i)2534 2127 y FC(\))2565 2001 y Fw(\021)2658 2111 y Fy(r)p Fz(')28 b Fs(6)g Fz(C)3031 1997 y Fw(\015)3031 2056 y(\015)3031 2116 y(\015)3087 2017 y(X)3247 2111 y Fz(")3293 2126 y FD(i)3321 2111 y Fz(\037)3382 2127 y FC([)p FD(a)3439 2137 y Fp(i)p Fl(\000)p Fm(1)3544 2127 y FD(;a)3601 2137 y Fp(i)3628 2127 y FC(\))3659 1997 y Fw(\015)3659 2056 y(\015)3659 2116 y(\015)3714 2180 y Fv(1)3817 2111 y Fs(6)g Fz(C)-246 2511 y FA(La)36 b(fonction)i Fw(e)-58 b Fz(\021)s Fx(\()p Fz(x)p Fx(\))34 b(=)f(lim)741 2526 y FD(y)r Fv(\045)p FD(x)909 2431 y Fw(R)992 2511 y Fz(\037)1053 2526 y FC(\()p Fv(\0001)p FD(;y)r FC(\))1311 2511 y Fy(r)p Fz(')j FA(est)g(la)f(limite)e(simple)h(des)j(fonctions)e Fz(\021)t Fx(\()p Fz(x)25 b Fy(\000)f Fx(1)p Fz(=n)p Fx(\))p FA(,)36 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y(terminera)c(puisque)i Fz(h)e FA(est)i(de)f(v)-5 b(ariation)27 b(\034nie\).)h(Si)g Fx(\010)g Fy(2)g(B)s Fx(\(\()p Fy(\000)p Fx(1)p Fz(;)17 b Fx(1\)\))p FA(,)29 b(on)g(d\351\034nit)f Fz(B)5 b Fx(\010)30 b FA(comme)e(la)f(fonction)h(\351gale)-246 454 y(\340)34 b Fx(\010)g FA(sur)h(les)e Fz(a)299 469 y FD(i)328 454 y FA(,)h(et)g(a\036ne)g(sur)g(les)g Fx([)p Fz(a)1150 469 y FD(i)1179 454 y Fz(;)17 b(a)1274 469 y FD(i)p FC(+1)1392 454 y Fx(])p FA(.)34 b(L'op)s(\351rateur)g Fz(B)39 b FA(est)34 b(con)m(tin)m(u)g(\(on)g(a)f Fx(V)-8 b(ar\()p Fz(B)5 b Fx(\010\))30 b Fs(6)h Fx(V)-8 b(ar\(\010\))p FA(\),)34 b(et)g(de)-246 574 y(rang)e(\034ni)g(\(puisque)i(les)e (\351l\351men)m(ts)h(de)g(son)g(image)e(son)m(t)i(d\351\034nis)g(par)f (leurs)h(v)-5 b(aleurs)32 b(en)h Fz(n)22 b Fx(+)g(1)33 b FA(p)s(oin)m(ts\).)-246 730 y(L'op)s(\351rateur)g Fz(A)357 745 y FD(h)402 730 y Fz(B)38 b FA(est)33 b(de)g(rang)f(\034ni,)h(on)f (v)-5 b(a)33 b(v)m(oir)f(qu'il)f(appro)s(c)m(he)j(bien)e Fz(A)2593 745 y FD(h)2638 730 y FA(.)h(Fixons)f Fx(\010)c Fy(2)g(B)s Fx(\(\()p Fy(\000)p Fx(1)p Fz(;)17 b Fx(1\)\))p FA(.)33 b(Notons)-246 850 y Fz(\037)k Fx(=)f Fz(B)5 b Fx(\010)26 b Fy(\000)g Fx(\010)38 b FA(:)f Fz(\037)h FA(est)g(n)m(ulle)f(sur)h(les)f Fz(a)1276 865 y FD(i)1305 850 y FA(,)g(et)h Fx(V)-8 b(ar)16 b Fz(\037)36 b Fs(6)h Fx(2)17 b(V)-8 b(ar)16 b(\010)p FA(.)38 b(Soit)e Fz(m)2526 865 y FD(i)2592 850 y FA(le)h(maxim)m(um)e(de)j Fz(\037)g FA(sur)g(l'in)m(terv)-5 b(alle)-246 971 y Fx([)p Fz(a)-168 986 y FD(i)-139 971 y Fz(;)17 b(a)-44 986 y FD(i)p FC(+1)74 971 y Fx(])p FA(.)43 b(Comme)f Fz(\037)g FA(est)i(n)m(ulle)d(sur)i(les) g Fz(a)1426 986 y FD(i)1454 971 y FA(,)1524 896 y Fw(P)1646 971 y Fz(m)1731 986 y FD(i)1804 971 y Fs(6)i Fx(V)-8 b(ar)17 b Fz(\037)p FA(.)42 b(De)i(plus,)e(le)g(calcul)g(du)h(d\351but) g(de)g(la)f(preuv)m(e)-246 1091 y(mon)m(tre)33 b(que)336 1343 y Fx(V)-8 b(ar\()p Fz(A)599 1358 y FD(h)644 1343 y Fx(\010)22 b Fy(\000)h Fz(A)909 1358 y FD(h)954 1343 y Fz(B)5 b Fx(\010\))28 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3940 1735 V 3998 1735 4 66 v -246 2097 a FB(R\351f\351rences)-246 2327 y FA([Bai01])174 b(Baillif)25 b(\(Mathieu\).)k(\025)37 b(Kneading)29 b(op)s(erators,)g (sharp)g(determinan)m(ts)g(and)h(w)m(eigh)m(ted)g(Lefsc)m(hetz)h(zeta) 226 2447 y(functions)h(in)g(higher)g(dimension.)f(\025)43 b(Preprin)m(t,)33 b(2001.)-246 2589 y([Bal00])174 b(Baladi)30 b(\(Viviane\).)h(\025)42 b Ft(Positive)35 b(T)-7 b(r)i(ansfer)33 b(Op)-5 b(er)g(ators)34 b(and)f(De)-5 b(c)g(ay)34 b(of)h(Corr)-5 b(elations)p FA(.)31 b(\025)42 b(Singap)s(ore,)226 2710 y(W)-8 b(orld)31 b(Scien)m(ti\034c,)i(2000,)e Ft(A)-5 b(dvanc)g(e)g(d)34 b(Series)g(in)h(Nonline)-5 b(ar)34 b(Dynamics)p FA(,)d(v)m(olume)h(16.)-246 2852 y([BK90])175 b(Baladi)22 b(\(Viviane\))h(et)h(Keller)f(\(Gerhard\).)h(\025)29 b(Zeta)24 b(functions)g(and)h(transfer)f(op)s(erators)g(for)g (piecewise)226 2972 y(monotone)k(transformations.)g Ft(Communic)-5 b(ations)30 b(in)h(Mathematic)-5 b(al)32 b(Physics)p FA(,)d(v)m(ol.)g(127,)f(1990,)h(pp.)226 3093 y(459\025477.)-246 3235 y([BKRS97])49 b(Baladi)25 b(\(V.\),)k(Kitaev)e(\(A.\),)h(Ruelle)f (\(D.\))h(et)g(Semmes)g(\(S.\).)g(\025)35 b(Sharp)29 b(determinan)m(ts)e(and)h(kneading)226 3356 y(op)s(erators)k(for)g (holomorphic)e(maps.)i Ft(Pr)-5 b(o)g(c.)35 b(Steklov.)f(Math.)h(Inst.) p FA(,)d(v)m(ol.)g(216,)g(1997,)g(pp.)h(186\025228.)-246 3498 y([BR96])179 b(Baladi)40 b(\(Viviane\))h(et)i(Ruelle)e(\(Da)m (vid\).)i(\025)73 b(Sharp)42 b(determinan)m(ts.)h Ft(Inventiones)f (Mathematic)-5 b(ae)p FA(,)226 3618 y(v)m(ol.)32 b(123,)g(1996,)f(pp.)i (553\025574.)-246 3761 y([GGK00])90 b(Goh)m(b)s(erg)32 b(\(Israel\),)g(Goldb)s(erg)f(\(Seymour\))i(et)g(Krupnik)f(\(Nah)m (um\).)g(\025)44 b Ft(T)-7 b(r)i(ac)g(es)33 b(and)i(Determinants)226 3881 y(of)f(Line)-5 b(ar)35 b(Op)-5 b(er)g(ators)p FA(.)31 b(\025)43 b(Birkh\344user,)34 b(2000,)d Ft(Op)-5 b(er)g(ator)35 b(The)-5 b(ory)p FA(,)32 b(v)m(olume)g(116.)-246 4023 y([MT88])161 b(Milnor)32 b(\(John\))i(et)h(Th)m(urston)g(\(William\).) 30 b(\025)47 b Ft(Iter)-5 b(ate)g(d)36 b(Maps)g(of)g(the)h(Interval)p FA(.)c(\025)47 b(Springer,)34 b(1988,)226 4144 y Ft(L)-5 b(e)g(ctur)g(e)35 b(Notes)g(in)f(Mathematics)p FA(,)f(v)m(olume)f (1342.)-246 4286 y([Nus70])155 b(Nussbaum)23 b(\(Roger)e(D.\).)i(\025)k (The)c(radius)f(of)g(the)h(essen)m(tial)f(sp)s(ectrum.)h Ft(Duke)i(Mathematic)-5 b(al)24 b(Journal)p FA(,)226 4406 y(v)m(ol.)32 b(37,)g(1970,)g(pp.)h(473\025478.)-246 4549 y([Rue91])151 b(Ruelle)31 b(\(Da)m(vid\).)h(\025)43 b(Sp)s(ectral)31 b(prop)s(erties)h(of)g(a)g(class)g(of)g(op)s(erators)g (asso)s(ciated)g(with)g(maps)g(in)f(one)226 4669 y(dimension.)g Ft(Er)-5 b(go)g(dic)34 b(The)-5 b(ory)34 b(and)h(Dynamic)-5 b(al)33 b(Systems)p FA(,)f(v)m(ol.)h(11,)f(1991,)f(pp.)i(757\025767.) -246 4812 y([Rue94a])102 b(Ruelle)23 b(\(Da)m(vid\).)h(\025)30 b Ft(Dynamic)-5 b(al)25 b(Zeta)i(F)-7 b(unctions)26 b(for)h(Pie)-5 b(c)g(ewise)26 b(Monotone)h(Maps)g(of)g(the)g(Interval)p FA(.)226 4932 y(\025)43 b(American)31 b(Mathematical)g(So)s(ciet)m(y)-8 b(,)32 b(1994,)g Ft(CRM)j(Mono)-5 b(gr)g(aph)34 b(Series)p FA(,)e(v)m(olume)f(4.)-246 5074 y([Rue94b])97 b(Ruelle)24 b(\(Da)m(vid\).)i(\025)32 b(Extending)26 b(the)g(results)g(of)g("sharp) g(determinan)m(ts")f(to)g(discon)m(tin)m(uous)h(w)m(eigh)m(ts,)226 5195 y(1994.)31 b(F)-8 b(ax.)-246 5337 y([Rue96a])102 b(Ruelle)35 b(\(Da)m(vid\).)j(\025)56 b(F)-8 b(unctional)35 b(equation)i(for)f(dynamical)f(zeta)j(functions)e(of)h(Milnor-Th)m (urston)226 5457 y(t)m(yp)s(e.)c Ft(Communic)-5 b(ations)33 b(in)i(Mathematic)-5 b(al)34 b(Physics)p FA(,)e(v)m(ol.)g(88,)h(1996,)e (pp.)i(63\02588.)-246 5600 y([Rue96b])97 b(Ruelle)42 b(\(Da)m(vid\).)i(\025)77 b(Sharp)44 b(zeta)g(functions)g(for)f(smo)s (oth)g(in)m(terv)-5 b(al)42 b(maps.)i Ft(In)61 b(:)89 b(International)226 5720 y(Confer)-5 b(enc)g(e)33 b(on)h(Dynamic)-5 b(al)34 b(Systems)g(\(Montevide)-5 b(o,)34 b(1995\))p FA(,)e(pp.)h(188\025206.)e(\025)43 b(Longman,)31 b(1996.)1832 6027 y(32)p eop %%Trailer end userdict /end-hook known{end-hook}if 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