The max-plus Martin boundary
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. The max-plus Martin boundary Marianne Akian — Stéphane Gaubert — Cormac Walsh. N° 5429 December 2004. ISSN 0249-6399. ISRN INRIA/RR--5429--FR+ENG. Thème NUM. apport de recherche.
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(18) . Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.
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(133) px¡ VUk~udprk~c¸¤¥lyy·*V |VUgipwdfV;| ba ⊕ urgi| }Uprgi},udfV,giudlzpg4£\kV,cfx*V,}ªdlzVUyar prklzg?csd~uwgi}UVr£\lz A ∈ R £ £§|VUgSprdfV;ctu T udkflzvA®ËprktÁVUkgSVUyׯª£*uwgi|2l¬ £ |SVUgSprdfV,cZu rV;}ªdprk;£S¤V|VUgSprdfV (i, ba j)Au7→∈AR £ i 7→ (Au) £dfPiVrV,}dfpkQ|SV«igSV;|=a u ∈ R i 7→ u . . max. max. S×S max. S max. ij S max. i. i. (Au)i :=. M. Aij uj ,. ¤¥PSV,kfV·dfPiV!ceabT¦*pry ⊕ |V,gSpwdV,c¨dfPiVhiceh?uwy4cfhSxSkVUT¦hiT V¦gSp¤lg=dfkpb|Shi}VdPSV
(134) TWuwvb¡NxSyhicZuwgiuryzprhSV!prdfPSV T udkflzv A ∈ R £i¤V|VU«igSV j∈S. %® ·kVUV,g2ÁrV,kfgSV,yn¯ · lrV,gÀuwgba. "$ A 3R
(135) Q Q . S×S max. S×S A∗ = I ⊕ A ⊕ A2 ⊕ · · · ∈ Rmax , S×S A+ = A ⊕ A2 ⊕ A3 ⊕ · · · ∈ Rmax. ¤¥PSV,kfV I = A |VUgSprdfV;cdfPiVtT uvb¡Nxiyzhic0ln|V,gdl¬dsa
(136) T udkflzv©£uwg?| 7OQPiV ¢pryyzp¤¥lgS
(137) ¢prkT¦hSynuwV uwkV prbblzphic G A 0. Ak. |VUgipwdfV;c0dfPiV kdfP x*p¤¨VUk7pr?dPSVtT udkflzv. wu g?| A = A A . <-dQTWuÑaW*V hiceVU¢hSy*dfpÁrVUV,x¸lg¸TWlgi| dPSVZkurxSPkVUxSkV,cfVUg=d~udflprgWpr9T udfkln}V,c GdfpWuwgbaWTWuwdfkl¬v lnc!urccep}UluwdfV,|µu|lkfV;}ªdV,|µrk~uwxSPµ¤¥lzdfP±cfVd!pwQgSp|V;c S uwgi| urgµurk}
(138) ¢kprT i dfp j l¬QdfPiV ¤V,lzAP∈d AR lnc |l *V,kfV,g=dt¢kfpT ©tOQPSV!¤¨VUlrP=d\pw7uWxiuwdfP2lnctba|V«igil¬dlzpgdfPSV¦TWuwvb¡NxSyhic\xSkfp|h?}ªd®¢dfPiuwd\lc,£?dfPSV¦cfhST¯ pwdfPSV·¤¨VUlrP=dc¨pwl¬d~c¥uwk~}Uc,%OQPSV,g4£ A urgi| A kVUxSkV,cfVUg=dDdPSVcehixSkfV,T¦hST pr4dfPiV·¤¨VUlrP=d~cDpr%uwyy§xiudPic ¢kprT i dfp j dPiud\urkfV£=kV,cfx*V,}ªdlzVUyar£pr%x*pcfl¬dlzVuwggSprgSgiVUuwdflrV·yzV,gSwdP4 [prdflÑuwdfV;|¸baWdfPiV urgiuwypraW¤¥l¬dP#x*pwdVUg=dflnuwy§dPSVUpkfa£=¤¨V ¤¥lzyy4cuÑadPiudtu
(139) rV;}ªdprk lnc ®¢T uvb¡ xSyhic~¯ $ "$ A VZlz Au = u urgi|X U$ "$ A VZl¬ Au ≤ u tX\pwdfV!dfPiuwdZ¤¨V!kfV;m=hSlzukV ∈dfPiRV¦VUg=dkflV,c\pw7u PiurkfTWprgil}¨prk7cfhSx*VUkf¡NP?uwkT
(140) pgSln}DV,}ªdprk0dp?Vt|lncedflgi}ªd¢kprT +∞ VtcfPiuwyyScuÑadfPiuwdDu rV;}ªdfpk π ∈ R lncDyV×d ®¢T uvb¡-xSyzh?c¯0P?uwkT
(141) pgSln}Ql¬ = π £ π *VUlgS!dPSprhirP=dDpw4ucDukp¤HV,}dfprk; 9lzÁVU¤¥lnceV£r¤¨V\cfPiuwyy?cuÑa dfP?ud π lnctyV×d®ËT uvb¡Nxiyzhic~¯¥cfhSx*VUπA kf¡NPiurkfTWpgSl}·lz πA ≤ π t_bhSx*VUkf¡-PiuwkTWprgSln} rV;}ªdfpkcQP?uÑrV dfPiV¢pryyzp¤¥lgS VUyVUTWV,gd~uwka¸}~Piurku}ªdfV,kflncudflprg9 ¾ ¾ ¢ ¾
(142) #V "$ u ∈ R AQ UA" V. T $ &," ( T u = A u 0" "?T E<- lnccfhSx?V,ke¡-PiurkfTWprgil}r£§dfPSV,g A u ≤ u ¢prkuwyy k ≥ 1 £©¢kprT¤¥PSln}~Pµl¬d¢pyzyp¤tc·dfPiuwd uOQ∈ R}UprgbrV,kcfVZurycfpWPSpryn|Sc,£Scelgi}UV S P ! V u=A u AA u = A u ≤ A u kprT gSp¤ pg4£¤V T uwÁV·dfPSV ¢pyzyp¤¥lzgiWucfcfhSTWxdlzpg4 ¸; ¾ I 0 0
(143) T# U$ "$ A V #V "$ T#% ' A" X"$ AQ "$0&J. 0" #V "$ #AV#S $ π∈R π ≥ πA °¨a¦uwxixSyzablgSo7kprx*pcfl¬dlzpgWS; 4¨dpZdPSV¥dkurgicex*pcfV¨pw A £r¤¨Vt}pgi}yhi|V¨dPiud π = πA _lzgi}UV π P?urc0gSp }pTWx?pgSVUg=dcDV,m=hiuryidp §£b¤VZceV,V¥dfPiuwdpgSVZ}UprgicfV,m=hSV,gi}V¥pr©dfPSV·uw*prVturccfhSTWxdflprg lnc7dPiud A ∈ R ¢prkturyzy i, j ∈ S (¢prkfdflprklÒ£ A ∈ R ¢prktuwyy i, j ∈ S OQPSV¨}~PSprln}Vpw ¤¨VDT urÁrV7¤¥lyzy|VUdfV,kfTWlgSV¤¥Pil}~P¦cfVd0pwSP?uwkT
(144) pgSln}7rV;}ªdprk~c4lnc4dPSVD¢p}Uhic9pr?udfdfV,gdlzpg4 <-dt¤¥lyzy4*V dfPiV!ceVUdtπpwP?uwkT
(145) pgSln}ZV,}dfprk~c dfPiuwd\uwkV 8 A8 '0
(146) Q;U£STWV,urgSlgS
(147) dfPiuwd ¶\0}UprhSk~cfVr£ dfPiV ?phSgi|SurkfadPiud!¤¨V |V«?gSV ¤¥lzyy7uwyncfpu|SVUx*VUgi|µprg π π £9lgArV,gSVUk~uwyN prkSkVUblzdsarπu£¤<V ∞cfPiuwyyprTWlzd·dPSV A∗ = I ⊕ A + ,. A+ = AA∗ = A∗ A,. ∗. ∗. ∗. S×S max ij. . + ij. ∗ ij. S max. S max. S max. . ∗. S max. . ∗. k. ∗. +. ∗. . . . . S. ∗. . ∗ ij. ij. max. max. . Ü¢ÝÞ9Ü×ß.
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(149) !#"$% &'$(. 1. V vxSyln}lzd¨|V,x?V,gi|VUg?}V\pg π pw§dPSVZm=hiurgdl¬dlzV;cdPiud¨¤VZlzg=dfkp|hi}UV\uwgi| cfPiuryzy?prTWl¬ddfPSV·urccehiT
(150) xSdflprg prg π lg dfPSV cedudVUTWVUg=d~c7pw©dPSVtdPSVUpkfV,T cU V·|VUgipwdfVZba urgi| £=kV,cfx*V,}ªdlzVUyar£rdfPSV ceVUdDpw π¡-lzg=dVUrk~uwiyzV PiurkfTWprgil} uwg?| π¡-lzg=dVUrk~uwiyzV cehix?V,ke¡-PiuwkTWprgSln}ZrV,}dfpkHc, S <-d\lnctpr×dfVUg2}UprgbrV,gSlVUg=d¥dfp¸}~Pip=p=ceV ¢prk·cepTWV ® V¦hicfVdPSV!gSprdudlzpg urgi| dfp2|SVUgSprdfVr£kfV;cex*V,}dflrV,yza£©dfPSV idPµkp¤ πur:=gi| AidP }pryhSTWgµpwb¥∈uwgbSa2T udkflzv M ¯ VcfPiuwyy7cfMuÑadPiud Mb lnc u #J A" 7¤¥PSV,gÀdPSVWrV,}dfpk |V«?gSV,| lzg dfPilc ¤QuÑaPiurcZ«igil¬dV VUg=dfklzV;cW®¢lg xiurkedl}UhSynuwk;£4u¸iuceV,x?plzg=d Piuctur},}V,cc¨dfp¸VUrV,kfa¸gSp|Vlzg Sπ¯ª lzdfPdPSlnc\}~PSpl}UVpw π £iVUVUka¸cfhSx?V,ke¡-PiurkfTWprgil}·rV;}ªdprk u ∈ R lnc uwhSdfprT uwdfln}Uuwyya π¡Nlg=dfV,rk~uwSyVcelgi}V£ba¸o7kprx*pcfl¬dlzpgS; 4r£ πu = (A u) = u < +∞ D_pi£SlgdfPSlnc\},urcfVr£ }Uprlgi}ln|V;c¤¥lzdfP#dfPiV cfVd¥pr%uwyy§PiuwkTWprgSln}tV,}dfprk~c,%OQPSlnc¥}pgi}yhicflzpg¸kVUT uwlgicDdfkhSV ¤¥PSVUg £ H ¤¥PSV,kfV σ lcturgbakfp¤ rV,}dfpk¨¤¥lzdfP«igil¬dV!cehSxix?pked;£SlÒ Vr£S¤¥lzdfP σ = #VvS}V,xd¥¢prkQ«?gSl¬dVUya¸T uwgbπa :=i σA V!|V«igiV dfPSYV IR
(151) K ¤¥lzdfPkV,cfx?V;}ªdQdfp π G ¢prk\uryzy i, j ∈ S . ®Ò¯ K := A (π ) _blgi}V π A ≤ (πA ) = 𠣤¨VPiuÑV ¢prktuwyy i, j ∈ S . ®Ë=¯ K ≤ (π ) OQPSlnc0cePSp¤tc4dfPiuwd9dPSV¨}UpryhSTWgic uwkV7*prhSgi|SV,|¦ur?pV7lgi|VUx*VUg?|VUg=dfyapw 0°¨a Oa}~PSprgip
(152) c4dfPSV,prkVUT£ dfPiV
(153) cfVd·pwD}pyzhiT
(154) g?c K := {KK | j ∈ S} lncZkfV,yuwdflrVUya}prTWxiu}ªd·lzgdPSV¦jxikfp|hi}d\dprx*pryprrapw R OQPSXV
(155) V M lnc |VU«igSV,|±dfpA*V#dPSV}ypcfhSkV¸pw K V}Uuryzy B := M \ K dPSXV #"% &'( kprT ®Ë=¯turgi| ®¢b¯ª£§¤V¦rVdZdfPiuwd uwgi| ¢pk·uryzy w ∈ K ·_lzgi}UV!dPSVcfVd·pw rV;}ªdprk~c¤¥lzdfPdPSV,cfV ds¤pWxSkprx*VUkfdflV,cQ},uwg?V¤¥Awkl¬dfdf≤V,g w πw ≤ uwg?| π w ≤ ¢prkturyzy i, j, k ∈ S} {w ∈ R | A w ≤w uwg?|¸dPSlnctceVUd¥lc¥pb=lprh?ceya}ypcfV,|¸lzgdfPiVxSkfp|h?}ªd¥dfpx?pyzpra pr R £i¤V PiuÑV\dPiud urgi| πw ≤ ¢pktuwyy w ∈ M . ®Ò¯ M ⊂S ! Q +!@ +D 00 Ñ% ¦ 0@ +#%%¥ "#¥ ¶\x?uwkfdfln}hSynuwkQlgdVUkV,cedQurkfVZdfPipcfV·}UpryhSTWg#V,}ªdprk~cDpr dfPiuwdtuwkVZPiurkfTWpgSl}rO9p
(156) lzgbV,cedfludVtdPSV,cfV ¤V ¤¥lyzySgiVUV,| cfprTWVQiuceln}¨gSpwdlzpgic7uwg?|¦Ëu}ªd~c0¢kfpTT uvb¡-xSyzKh?ccex*V,}dfk~uwydPSVUpkfa ZV«igSVtdfPSWV
(157) VQ 0V % $ pw dfp *V A M dfk ρ(A) := ( A ) , ¤¥PSV,kfVZdfk L 7OQPbhic,£ lncdfPSV T uvlT¦hSTK¤¨VUlrP=de¡Ndfpr¡NyVUgiwdfP¸kuwdflp!¢prkQuwyy*dfPSV }lk~}hSlzdc¨pw dfPiVtrk~uwxSP
(158) Apw= A OQPSV¥AVUvlcedfV,gi}V¥prρ(A) §ucfhSx*VUkf¡NPiurkfTWpgSl}¨kp¤ V,}ªdprk¤¥lzdfP
(159) ¢hSyy?cfhSxSx*prkfd,£ \ccehSTWxdlzpg¸Æb£ lT
(160) xiyzlV,cDdfPiuwd ρ(A) ≤ ®ÒceV,V¥¢prk¨lzg?csd~uwgi}UVto7kprx4=SÆ pw - Zhi|S=J 5§prk 9VUTWT uÆ pr - r 5ׯ \VU«igSV dfPiV " Q#& $U A˜ = ρ(A) A OQPSVTWuwvb¡NxSyhic urgiuwyprhSV¸pr\dfPiVgSpwdlzpgHprZkV,}hikfkVUgi}UV#lnc |VU«igSV,|l!g - r 5 G º ¢ ¾
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(164) |VUgSprdfV ba N (A) dPSV ceVUd!pwkV,}UhSkkfV,gd·gSp|V,c, V},uwyPy 0#VQ%0 AVQ;i#QQZpr A dPSVWV,m=A˜hSlÑ=uryzV,gi}V }UyucfcfV,c·pw ¤¥lzdfPdfPiVkfV,yuwdflprg R |VU«igSV;|#ba iRj l¬ A˜ A˜ = © N (A) OQPSlnc\cfPSprhiy|*V}pTWxiuwkV,|¤¥lzdfPdfPiV¦|V«?gSl¬dlzpg2pwkV,}UhSkfkVUg?}V·¢prk\[ÀuwkÁrp¸}~Piurlzg?cU£?¤¥PSVUkV!u gSp|V lnctkV,}hikfkVUg=d¥l¬7pgSVkVdfhikfgic¥dfplzd\¤¥lzdfP2xSkpr?uwSlyzlzdsapgSVr@ 6tV,kfV£?uWgSp|V¦lc\kfV;}hSkkfV,g=dQlz¤V}UuwgkVdhSkfg dfp lzdt¤¥lzdfPkfV,¤¨urk| lg A˜ _blgi}V = A ≤ A £9V,rV,kfa2}UpryhSTWgµpr A lccfhSx*VUkf¡NP?uwkT
(165) pgSln}w¶ZgSyza2dfPSp=ceV }pyzhiT
(166) g?c pw A }pkfkV,cfx*prgiAA |lgSdfpWkV,}hikfkVUg=dQgSp|V,c¥ablVUyn|¸P?uwkT
(167) pgSln}ZV,}dfprk~c G ∗ b·. i·. ∗. ·i. S max. b. b. ∗. . i. ∗ ij. ij. i. ∗ ij. ∗. j. j. −1. j. ij. i. −1. ·j. S max. ·j. . S max. . ij. j. i. k. k. S max. . k 1/k. k≥1. ii. i∈S. . −1. + ii. r. + ij. r. + ji. . . . ∗. Þ9Þ Ó$[Së#\Q]^. +. ∗. ∗. ∗.
(168) /. C$ R $ 8 A O A# "'V
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(171) MA V#"%Y #V "$ A @" VW T &Y"$ A( & W0#VQ% Q ρ(A) = i OQPSV¦cuwTWV!lnc¥dkfhiV¢prk¥dPSV}UpryhSTWgictpr K celgi}UVdfPSV,aurkfVxikfpx?pkedlzpgiuwy4lgdfPiV¦T uvb¡-xSyzh?ctceV,gicfVdfp dfPipcfV·lz A OQPSV¢pryyzp¤¥lgSZds¤p·kV,cfhSy¬d~c0cePSp¤AdPiudlzdTWurÁrV;c%cfVUg?ceVdfp·l|SVUg=dflz¢a!VUyVUTWV,gd~c0lzgdfPiVQcuwTWV¨kfV;}hSkkVUgi}UV }ynurcc, ¾ ¾ ¢ ¾ 3 # & U VQ T.$ &," (M T & & K =K ρ(A) = i j L AW .0#VQ%0 i,V#j∈VQ;S # 0" "?T 4VUd ?Vcfhi}~PHdfPiuwd K = K OQPSVUg4£¨lgHxiurkedl}UhSynuwk;£ K = K £¨urgi|Hcfp A = i, j ∈ S. b _ b a
(172) T W T U V f d k l , } uwyyza£?¤¨V!prSduwlg A = π (π ) ·OQPSVUkV¢pkfV£ A A = § <- i 6= j £idPSVUgÀdfPSlnc π (π ) lT
(173) xiyzlV,c
(174) dPiud A ≥ A A = A A = §£lzgH¤¥PSln}~PH}UuceV ρ(A) = §£ i lc
(175) kV,}UhSkfkVUg=d;£0urgi| i urgi| uwkV
(176) lgAdfPiV¸cfurTWV kfV;}hSkkfV,gi}V }Uyucfc,#OQPSlnc!cfPSp¤tc dPSV¦²spgSyaÀlz´³xiuwkfd!pwQdfPiVxSkprx*pcfl¬dlzpg4¸Xtp¤ yVd j urgi| i urgi| j *V!lgdfPSV¦cuwTWVkV,}UhSkfkVUg?}V}ynurccU¨OQPSVUg9£?ur},}pk|lgSdfCp - rS£ioDkfpx4*S 5N£ ρ(A) = £©uwgi|cfp \°¨hd·cflgi}V £*¤¨V!PiuÑrV!dfPiuwd £§uwgi| A dfPiVUk=V¢prAkV AK ≤ K 7OQKPSV =kfV,rKVUk~cf(πVZlgS)V,m=hiπuryzAlzdsa ¢pyzyp¤tc¢kfpT πuW=ceabπATWTWVdkfln}Uury§urkfhSTWVUg=πd, ≥ π A ¾ ¾ ¢ ¾ . ##%W $ ρ(A) = MAQ TQ"$ u ∈ S & i, j L AW $ 0#V %0Q V# V ;J# πu =π u 0" "?T ·_blgi}V £%¤V},uwgA}pgiceln|V,kdfPSV V,}dfprk #OQPiuwd lnc¦cfhSx*VUkf¡NP?uwkT
(177) pgSln} }UurgA?VVUvxSkfV;cfπcfV,∈|µRurc π ≥ π A £4¢pk¦uwyy i, j ∈ S ¸πOQPilc:=lnc!V;(πm=hSlz)uryzV,gd dp (π )π ≥ A (π ) lzg pwdPSVUk¥¤¨prk~|Sc,£rdPiud π £ScfVUV,g#urc¥u
(178) }pyzhiT
(179) grV;}ªdprk;£=lncQcehix?V,ke¡-PiuwkTWprgSln}w%o7kprx*pcfl¬dlzpgS ¦pw - r$ 5 cedudV,ctdfPiuwd\dfPiV!kfV;csdkfln}ªdlzpgpw7uwgba¸ds¤p ρ(A)¡´cehSx*VUkf¡-VUlrVUgbV,}ªdprk~cQpw A dp¸uwgba¸kV,}UhSkkfV,gi}V!}Uyucfc¥pr A uwkVtxSkprx*prkfdflprg?uwyNOQPSV,kfVU¢prkVr£rV,l¬dPSVUk Wpk7dPSV\kV,cedfkl}dflprgic7pw urgi| dfpuwgbakV,}UhSkkfV,gi}Vt}Uyucfc uwkV xSkprx*prkfdflprgiuryÒD <´gVUlzdfPSV,kt}UurcfVr£bdPSVu T =uwx i ∈ S 7→ π u lnc¥}pgiceduuwg=d¥prgπV,u}~P#kV,}UhSkkfV,gi}V }ynurccU E 0RWi <-d0¢pryyzp¤tc9¢kfpT dfPSV;ceVds¤¨pZxSkprx*pcflzdflprgic9dfPiuwd,£¢pkuwgba u ∈ S £ÑdPSVQTWurx S → R , i 7→ lgi|hi}UV,c uT urx OQPbhic,£©u¸cfhSx*VUkf¡NPiurkfTWpgSl}!rV,}dfpk\T uÑa?V
(180) kfV,urk|V;| π urc¥uu¢hSg?}ªdflprg|VU«igSV;|#Kpg →K R , K 7→ π u 4VUd u ∈ R *V!u π¡Nlg=dfV,rk~uwSyVZV,}dfprk; V!|SV«igSV·dPSVT uwx µ : M → R =a ¢prk w ∈ M , µ (w) := lim sup π u := inf sup π u ¤¥PSV,kfV dPSV!lg«iT¦hiT lncQdurÁrVUgpVUk¥uwyy4gSV,lzP=*prhikfPSpbp|Sc W pw w lzg M OQPSVkV,ucepg#¤¥Pba¸dfPSVylzT cfhSx uw*prVW}UurgSgSprdZd~uwÁrVdfPiV
(181) uryzhSV +∞ lcZdPiud π u ≤ πu < +∞ ¢prkuryzy j ∈ S OQPiV¢pyzyp¤¥lgS¸kV,cfhSyzd cfPSp¤tcQdfP?ud lnc\urghixSx?V,k·cfVUTWln}pgdlzgbhSphicQVvbdVUgicflzpgpw0dfPiV!T uwx¢kprT K dfp R lgdkfp|h?}V,|lµg{t: VUMT uw→kÁ¸RSÆb Dº. 3 u I π 8 '0' ;Z AQ UA" A V #V "$ µ (K ) = π u TQ"$S#'V# . T. . ∗ ·i. . ∗. . . ·i. . ·j. . ·i. j. −1. i. + ii. + ij. ·j. ∗ ji. + ji. ∗ ij. i. . ∗ ji. j. ∗ ij. −1. . ii ∗ ji. ∗ ij. ij. . . ∗ ·i. ∗ ·j. ∗ ji. ·i. ·i. i. ·j. −1. ∗ ji. j. ∗. i. ∗ ji. j. ·j. . . . i i. . j j. S. −1. j. i. −1 i∈S i. ij. i. −1. ij. j. −1. −1. . −1. i i. . max. max. i i. i i. ·i. S max. u. u. K·j →w. j j. W 3w K·j ∈W. max. j j. j j. u. i∈S. &. 0" "?T. µu (w)w ≤ u. max. max. TQ"#
(182) V . . . . C"$0#" Q . u. Z°¨a¸o7kprx*pcflzdflprgS;4£ A u = u t6 V,gi}V£¢prkturyzy i ∈ S £. i i. w∈M. w∈K. . ·i. w∈M M M µu (w)w . µu (w)w = u=. ∗. ui =. M j∈S. A∗ij uj =. M. Kij πj uj .. ®UF¯. j∈S. Ü¢ÝÞ9Ü×ß.
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(184) !#"$% &'$(. V
(185) }Uprgi}Uyzhi|SV!dfP?ud lgSV,m=hiuryzlzdsar£¤¨Vprd~uwulg¸≥dPiuKd i. ij πj uj. ¢pk·uryzy i, j ∈ S °¨a#d~uwÁblgSdPSVylTWcfhSx2¤¥lzdfP kfV;cex*V,}d\dp j pwDdfPSlnc. ® 1w¯ ¢prk!uryzy w ∈ M uwgi| i ∈ S WOQPSlccfPSp¤tcZdfPiVWcfV,}Uprgi| xiurked pwdfPSVW«ik~ced ucfcfVUkfdflprg pwDdPSVWyzV,TWTWui
(186) Op xSkprVZdfPSV «?kcedtxiuwkfd,£¤¨VuwxSxiyza dPSlnc¥lzgiV,m=hiuwylzdsa¤¥l¬dP w = K VVdQdfP?ud u ≥ K µ (K ) _blgi}V £?¤¨VcfVUV!dfPiuwd ZOQPSVkfV,rV,kcfVlzgiV,m=hiuwylzdsa¸¢pyzyp¤tc¥¢kprT dfPSVW|VU«igSlzdflprg2pr K 7= (π )giury9csd~udVUTWVUg=d¥pw%πdfPSuVy≥ µT (K )yzp¤tc¨¢kprT =Dm=hiudlzpgµ®UF¯Quwg?|¸dPSV·«ik~csd\ceduwdfVUTWV,gd; Q O S P · V i « U V W T u ¢ r p y µ # U! !9U 4 +¨ <´gxSkpriurSlyzlncsdl}·x*pwdVUg=dflnuwy9dfPSV,prkar£SpgSV!|pbV,c¥gipwdtgSV,V,|dfPiV!VUg=dflkV?phSgi|Surkfa dp*V¦uwSyV dfp kVUxikfV;ceV,gd PiurkfTWprgil}DrV,}dfpkc,£u\}V,ked~uwlg¦cfhSicfVdce%h N}UV,c, VQcfPiuryzybceV,V7dPiud0dfPiV¨cfl¬dhiudlzpg!lg¦dPSVT uwv=¡-xSyhic%ceVUdedflgS lnc¨cflT
(187) lynuwk;0Op|SV«igSVZdfPiVW®ËT uvb¡Nxiyzhic~¯TWlgSlTWury4[2uwkfdflg¸cfxiur}UVr£b¤¨V\gSV,V,| dplg=dfkp|hi}V uwgipwdfPiVUk¨ÁrVUkgSV,Uy G ¢prk\uryzy i, j ∈ S . K := A (π ) XtprdfV dPiud K = AK lnc¥u¢hSgi}dflprgpw K pktuwyy w ∈ M £¤Vurycfp |V«igiV w ∈ R G ¢prkturyzy i ∈ S . w = lim inf K OQPSV ¢pyzyp¤¥lzgiyVUTWT u cePip¤tcDdPiudtgipWurT¦Slrhil¬dsa¸urkflncfV,c¢kfpT dfPilc¥gipwduwdflprgcelgi}UV (K ) = K Dº. .
(188) YY w = w TQ" w ∈ B $ & w = K = Aw TQ"$ w = K ∈ K 3"$Z ui ≥ lim sup Kij πj uj ≥ lim inf Kij lim sup πj uj = wi µu (w) , K·j →w. K·j →w. K·j →w. i. ·i. ii. i. −1. i i. u. ii u. ·i. ·i. u. + ij. [ ij. [ ·j. ·j. −1. j. [. ·j. [ i. S max. [ ij. K·j →w. ·j. w∈M. 0" "?T. . . . 4VUDd _bp . [. w[ ∈ S. w ∈ B. $ &. [. πw[ ≤. OQPSV,g4£¨¢prkV;ur}~P. K·i 6∈ W. . i ∈ S. [. [ ·j. . [ ·j. ·j. £¨dfPiVUkV2Vvlncsd~cuAgSVUlrPb*prhSkPSpbp|. W. pw w cfhi}~P dfPiuwd. [ wi[ = lim inf Kij = lim inf Kij = wi ,. xSkpblzgSdfP?ud w = w Xtp¤(yVd w = K ¢prk\cfprTWV j ∈ S ¥O%uwÁblgS dfPiV¦ceV;m=hSVUgi}UV¤¥l¬dPÀ}pgiceduwg=dtuwyhSV K £i¤¨V¦cfVUVdPiud 7O9pWV;csd~uwSylcfP#dfPSVpxSx?p=celzdfV lgSV,m=hiuryzlzdsar£¤¨Vpr?ceV,kfVtdfP?ud w ≤K ¢pktuwyy i ∈ S , w = lim inf AK ≥ lim inf A K = A w prk;£lgprdfPSV,k¥¤pk|icU£ w ≥ Aw DOQPSV,kfVU¢prkV ¤V PiuÑV·cfPSp¤¥g#dfPiuwd w = K OQPSV ynurced\urcceV,kedlzpg¸pwdfPSVyVUTWT u
(189) ¢pryyp¤tcD¢kprT ®Òr¯Qurgi|¸dfPSV Ëu}ªd¥dfP?ud π lc¥cfhSx*VUkf¡NPiurkfTWpgSl}r XtVUvbd,£S¤¨V|V«igiV ds¤pWÁrV,kfgiVUync H uwg?| H pVUk M K·j →w. K·j →w. [. ·j. [. ·j. [ ·j. [. K·k →w. ·k. K·k →w. ik. ·i. ·i. i. [. [. [ ·j. [. H(z, w) :=µw (z) = lim sup πi wi = lim sup lim πi Kij K·i →z K·j →w. K·i →z. [. Y\cflgSWdfPSV Ëu}ªdQdfP?ud. H (z, w) :=µw[ (z) =. [ = lim sup lim inf πi Kij . K·j →w. urgi|,<´gSV,m=hiuryzlzdsaÀ®Ë¯ª£S¤¨VrVUdQdfPiuwd ¢prkturyzy w, z ∈ M . H (z, w) ≤ H(z, w) ≤. K[ ≤ K. [. Þ9Þ Ó$[Së#\Q]^. lim sup πi wi[ K·i →z. . K·i →z.
(190) , 4. C$ R $ 8 A O A# "'V
(191) 0. - w ∈ M £wdfPSV,g*pwdP w uwgi| lg 9VUTWT uWS F£S¤¨V·Vd¥dPiud <. uwkVDV,yzV,TWVUg=dc%pw S =a®Nr¯%uwg?| 4 V,TWTWuZi4w0Y\cflzgi\dPSV¨«ikcedurcceV,kedlzpg. w[. ®U/¯ ®Ò¯. H(K·i , w) = πi wi H [ (K·i , w) = πi wi[ .. ´gxiuwkfdfln}hiyurk <. H(K·i , K·j ) = πi Kij = πi A∗ij (πj )−1 [ −1 H [ (K·i , K·j ) = πi Kij = π i A+ . ij (πj ). OQPSV,kfVU¢prkVr£©hSx dfpu|lurrpgiuwy0celTWlzynuwklzdsar£ H uwgi| uwg?| A kV,cfx*V,}ªdlzVUyar Dº. . 3"$. w, z ∈ M . urkfV
(192) VvbdVUgicflzpgicZdfp. H[. pw7dPSV ÁrV,kfgiVUync. M ×M. ®?4;¯ ®?4
(193) 4;¯ A∗. +. . ¤¥PSVUg w 6= z pk w = z ∈ B , pwdPSVUk¤¥lcfV . 0" "?T E<- £?dPSVUg w = w ba 9VUTWT u?;4r£*urgi|dfPiV¦V,m=hiuryzlzdsapw H(z, w) uwgi| H (z, w) ¢prk·uwyy w∈B ¢ r p y z y p t ¤ ¨ c l W T T
(194) V;|lnudfV,yza z ∈ 4M VUd w = K ¢prkcfprTWV j ∈ S urgi| yVd z ∈ M *VA|Sl §VUkVUg=d¢kfpT w OQPiVUg4£ZdfPSV,kfV Vvlncsd~c uµgSVUlrPb*prhSkPSpbp| W pw z dPiud|pbV;cWgSpwd¸}Uprg=durlzg w txSxiyzablgS 4VUTWT uµi 4uw=uwlg4£7¤¨VVd dPiud ¢prktuwyy V!|SV,|hi}UV dfPiuwd w) = H (z, w) lg#dPSlnct}UuceVuwyncfpi w = <´gKdfPSV =«?giKuwy4},= urcfVrw£¤¨V·P?uÑrV iw∈ =Wz ∈ K 7OQPSVkV,cfhSy¬dQH(z, ¢pyzyp¤tc¨¢kfpT =Dm=hiuwdflprgA?® 4;¯ V!|V«igiV dfPSV Y A H$U U A'V#¥dfpW?V H(z, w) =. . (. H [ (z, w) . [. [. ·j. [ i. [ ij. ij. [. i. kprT 4VUTWT uWiÆb£¤¨V!ceV,V·dfPiuwd. . Dº. 0" "?T. . Q( VPiuÑrV. . M m := {w ∈ M | H [ (w, w) = } .. ®?4Ñr¯. . {w ∈ M | H(w, w) = } = M m ∪ K . w ∈ Mm ∪ K. . πw =. . . πw = sup πi wi ≥ lim sup πi wi = H(w, w) = .. °¨a =Dm=hiuwdflprgA®Ò¯ª£ πw ≤ §£Suwgi|#dPSVkfV;cehiy¬dQ¢pryyp¤tcU ¾ ¾ ¢ ¾. (ZQQQ P"9T $ "$ A V M 0" "?T E<- }prg=d~uwlgic¥uwg#VUyVUTWVUg=d £dPSVUg9£b¢kfpT =Dm=hiudlzpgµ®94'4ѯª£¤¨V cfVUV·dPiud ρ(A) = ¸urgi| K ∩M n l Q c f k ; V } S h k f k , V = g , d D <-d¥¢pryyzp¤tc¨¢kprToDkfpx?p=celzdflprwgSƦdfPiuwd lncQPiurkfTWprgil}r w <-dWkfV,TWurlzg?cdfp xSkprV dPiud
(195) dfPSVcfurTWV¸lnc!dkfhSV#¢prkV,w u}~P±VUyVUTWVUg=d w pr B ∩ M 4VUd i ∈ S *V cfhi}~PÀdPiud µuwg?| ucfcfhSTWV
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