J OPER OR oo © Copyright by INCREST, 1985
ABTOMOP®U3MbI HHBEHRTMHBHHX ®ARTOPOB THHA III,
B.H FO7IO/HEH
BBBHEHH1
ƠNHỌï W8 LI€HTPA7IbHBIX 387A 5B reopuHn aJreÕp don Heiimana apuaerca 8/448 H3VW€HHN W OIHCAHHH BTOMODjHSMOB ajreốp HanpHMeP, KJIAaCGH- N@GOH8H 3DT0/HWGGHäH TODHH M3ÿ486T 8BTOMOPpHSMEI HpOGTpaHecrsa JleGera, MJIH BTOMOP(WHSMBI £OMMYTATMBHoH aareðpÐL don Heiimana B nocnequue TOMbI TOWYUCHI 8HAH4HT©JIbHBI© TIDOHBH?RGHHN B HSÿWEHHH H HJIACGHIỦHEBHHH ABTOMODŸMSMOB AHIDOECHMATHBHO HKOHGHHBIX (MHb€RTHBHBIX) àTeỐp (0H Heiimana [1], [2], [3] [saa daxropos runa Iu WI,, 0<A< 1, pacnagaw- muxca B ỐGGKOH€HHO€ T©H30DHOU HĐOH8B6IGHH© KOHGWHHIX (aKTODOB THIA I,,n <0co (agropos Apagn-Bynoa [4]) HalN€HO OHHGAHH© KJIACCOB BH€IIH€ COMPARCHHEIX ABTOMOPPH3MOB, XOTA /1ORA8AT©JIbGfBA NI Giydan III, He ny6umkOBaIuch ABTOMOP@U3MEL HHbCKTHBHEIX MakTopos tuna Il, msyua- IUCh MAIO, & Balata WSYFeHUA MHHBADHMAHTOB BHOUHIHGTO COHDH?ZMGHHH ABTO- MOPOH3MOB JIA TAKUX PakTOpoB He paccMaTpuBastace
Hacronnjan Crarba NOcBAIeHa U8y4YeHHO TAaKHX BOMpOCcoB, OHA ABIIA- eron epepaØorkoii namero npenpwnra [19]
Trang 24 8 1 PO2IO71EH
B [7] HORA38HO, WTO GYHI©OCTBYT R8HOHHWGCHHỦH TOMOMODÙHSM 8 — modf us AutW*(A,a,Z) B C(W), aAapy koroporo HDHHA/UI6?7RHT
Int W*(A, a, Z) (Int M ompeseeno s [1])
Tenepb MEI MO?ReM GÿODMY:THpOBATbB BOHIPOCBI, KOTOPBIS DACCMATPHBA- IOTCA B OTOH craTLe
1) aa neaxoro un ốc C(W(M)) cymecrayer ye AutM raKoii, aro mod y=: 6?
2) Ilyerm 6; € Aut M, i= 1,2 uw modf, = modf, == W(M), t #0 nam mod f, == mod £, = id p,(B,) = p, (Bs) = 0, rie p, (8) — acumurotugeckuil nepnog B [1] Caeyer -m orcioga, ro fp, u By BHeNTHE COIDĐH?R@HEL OTHOGITGIbHO Ấƒ, T.e Jye AutM u ue UCM), rae U(M)—vyunrapnaa rpynna M, ranue, aro
By == Adu-y-Bgry7? ?
(SameTiM, YO OTBET Hà HO7OOHBI BOIPOC HJI HHb€RTHBHOTO aKT0pa Ta TÍT,, O0 < 2 < |, /AGT THOHYH KCIACCH(ĐHEAHHI KJIAGGOB BH@IHHĐ GOIDR- JK£@HHHX 8BTOMODpH3MOB 6 p, = Ơ [2].)
3) Ilyers B;€ AutM, /-:l,2 uw JyeC(W), ragoii, ro mod, =: c= y-modf,-y7! Crenyer cm orcioga, ato ¡ H By BHOIHH€ GOHIDH?RHB oTHocHTembHO M, ecu p,(B;) == 0°
B uacroatteit crarbe /IAHBI OPBGTBI HA H€PBbI€ /1Bä BOIDPOG8 HZIH HpOHđ- HO.IbHBIX HHbOKRTHBHBHX (}aRTOPOB THUA I[I, Hm Ha Tperiii Bompoc — aA HPOM3BOJIBHBIX HHBCRTUBHELX daxtopos M tua Ill), y KoTOpHX HHBBAPHAHT T(M) = (1:6, € Int M1) #0 (cm [8])
Pemenne sonpoca 2) (cm §2), CBO/IHTCH K JoKasareibeTBy TOTO, f0 HIDOM OTOỐPp8/REHIN ƒ > modf? sapcinerca Int Ä
IÏepBEU BOHDpOC PGHIOH B H€€RO:IBRO ð0:160 OỐIHeii (bOpMe, em chopmy- JIHpOBAHO lĨy€rb [Z] — HOIHađØ TpYHHA ABTOMODŸjHSMOB Á, HODO?RKIHHaR 2 [6], a ![z] — ee Hopmadmaarop, re .#[z] = {y€ AutA : y|a]y~! = [a]} lÍonnrHo, dro 66 ye V[a], To y mwanynupyer y, € Aut W*(4,a, Z) B u.3 §1 oRaaaHo, wfo ;In ðe C(M) cymecrpyer y€.fT[e] TaROi, ro mod ? = 6 (em [19]) Bamerum, WTO /IDYTO€ ĐGIIGHI© ĐTOTO B8?RHOTO BOHpOGA vie Woctvaero B [21] (cm rTaRzge [22])
Bonpoo 3) paceMoTpeH ;ữ1 ĐaRTopOB ă ¢ T(M) #0 Ms orpeta ua oror BOlpoc cclenyeT, GTO BCHRHII ye AutAY 6© Pp,(y) =Ũ BHGIHIH€ compamen ¢ yy & Aut M, ipnseM (4) = 4 w | 6 #7[z] B [22] npusegeno nomnoe pemenne BOinpooa 3) n1 aBTOMODÙHaMOB H3 f [2] C HOMOHPbBIO peayabraros [22] MOsKHO JJaTb WOTHOe pettientre BoTIpoca 3) H ;18 @akTopow Tuna Hil, , yro noA- TOTABZHIHAGTGOWT ft IIYỐ/IHRAHHH
Kpome ppeyeHia cratba C0/16pZRHT TpH uaparpada B §I pacemarpn- BAlOTCA HOPMadAHZaTOpbi MO-THBIX TPVIIN 11 aCCONMMpPOBAHHELe NOTOKU, BI 3, § I nam onner na Bompoc I B §2 usyyarorca a € Aut M, y Koroprrx moda = W, nin fe R, B§3 pacemarpuBaiwtca asromophu3smEr daxtopos M c T(M) # 0
Trang 3ABTOMOPĐH3MBI WHHbEHTHBHDIX ĐARTOPOB THHA ITI, 5
1 AIHPORCHMATHBHO KOHEYHDIE VPYIIibl THUA III, M WX HOPMAJIM3ATOPHI
4 llyorb Ớ — 04GTHÀ H€GHHTYZIHĐHAH TPYHHa HP€OỐPA80BAHHH TpO- erpaHcrna JleGera (X, B, uw) Uepes (G| o6osnagum nomHy10 rpynny Hpe€- oỐpasopanni (X, B,u), nopompenuyio G (cm [6]) Honnste aproquaecnue TpyEnm [ỚŒ] 1OHYGRKAIOT H/IAGCHÙHRABIIO, AH80THMHVIO TỌ, OT0DAH XOPpOIHO M8B©CTHA J1H (baRTopoB jon Heiimana B nazIbH6iiHIM MEI YHâM HpđHHO/IA- rarb, ro [Ớ] umeer rum IHI, [S] OnpenenM ypoiicrnennyio Gy rpynny IpeoØpaaosanuli nun rpynneri Ớ ([9], [10]), meitcrByroutyio Ha (Xx R, Bx B(R), dự x< du) c0TJraocHO (ĐODMYI
(1.1) #a(X, M) =: (ex, w ~E log HED ), (x, u)EeXXR
du(x)
llyerb Z — usmepumoe pasdnenne XxR [11], mopompenuoe Gy — MHB8DH8HTHBIMH H3MGDHMBIMH MHO?R©CTBAMH PaccMOTpHM Ha XXR HOTOK T(x, wy) = (x,u+s), (x,u)€ XX<R TaK KAR Tý HROMMWTHDVT © g,€ Gy, TO MO?RHO paccMoTperb (arrop-norox W,(G) na XxR/Z Torna M,G) — ne- CHHT'VJIIDHEI HaM€DHMbLđ ToToE [9], Ha3bIBA6MBIÍ IOTOROM, 8GC0I1HDOB8HHEIM c G Cornacno [6], 6074 Ở; — AIIĐOROHMATHBHO KOH©HHH© (A.K.) TDYHHHI npeoốpaaosnannk (X;,, 6) Tuna Ill,, TO Ớy HH G, cnado skBMBaNeHTHE TOTHA MW TOUKO norxa, Korna W{(G¡) u W(G,) usomopdutt
JIEMMA 1.1 Heau ae W[G], mo % onpeDeasem asmomopduasm moda npocmpancmsca Xy=XXR/F, npuren modae C(W(G)) (cw Beedenue } Omo6pascenue « > moda — zomomopGuss, nOpo Komopozo codepacum, no kpaũnecú atepe, [G]
JH HORA8ATGJIbOTBA /OCTATOHHO 8AMGTHTb, TO ©0GIH œ€.Ý[GŒ], TO a(x, u) = (ax, + logdu(x)/du(x)) npnHanezmnr V[G,] Ocranmbunie pac- OVHIĐHHf HIOHHTHEI C7IENGTBHB 1.1 Beau fe L°(XxR, wxm), 2em — mepa Te6eza na Ru du(gx) 1.2 + log -—2-~ | = fix, (1.2) ⁄ Bx, og aux) ) ƒ(x, m) mo F(x, u) = f(ax, u + log du(ax)/du(x)), 20e a € W[G], maxatce yOos.semsopacm ypasnenuio euda (1.2)
HanoMHHM, jlanee, uro corzacno [6] senRan IoHan a.R rpynna [G]
Trang 4H©GIIDG-6 B H LOTO/EEH phipHad Mepa Ha X, v — o-KROHeuHAM Mepa Ha Y,o=pxXv~o' S.(x, ¥) = (x, S,), Q,(x, ¥) = (Qx, Uy), (13)
rye S—csodoquo /6ÏioTBVIOHIMH 9PTOHMd6OKHE aBToMOpÙHAM (Y, By, v), 00XPAHHIOIHHH v, Ứ, — H8M€pHmMO€ nozie asnroMOpQHawos (V, By, v) (cm [6], §2), npnseMw , e.Ÿ[S], gaa u.p xe X w ve U, = exp G(U,)v, a O — spro- JH4©CKHH H@CHHTVJIHPHBIÏ CBOỐOHHO AelicrByiomui apromophusm (X, By, pM)
S,,Q, mopompaiot [G], ao = uXv BEOpana TakuM OOpazoM, TO
> 0
We 2s (x,y) = OU, + log MY 5 5
1.4 =!
(1.4) p(x) = log aul)
Tenepb onmmem aGGOIHIHpOBBHHHII HOTOR W, /IH TPVHHEI (S,, Q,)- J[polierBeHHan rpynna Ớ, jelicrsyer B upocrpancrpe (QxR,oxm), a 0G0HHHPOBRHHELÍ HOTOKE B HOHHPO©TpaAHoTBS (2X, X??), HHBADHAHTHOM OTHOCHMTGJIBHO (Ởẹda, Š;d TAR RAR Š¿ GOXDAHHET M@DV, TO ,Š;4 — HHBADH- AHTHHIG (ÌYHKHHH YNOBJITBOPfHIOT YCsOBUIO
SOY, u) = f(x, S„, 10)
Ho S yeiicrsyer sproqugeckn B (Y, By, v), mostTomy G, — wHBapu- AHTHLIe PYHRUMI He 8aBUCAT OT Y WM YHOBIeTBOPAIOT YC.TOBINIO
(1.5) ƒ{Qx, u + @(3)) = ƒQ, n) Tenepbe s8 (XXR,# X7) pACCMOTPHM aBromopdu3m
(1.6) Q(x, u) = (Ox, ut y(x))
B cusuy yexosua (1.4) sto upeodpasopanue umeer run I, T.© pA8ÕI@HH€ XxXR Ha ero TpaexTopun usmepumo lÏycre Z(Œ,) — pasốueHHe XX< Ha TpaektTopun Q, , Torya PakTop-mpocrpancrso Xx R/.¥(Q,) us0mophHo woAMHO- mectBy X), co XXR Buna (x,Z)€ Xs;, ecnn 0 <u < g(Q-'x), Mepa py Ha q G€OHfữa/taer 6 OTpaHHnweHHew xi Ha Xạ lÍOHHTHO, HTO HOTOE 7,(x, £) = == (x,u +5), KOMMyTHpyiomnit c Q,, olpefenner moron W, Ha XX R/¥(Q,) (a aHagnT Wu Ha X,) CaegqoBateIbHO accolyupoBaHHblli noTOR WG) aBainerca CHEWMAABHBIM NOTOKOM G IOTOIONHOH PyHK Wel @(Q~!X) HH ỐA3HOHBIM aBTO- MopdmsMow @~1 [6]
Trang 5ABTOMOPA1)MDI HUƯEERTHBHBIX GAIRTOPOB THA lHIạ 7
te [G] maxot, umo B = t-1x umeem suƠ
(1.7) BOX, y) = (x, VAY),
20e V, — usmepumoe nose asmomophusmoe (Y, By, v) [6], npuren Ve W[S] u
OV.) = 5 Onn noumu ecex x
Aoxazamenocmeo Ilycth moda = id, torga a onpefennet mpeodpaso- panne XX Y: (x, y) > (x(x, y), v(x, vy) Sametum, uTo @yHRuMH x, uy, WeaMe- PHI, 4@liGrBHT©ZIEHO, ©GJIM E — M8MPHMO© HOHNMHO/R©OTBO ÄÝ, TO H8M©DH- MOCT X„ BbIrekaer I8 ooorHoImeHmn {(x, y)€ XXY : x¿(x, y) 6€ E} = œ~1(EXxY) Tax Kak moda = idua ¥XR/S(Q,), TO AA Boex PyHKIMi M3 L°(XxR), YROBueTROpAION(Mx (1.5), BbUIOMHEHO paBeHcTEBO f(x,(x, y), ứ + Ủ„(x, »)) =f 4),
rne W(x, y) = log (do(a(x, y))/dø(x, y)) Ho Torna (x¿(x, y), # + Wax, y)) H (X, #)
IIPHH8716?&HT OHOïÏi rpaexropuu Q, (cm (1.6)), m.e x(x, y) = Q"%'x, n(x, y) € Z, (1.8) W(x, y) = Z(n, Y, X), n—1 Z(n, 9, x) = W 0(Q'x) npm n > 0 ¡=0
llyerp £ƒ — pasỐneHne XXÏŸ Ha MHO?R€CTBG £,,n EZ, Take, WTO JIDUH (X, y) E„, BBIHIONH€HO DpABEHOTBO x,(x,y) = Q*x ÏÏOHHTHO, WTO È„ — IIOIAĐHO Hellepecekaiommeca MHO?R€OTBA HỆ | }E, = XXY Paccmotpum
n
velepb MHomecTBa a&, u QFE B cusy (1.8) X — HocnTenm y Hux coB- IANAIOT, H3 HOGTDOSHHH G2I€HV€T, uTO mMHomecrBAa {vy : (x,y) EaE,} m {y 1 (x,y) € QZE,) HH HOWTH BCexX xX UMeIOT OHHAKROBWIO Mepy v Orcioma enenyet (cM semmy 4.2[6]), aro cymecrsyer s, € [S,] Takoli, wo 5,Q7E, = aE, Pacemotpum tTenepb mpeobpasopanue XxY BH7a
q9) f(x, y) = s,@s(x, y) mpm (x,y) € E,
Tak KHaR ÈF„ — HOHAĐHO He mepecerawrca mu | J]Z„,=XXxỲ, a TâAE?R SaOsE„ (=«E,) — nomapHo He Tepecekalorca u Js, Ore, = XxXY, To B curly 4ieMMEI J[am [12] ? e [G] Ho rorna £~1+ e [G] Ht, GOomee toro, BcIy (1.8) ? 1œ coxpaHneT Mepy o=yuxyv CJI@10BAT€IbHO, / lxe.f[S,]} a Tak Kar (t~ta)(x, y) == (x, yi(x, y)), TO tte = \ @ V,du(x), rne Vy — namMepumoe nome
x
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2 B stom uyHktTe Õ0:166 HOTADÕHO paccmorpumM nomHyio rpynuy [G] c GOpa3yiomuMy Biya (1.3) B Ip€7U10210?R€HU”H, 410 @(x) = @ (Const) (cm (1.4)) B orom exyyae Oygem ropoputb, WTO Aelicrpue Gua XXY aBaercs# ueprio- IHW€6GRMM
BO3HHR86T BOIPOC HH RARHX OIOBHHX TĐYHHA Ở MO?ZRGT UMeTb I€DHO0/I460RO© 716ÏiGTBH€., TB@T BBID3?EA6TCH B T©DMHHaAX HHBADHaAHTA 7(Œ) (HE HHaue, 7 — MHOZReerBa) 4n Œ |9]: T(G) comepamr Boe te R, y21H KOTODEIX 0VHISOTBV@T B@IHI©CTBGHHAf HSMDIHMANH (yHKHHH ¢(@) TARAH, WTO exp i(€(gw) — €(w)) = exp itlog(do(g@)/do(m)), g€G, roe (2,0) — npocrpaH- crso vledera, B KOTOpoM jeiicrsyer G IloHaATHO, WTO T(G) coBuaqaeT c mHBapnanrom HoHHa 7(M) [2], rae M — darrop suga M = W*(A, G), A == L*(Q, 0)
TEOPEMA 2.1 Hyems G — cuemnan epynna ve cuneyanpnoxe npeodpa- 80eanul npocmpaHemøa Jledeza (Q,0) Ecaw Te T(G), T #0, mo G uneem
nepuodureckoe Oeticmeue
Hoxazameavemeo Paccmorpum Mepy P na 2, nonomus dP(w) : =
=exp(—(w)/T)do(o), log (dP(gw)/dP(w))=! EN OE / a
°pC -c6@)J1)4966), *orna log(4/0)14760) s[ exp(—š(@)/7)_ dø(œ) 2nn(@) ;
=-—-—+*, mie n(a)€ Z CaeqopareabHo Mepa P ABOAeTCA ¿IARVHADHỌL H, Kak 1 B [6], MomHO onpegeanth QO, u S,, jelicrBylome Ha (Xx Y, ux v) npuywemM
XXY=Q, uxv = Pu dP(Q,@)/dP(@) = = n, neN,
dP(S,w)/dP(@) = 1
Ec:n Gy, — ,B0licrnenHaa rpynma qua G, To G, jelicrByer B Wpo- crpanctpe XXYXR c mepoti duxdvxe~"du, we R
llycTb Z — HONHHPOCTPAHCTBO, HOPO/ZRI€HHO€ H8M@ĐHMBIMH HO/UMHO- mecTBama XX YxR, HHBAPpHAHTHHMI OTHOOHT€IEHO G,(Q,, Sy) Eccm fe LZ), to
T(x, wu =f(Ox, u+(22n/Tyn(x), xe X, ue R,
rae n(x) e Z (om (1.5)), n(x) > 0 B dacTHOGTH, ecam e,(x, u) = expiTnu, néZ, To e, € L*(Z)n L?(R)
IIlycts Tenepb p(t), t¢ R, — usmepumpii norok Ha S=XxYxXR, T©ÏiCTBVIOHIHHI ©OTJIAGHO dopMysie
p(Œ)(x, Vs 1) = (x, y, u + t),
Trang 7ABTOMOPOM3MbI WHLBEKTHBHDIX MAKTOPOB THITA Ig >
== p(2n/T) Ha atoii anredpe rpynna p(t) copnayatonjan, ovesugquo, c p(t) HBJIHĐTCH H@ĐHOJHWGCROII c mepuopom 22/7, a nockombKy p(t) — 9pTO- awueckuit noToK Ha Z, ro p(t) — aproxuyecknit noroK na Z,, mocmequee: osHayaeT, uro L(Z,) ~ L™(R/(22/7T)Z)
llyerp Ï — rpynHa aBTOMOPÙM3MOB S, nopommenHaa G, u_ p(t) PaccMorpMuM waMepHMHH rpynnown SXxPƑ [lá] Mnozseerso S¿ = XxY x {0} HIBJII©TGH, O4@BHJ[HO ©TO HOJIHBIM JIARVH8DHBIM OH€THBIM G©@W€HIHM (GM OHIPG eeHHe 2.1 [14]) Tax KaH Ha Š; I1©ÏOTBV€T AHIPOROHMATHBHO KROH€WHAHẳ rpynna G(S,,Q,), To SXIF TAR?R© HIPORCHMATHBHO KOH€HHHII TDYHHOHJL (cm ompey 6.1 w reop 5.3 [14]) nonoốen rpynnonTy SạxŒ
C HDYTỌ GTOĐOHBI, HOHTH RA?£1Ọ TOWR© Z € Z¡ OTB@WA@T H8M€@DHMAH Ò0/i0dEA S, opouret rpynmpr ©, nopompennolt G, u p(2n/T) B cuny paccym7eHuil, HHB€JICHHBIX BBII©, S, Take ABSIAeTCA T0JIHBIM JIAEY- HADPHBIM OH4ƠGTHBIM O@W€HHM JIA rpyntouga Sx, nosromy Sxl TâHRe 1010ðeH Š.X<Ƒ; Ho rDynna Ï;¡ Ha S, wMeer mepwoggeckoe jeiicrsue, Tar Kak G, coxpanaer M€DY, Ø(2m/T7) /I@loTBV6T 3DTOTHUGGRM Hà 8JT@ỐPp€ H3MG- DHMBIX HO/MHO?/KROCTB, HHBADHAHTHBIK OTHOCHTeAbHO G, H YMHOMAeT Mepy H, Ha S, Ha e~*7/T, npuyem p(2x/T) € W[G,)
ƠGTaA7I0Gb 38M@THTE, W†O IOCKOJIERY TDVNHOIHJI SạXỚ H S,X TT) H0TOỐHM,
TO HX 8000HHHĐOBAHHBI6 IHOTORI H30MOĐÙHH 2
IHIPE7UIO?RKEHHE 2.2 Hyeme W, — cnetuasonoii nomorn © Ốđ8HCHbiat asmomoppu3zmMom Q-! u nomosounok Gynnyuett p(x) = @ Ecau % — Gopenes-
ckul, acmomoppusm npocmpancmea (Xy, Mo), 20 Xo = {(x, u) | x€X, 0<u<ø0},
a duty = duxdu, rommymupynoumuii cW, mo % umeem eud %g(x, w) = (ax, u + p),
2de a € Aut(X, nw), eO = Ox, p = Const
(IHoxoốnoe yTrBep;£eHne ZoKazaHo B [21], cm raxme [19].) Hoxazsamessemeo Hanomuum, ato
W(x, vw) = (O-"x, u + t — Zn, @))
IIM Z1, @) — w S † < Z{n + Ì, 0) — u, rđ1e Z(n, ø) = @(Q~1x) + + 0(Q~"xỳ (quan 2 0) Noaromy ecau
(2.1) a(x, u) = (x',u,), O< U<@, TO TOCKONBKY W% = aW,
Trang 810 5B Ø, TO¿ZIOƑTEITI
Oreclo/a € HOMOHIBIO AHA7IOPHMHBIX paccyz:xeHnili MO?RHO IIOKA8ATb, HTO
(2.2) %(Q~"x, 0) = (Q—"x',u), ne Z
djatee, Tak Kak 2, — ỐOD@/I€RGERHI @BTOMOpẰH3M (Xạ, Hạ), TO MHO- 7R©GTBO (X/,x) (=4%(x,u)), roe x eX, ecrb Gopexepckoe HO/TMHO?R€CTBO Ð Xọ Boxee roro, cayuaii %(x,,0) = (x', ứụ), %g(Xs, Ư) — (X”, ve), THE ty F Ue uw O<4,< 9, UWcKMOUeH, HOGRKOIbRV (x’, v2) = Wu, —u(', u,), @& 3HAYUT (x;, 0) = %ạ "W4, —u đo (x,,0) = (41,4 — 4) HỆ Xị =Agz, =y, Ho Torxa X” — 1z ĐCTb ÕOD@zIEBGRAR (JVHRIHIIT, a BBHTAY (2.2), yHHTBIBAR 9ÐTOTNHNWHOCTb
Ĩ, MO?RHO C716/IATb BBIBOI 0 TOẠI, WTO „ =: p (Const), Ư < p < ø
Pacemorpiim Tenepb asromopdusm ƒ = W „xạ Jlerko BH/I6Tb, WTO B(x, u) = (x0), re x” onpeneneren Tak se, Kak H B (2.1) Ho rorya x—>x' == a(x) ecth aBromop®@usm (X,), KOTOpHH BBHNY (2.2) HOMMY-
THpV€T 6 Ở A
TEOPEMA 2.3 Iyems G yOosrembopsem ycnosuas 1emuot 1.2 u, Goce mozo, G umeem nepuoduueckoe Oeticmeue Ecau xe N[G], mo cywyecmeyem té[G] maxott, ymo B = t-1a w.ueem eud
(2.3) B(x, ¥) = (&x, Wy),
2de % — acmomoppuan (X, 2), 2O =Q2, a M„ — UIMCPUMOE NOI G6H10- u0p(uamoe (Y, By,v), npuuen Wye.W[S] 019 noumu ecer x u O(W,) =
= p — log(du(&x)/du(x)), 20e p — Const
SAMEYAHUE 2.4 Ilverh S — aprognaeckiii apromoppusm (Y, By, v), CcOxXpaHAlMii o-RoWeINyIO Mepy v (rpynua Tuma II ) Torga W[S] e046PZRHT O;IHOIADAMGTDPHHGCHRYIO HellpepbIBHy!O Tpynumy aBromopdusmos p(t), te R, TaRKy© ro p(t,)p(t.) = p(t, + t2) m v-p(t) = e'v I[pumep TpYHHBL © TARHATI CHỌOTBAMH MO/RHO HOGTDOHTb C/I6AYIOUHIIMI OỐpA80M lÏycTme G rpynua Tunø lHÏị, a Ớa — 7BolicrBeHHan Tpynna (cm (1.1)), torga G, 0Ố:18/1A6T H?RHHIMH CBOIIOTBAMH ÏÏOHRTHO, WTO Ớ, I€ÏÏCTBY€T ĐDTOTHMĐGEIH, GOXPAHRET Mepy dw(œ, +) = e”“du(œ)du ma Y =(@x<R lĨoTroK OHD€NGIHM COTZIAGHO gopmyue p(s\(@, u) = (w,u — s), torga p(s), se R, u Gy, KommMytTupynr
Horasameascmeo meope.we 2.3 Hapagy c rpynnoli G = (S,, Q,) pacemo- tpum rpynuy G’ = (S;, Q,), Koropad welicrayer B upocrpancrse (Xx Y’, nx v’),
Tae (Y’, v’) — Takoe we, Kak I B 8aM@waHum 2.4, a (X, ), KAK H ÿ TPYHNMI Ớ
Sex, y) = (x, Sy),
Trang 9ABTOMOPG@HOMbI HHBbERTHBHBIX (ĐARTOPOB THHA HH 11
rne Ủy = ø(®(U,)), p(s) — aproMopjwaM n3 f[SŠ], pAaOGMOTp€HHHH B 3aMme- waHHH 2.4, a &(U,) = @ — log(du(Qx)/du(x)) [lonnrHo, aro y G’ acconuupo- BAHHEBLH notroK W(G’) HM@ĐT B KAWECTB© HOTO/IOHHỌ (YHEKHHHM @(X) = @, a B taqeGrBe ỐasancHorO aBToMop(HaMa Œ, T.e W(GŒ') = W(G) Ho rorna B cuny peayabTarop [6] rDYHHBI G u G’ cua6o skBuBaneHTHE Tlosromy 6e3 orpa- HUYeHHA BE OỐHIHOOTH MO?RHO HD€HHO2IO#Tb, uro U, = p(®(U,))
Ilyerh tenepp œ„€.f[Œ], Torna moda, — asromopdusm (Ấp; Hạ), KOoMMyTupylommi c WG) B cuny upeqnonomenun 2.2 (moda,)(x, u) = = (ax, u-+ p), re x — apromopd@usm (X, 4), KomMyTupyromuit c QO Tlonomuu
(2.4) B(x, y) = (ax, p(p — log (du(ex)/du(x))y)
B cusy cxBolicrs p(s) (cm sameyanue) Ø,€.#T[S] lloKazreM, aro
0,8, == P,Q„ Jleiicrsurenbuo, Tak Kak
du(Qax) du(ax)
fy) cs 1X, — log —=— — log ——— ?
(Ø;8,⁄x ») (ox o(o — oe u(x) Jp (? _ |
du(aQx) du(Qx)
x,y) = [aOx, pl p — log HE? — tog SHE),
(BQ, \X; y) (so: "|? og đo Mị» og uo )>}
TO B CHIY CBOHCTB HĐOH3BOHHBIX PAHOHA paste HAGTH aTux paBeHCTB ©oBna/taor, T.e O,8, = 8,Q, Ho torga „€.#f[S] m y„ = „6z°c ÝTG), upu4em mod y, = 1d C2I61OBAT@JIbHO, a, = ÿ„„ DB cwzry zIeMMBI 1.2 CyHIeCT- Byer /€[Ớ], raRoli, ro (~'y,)(x, y) = (x, Ứ¿ÿ), rne OV.) =0, Ho ToTHA t-ta, == (t~1y,)-B, BBMJAV (2.4) đ6JIA6M BBIBOIT O GIĐAB€7IHBOGTH (2.3) ZY
3 I[pu qonasaresperse 2.3 ỐEIIO HOKA88HO, B HACTHOCTH, ITO BCHKOMY aBTOMOpwaMYV « us C(W), rue M⁄, — GH@IIHAJIbHEUÍ HOTOR c mocToAHHOit
TIOT07104H0lf ÿHRIINGII, OTBeqaeT aBTOMOD(HäM a, € W[G] taxol, aro moda, =a, rye G — rpynma, WA KoTopoli W apuaerca CGOHHHDOBS8HHEIM HOTOROM dloxaskem sToTr pesyubtaT B oOmjeM Cary4ae
TEOPEMA 3.1 Hycmb (Xq, tig) — npocmpancmeo, ¢ Komopom Oeticmeyem cheyuatonei nomor WG), accouuuposanneiii c epynnoli G, asmomoppuanoe npocmpancmea Jlebeza, a « — acmomoppuas (Xp, to) us CCW), v.€
3.1) aoW,= Wa, teR
To20a cywecmeyem ở e [G] mawoă, wmo mod & => a
Trang 1012 B A POCTIOEIE
cRpeMmeHHes npouscedenues A = L°(XxY,uxv) na epynny G = (S,, Qg} (cst (1.3)) Toeda cywecmeyem % € Aut M maxot, umo mod % = &
(20 Hp/LI07R€HH© 7OKazaHO copmectHo c C WM BeayT:IHM.)
Aoxazameavemeo HanomHun, 470 X, ects {(x, ä) |x € X, 0 < ứ < ø(Q~1x)}, djto(x, u) =: du(x)du Paccmorpum nwa X, mepy &:
(3.1) dk(x, u) == e~*du(x) du, a B WpocrpanctBe (X)xY,kxXv) paccmoTpumM rpyniry
Sex, u 9) = (eu SY),
roe S — oproqnuecniii asromop@usm (Y, ¥), COXDAHHHIHHH o-KOHeUHYyIO Mepy v (Kak 1B (1.3)), npuuem Oynem npep~Noaarats, uro (Y, v) H S onpexe- JIĐHEI TAK ?©, RAR II B 3AM©SaHRHH 2.4
llyorb U, = ø(®(Ú,)), rne ø onpeneeno s8 3aMewuanmn 2.4, Ú, w„ ®(Ữ,) — rakiie ae, kak u B (1.3) Torga U, ROMMVTHDVIOT M€ZRHV GOƠỌ 71H H.B x € X Jlossromum Z0, U-, x) = J, Z,U-',x)= ][U `, Z(1LU-'x)= Ivu, i>i>o Ø9 x i>iza Ox Torga yaa / > 0 1 _ (3.2) %(Z(, U~1, x)) = Zl, UY), == %(0.~¡ ), inl *
aH82I0THWHO onpesennetca O(Z(/, U-}, x)) gaa 1] < 0
Trang 11.ABTOMOPЮII3MbI HHbERTHBHDBIX PAKTOPOB THITA Hy 13
B cuny Takoro onpefesenua V, moroK Wit) € W[S] ana te R, Gonee TOrO, Wt)S = SW(t), te R Haureli yesbw ỐVR€T pACHIIDGHH€ ở WO ABTO- Moppusma a, wpocrpancrBa (X)xY,kxXv) raKoro, WTO a, W(t) = W(t)a, " a, € V[S] Iponenaem BGHOMOTATGIIbHBI BBIHHGJICHHRH
W(t)—1d(k Xv) dk x v)(W (x, u), Vix + Oy)
d(k xv) 4
Trang 1214 8 H TO:I0:1EHN 1102IVHHM 07I81Vÿ1OII@6 OOOTHOIIIEHH€ O(V(x', uw +1) — V(x, u + 0) = dW, tk t — log (a(x, u)) + 1+ tog 24 & wy =—t— »u)) + t + log —~—— (x, u) = dk dư~1W 1xk qdW1k = log (x, wu) — log —~— (a(x, u)) = dW tk dak = log dk (WG, u)) + log — 2 — (ate, u)) + dx~1* dW 1k -++ log —-— 8 (x, (x, u) ~ lo 1) se (a(x, a(x, u)) = u)) lk 1ÿ =log- 5 ` @) — log —* (wx wp dk Takum odpasom, ecan O0Õ08HAWHTb ~Ik ⁄œ, 1) = log- da (x, uv), T0 (3.5) %(V(', uw + ?)) — OV G,u + 2) = fy, 8) — ƒ(W,(x, 0)) ]ÏQ110zWHM renepb B COOTBETCTBHM € 3amMeqanitem 2.4 (3.6) Pox —= p(—ƒ(x, 8)) Torna (x„ € ˆ [S] i (3.7) Pw cx V(x, u + OPE == VỆ, t),
rye (x’,u’) «= a(x, u) JleiicrsurerpbHo, nockoapRy V(x, u): > ZU(x, u), U-4, x),
a U, =» p(®(U,)), to V(x, u) = p(ZU, u), (Uz), x)) = p(P(V(x, u))) Oveiona,
yuureran (3.5) u (3.6), Baroy (3.7) Ho roraza
(3.8) x(x, H, y) = (a(x, 1), Pœ?)
Trang 13AHTOAOIPG®113A1BI 1IHUbBEHTHBHBIX MAKTOPOB THIEA IL, 15
(bartopa M ’ KoTopsili ABJIACTCH CKPeUICHHEIM WpousBesenHuem L*(X)x Y, kx v) na S, a satem Ha W(t), te R
Ilyctb — œKpemeHHoe npowsBexeHne L™(Xy)xY, kX v) Ha S, TOr1na N — Il,,-aure6pa, ee cueyq 0Ø0gHaqunM wepe3 r lĨoTOR W(t), tŒR, HHHy- 1npyer rpynny 0), £e R, àroMopjaMoB W, npHweM 8 ©HJ1Y (3.4) r - Ø(f) = e7't, 0JI910BAT©7IbHO, 718 M=Ww *(N, 0, R) MBI HMeeM H€IP€PEIBHOG pa3smomenne \looromy raaqKuii noroK Becos yaa M copnagaer c W(t) (cm ra IL memma 1.4 [7])
Ma nocrpoeHHH Đ cienyer, aro N — aHHPĐOKCHMATHBHO KOH€HHAR
ˆ
(uum unpbenTuBHad [13]) anreOpa Tuna IT Ho rorga M Traxme HHb€RTHBHHH MakvTop, WOCKONbRY OH ABUIACTCA CKPeleHHLIM mpousBegeHuem N Ha amena-
^
ỐØenemnyo rpynmy ero asnroMopwawos R [13] Ho rorna M ~ M, tax KâR VY HUX HOTORH B©OOB I80MOPÿHBL ( (cm [5], [6]), TaRHM o6pasom, BCAKOMYy
a € C(W) orpeyaer & € Aut M Z
Horxazameavcméeo meopeme 3.1 ipemye Bcero 3ametum, uro B cusy (3.4)
a
pas6uenue mpocrpancrna (X,XY,kxv) wa opbuTe W(t), te R, usmepumo Jleitcrsureabno, paccmorpum W(1) Torna pBBHxy (3.4) cormacHo aemme /lan (cm xemmy 8.8 [15]) pas6uenue (X)x Y,4x v) Ha Tpaextopun Wl) 18M6GDMMO QỐốoaHawIM 83P0 pA3ÕỐHeHwe 4©p©3 Z¡ H pA©€GMOTPHM Hã (AKT0D-HĐOGTDAHGTB© X)xXY/F, woTok Wit), te R Tak Kak Wil) Ha XạXY /Z¡ HBJIHGTCH TO?RH@GT-
A
BGHHEIM IpeoOpasoBaHnem, TO W(t) Ha XoXY | HBIIHĐTGH H€PHOJTHH6GRHM IIOTOROM 6 IepHoTOM paBHBIM 1 Ho rorna Ha XạXY/Z¡ GVII@CTBV6T MHHBA-
^ aA
puauTHaAd Mepa orHocuTenpHO W(t),0 <t < lu W onpenenaer ofHonapame- ĐPHW€CKVIO CHJIEHO HelpepbipHylo TlepnHogquyeckylo TĐYHHV VHHTAPHBIX OI©€PATOPOB 0G AMCKpeTHHIM cneKTpoM TCIOJIA MO?RKHO 8AKJHOHHTb, WTO
A
pas6ueHue mpocrpancrsa X,xY/%, na opouTn W(t), O< 4 <1, usmepumo llonatHO, ITO pasBueHne % = %,v.¥%, IPOCTPAHCTBA XạX Ý TAE?R© H3M€DHMO H 2TO DA8ỐH©HH© 00BHA/TA6T ©€ DA8ÕII€HH©M Ha TDA6ETODHH FIOTORA Wit), te R Ilycrn Q= X)X¥/¥ M ơ — Mepa Ha 2, HH/VHIHDOBAHHaAR &X v, TOPHA (XaxY, kx v) = (2x, dơxe~*du) (cm [16]) 11 W(t) đ©elcTByeT Ha (œ, ) @ < R cormacHo (ÙOPMVZI€
Wo, u) = (@,u + 0)
Trang 1416 B f1, TOZI107Ið
‘yorga ss — ABTOMODÙHäM (2, R), TOZ:1€CTBEHHO JelicrByiomyii Ha @ m ROMMYTHPY ION © Wit) Ho torga Sos = Tow) , THE T,(@, u) == (@,u + g(o)), -8 @ —- H3M6@JIMAH (byHtItH Hà ©, T.G So, wu) = (S0, tr @(G)) Ản8z10ptrqHOG, #, 0IP@/IG:THHEIÌT GOrzIaCHO (3.8) umeer Bi 8(@, uv) = (0,0, u -+- W(@)), rae a, —: aBToMoppusm @, upnuem 2, € A[S,], nockoabKy ae ATS] i & Wit) Wt ye, teR,
TÍDHBGH@HHbIS DAGGV?E/16HHH TIOK88BIBAIOT, 470 5, ne Z, ABUIAETCH jIBoli- ‘CTREHHOH rpyunoli (om nm 1) aa S?,n e€ Z, a notoK W(t) accoymuposan c SY, 'õo.1ee 'roro, quia « € C(W) cyulectByer apTomopdusM a, , Upocrpancrpa Q, rye Aelicrnyer S,, upnuem a, e M[S,] 1 moda, = «a ZB 2 HECROLIBRO TEOPEM O CHUEILLASIBHOTO BHA ABTOMOP®H3BMAX
HHEEERTHBHBIX ®AXHRTOPOB THIIA ITI,
llycTbÕ Äƒ — HHbeRTHBHMI (arroD trina lil, W, — ruayeuii moroK
Becos ana M Bia 1, 2 paccMaTpHBAIOT€R #, 8 € Aut M, y ROTOPEBIX mods =: ==modf = W,, te R, w /IORA3BIBA©TGH, WTO TâRH© 8BTOMOD(H3MBI BHeIIHe ‘CONPAAREHBL (ecu ƒ = Ơ, TO npennoziaraeren, 4To p,(@) = p,(B) =: 9) Bu.s nsyyaiorca «@ e€ Aut M taxne, uvo a(M,)-= ÂM, 7GIH H@ROTOPOTO clakyHapHoro Beca @ Ha M,
{ lÍaqHe@M © HOR88aT€¿Ibcrna
TEOPEMA 1.1 HỤecmb ME — unoexniucneii parmop muna IIl,,% € Aut M a moda: -id, mozda a € Int M °
(Kak OTMGTHGIE DEH€HSEHT, 0T0T P€3V¿IbTrar aHOHCHpOBAH B [20], qoRa- ‘BATEIBCTBO IIỸ/HIRY€TCØ BIIĐDBBI©.)
jJfơkbdaiaimebcmeo B cnry 1V.1.10 [7] na M cyutectryer tT Hopm erporo 11071ÿROHGHNBI JIARVHADHDHÏI BĐO My TAKOH, ITO Py % = Oy Ẩm := 1đ, TJI@ Zo, — WenTp Mo,» aag-= Adir-ứ JỤNH HeRoroporo € U(Mf) /Íazree, connacHo neopeMe 1.5 [17] @YH(eCTBV€T I0O8HTPHBHBBH & € Z, , WA KOTOPOLO @(+) == Po(k-) ABILAeTCH Q -~ IOYTH-IlepHOANGeCKUM TOUHLIM HODMAIbHBM CTPOTO M0-1Y- konequpm Becom ua M Tan kak My © M,, 10 Z, = Mạn MẶc Mỹ nM ~ Z4 (cm Teopemy 1.5 [17]), a UIOCEOIbBRV Á€ Z4 H a(k) =k, To
n
ˆ '
(LT) @-ä =: @(kĐ(-)) = ø, iz, = id
HGCHOZIb8OBB8BHIHCB T€HGB TOẠIH ?R{Ơ 000ÕĐ9/7R0HHHMI, GTO H IPH HNOR888TCHb- no TeODGMEL [Í.,Í [5] MO?RHO HOCTDOITTBE BOSDAGTAIOILVIO TIO071610B8T€7IĐHOCTE ieliwanoBCcEnX nonanreốp J⁄, c M tuna J co cBoiiernamn: (J M,)” = M;
Trang 15AB TOMOTPGIE3MBE THƯPENTHDBHNHDIX DPATTOPOB TITILA Hip 17
0Vm©einHe @ Ha AM, TO/IYROH€HHO; Â, — Ø”°,/€ R, HHBADHAHTHA, k€N; ec1n N= (Mi),, 1O T@HTP Ấy, COBHAjqAST © 24, (U Ny" = My; WIA BCHKOTO adberescro mpoerropa e € N, ¢ WeHTpadbHbmm woenteiem papubim Ï, @(e) < co; CVINCCTBYeT YeuOBNOe OMManHe E, wa Mua M,
llonanem, wro juris Besworo k € N cynecteyer VHHTAĐHEHI u,us M,, qa RoToporo &(x) = Adu,(x) mpu xe M, Ilyers Z, —nenrp M, , rorga Z, € Z2 lloisrno, aro B OM, GVH(G0TBV€T HOT AKETOP R, Tuma /, Takoli, ato R, u Z, moporjjaior M, HanomawmM, wro cornacHo wocrpoeHuio (em I] [5]) M, OŒTDb CHRDGII(CHHOO IIDOH3B6/(GHHI© N, Ha KOHETHYIO KOMMYTATMBHYIO rpynny r, asromopdusmor N, , elicreyroutylo cBobonHo Ha Z,, upnuem ecun u € U(M,) uAduwel,, vo of(u) = zifu, rye ze Z} (cm HoRagareabperso teop I] 1 [5]) lovromy 2, MOIHHO RBIOpaTh Tak, HTOỐBI ©TO MaTPIGNBIe €NHHUNB @;,, ijrol,2, , oOsayacim cBotictBamu: e;,€ N,,i==1,2, ,u ele;;)= P(e) <0; of(e,,;) = 27501; Tne z,,€ Zp ,i,/= 1,2, Ho rorya B cnay (1.1) 1€HTpA7ISHBI© HOCHTEIH @, HW &7"(e,,) BM, CoBmajaior Hu CYINeCTByeT YACTHUHAA WZOMeTPHA 1 n M, rakast, aro ww = ae) nw*w = e,, Een nomomnta u, = Sa-e wey;
j
ro uw, € U(M,) mu Adu,(x) = «"(x) ju xe R, Kpome toro, a74(z) = 2 = =Adu,(z) qn ze Z, Uvan, a(x) = Adu,(x) nan xe M, uw o-Adu, = @ Jloxarkem, wTro lim Aduj! = % OTHOGHT€IbHO Ø — TOHOZIOTHHI B Aut M
k
(cm [18]) llycrb 9ì = {x :x € ẤM, @(X*x) < CO}, TOFJA JHIH@ÏHBI© ROMỐHHAT (PYARTMOHATOB BUA W(x) = @(hxđ¿), rae h,c 9ì,/—=l,2,ax € ă HHIOTHEIB ÂÍ,, Kent £, yes1oproe omupanne BM ua M,,ahe, ro s-lim£,A = h Kpome
k›co
moro, naR kaK (E,h)*Eph < h*h, ro Eh e 9ìn My, woaromy saunetinsie ROMỐM-
HAIIHT (ÿHRIIOHAiIOB BHqNA /(x) = @(Nxổ;), re đ,ce9tn(J AM), a xe M K
1UI0THĐH B Aƒ, HO 7UIH KAHUIOTO TAROFO (YHEHIHOHA/IA4, BBH/[Y TOTO, WTO @-# =z @ 1L 0-Adw, cc @, KT, HỤIT NOCTATOHNHO O2BÐHHIX k uMeer mecTo TIABEHCTBO yar) = pla“ Ay )va he) =: (Ad u,(4,)vAd 1, (tg) =: = WiAdug(x), xe M Orciona crenyer, aro lim Adu? = « Bp — TOHO/I0PTHH B Aut M Z k
C.TENGTBHE 41.2 Zyeme M — maxoti sice, ra ue meopene, 0; € Aut M, j- 1,2, mod0,; = id wu p,(9;) = 0, mozda cymecmeyem a € Int M, 022 Komopoeo @, Adu-o+0,-071, 20€ we UM) Eeaw owe p0;))>0 u p,(0,) = p.@,),
062 id, mo 0 u 0, conpsarcerse
C2I0/I0TBIG BEiTekaer H3 TeopemnI 1.1 n reopeMEIl 2 [1]
Trang 1618 B ữ TOzIO/EH 2 llepeiiyem k paccmoTpenmw x € Aut M, y Koropsix moda = W,, rt # 0, roe W, — raaaqKuii noToK sBecos xqan M [7]
tIEMMA 2.1 ITyems M — maxoii oe, ran u npeocde, ecru x € Aut M, moda =: W,, t #0, mo p,(x) = 0
Aorazamenecmeo Ilycth p,(2) =m > 0, rorga a” € Ct M [1], Ho coraacHo [2], ctp 467, CtM = Rangedy Ilosromy mod a” = id, t.e W,,, = id 470 HCRJIOWGHO, TAK Kak HOTOR III,-akropa He ABIAeTCH H€PHOTIHG-
ckum [15] 2
TEOPEMA 2.2 ITyems M — unsexmuensiii giaxmop muna III, , 6;¢ Aut VW, ¡ = 1,2, mod; =: M,, ¡t # 0, moaÐa 0, w 05 enewHe conpsotcenr
Horazamercmeo Tak Kak M — daxrop Hpurepa, ro M@N ~ M, roe — HHb€KTHBHBHI (akgfop tuna I Ilyerh p(t) AutN ranoii, aro t-p(f)=e't, Te t — T.HOpM NoayKoHe4HEni cnoeq Ha N [lonatHo, aro id @ p(t) € Aut(M@N), p,(id@p(t)) = 0, mod (id@p(t)) = W,
PaccmorpumM asBromopdusmet », = 0; @ ø(—?) @ 0() € Aut(M @ W @ X) (= AutM) Torna @;@p(—t) € Aut(M@N) (=AutM) u mod(@;@p(—d) = id, B city Teopemst 1.1 0;@p(—‘)e€ IntM Jlanee vax, Kaw p,(o(—?)) == 0 To p,(0,®@p(—t)) =0, Ho Torga no Teopeme 2.3.1 [1] y; BHelniHe coupAKeH id@id@p(t) C XpyTOH C€TOPOHEHI, AHAJIOPHHHEIS DAGOY?RIEHHH IIORA8BIBRKRT, MTO ?; BH€IHIH€ COHĐHN?ReH Ø,@id@Id (a 3HaWnr H Ø,) C:I1€10BATGIbHO, Ø,,
j=: 1,2, puemHe conpamenst %
Trang 176OIDHZRGH-ABTOMOP®H8SMBI HHDERTHBHBIX (®ĐARTOPOB THHA Til 19
HBIX ABTOMOD(ĐM8MOB 7718 ă GOBH8/86T G 3 llookozregy M*(R;, >)@ W*(R;, Ð) ~ ~M*(Đ¿, Y), TO MOX:HO /UIH EJIACGOB BH€HIH© GOHDH?Z4GHHBIX 8BTOMOPÙH8MOB OIID©H©JIHITĐ VMHO?R€HH© B COOTBETCTBUM CO ©JI6XVIOHIHM VMHO?R€HH©M GơMHX aBTromopusmos: ecau a, Be AutM, tro z@ec AutAƒ@M (SAut22) Ecan moda = id, p,(@#@) = 0, ro «@f BHeHmmHe conmpaxken f (B cumy teopemp 1.1 u TeopeMEI 2.3.1 [1]) C2I@1OBAT©JIbHO, KJIAGG TARHX aBTOMOPH8MOB 2, OOpasyer €QMHULY OTHOCHTeIBHO BBeeHHOrO yMHomenUA Jlamne, ecm ở, 8€ Aut M u mod # == mod 8~', ro mod(#@f) = id u a@P € Int M@M (v.c c@P upunanze- ?HMT GJIIHHWNHOMY K¿IaOOV) Arak, orHocuTebHO BREeHHOTO YMHOMKCHMA MHO- *KECTBO KJLACCOB BHGHIH€ COMP AKCHHBLX ABTOMOppu3sM0B M=W*(R,, 2) c p,(@)=0 TIDEBDAHIA©TGH 5 TpYHNHY, 30MODÙHY1O 3, 1 # = mod 00ÿHIĐCTBJIHGT 8TOT W30- MOpjQH3M OỐIHMÌ CJIydWAÌ MOZi@T ỐBITb DACCMOTP€H AHA/I0PHMHO
3 Ilycrs M — mwbeRrwbHbrii Đagrop rana IIlạ Corxacno [5]jaRrop M wsoMopen jagrony W(A, s, Z7), rxe 4 = L®(0,ø), a (@, ø) — TIp00TpAHGTBO d[eOera, œ — GBOỐOHHO ;I©OTBVIOIHHH 2pP0/MW©CKHH apromoppusm (Q, 0) Cuenys [6] npocrpaHorBo (,ø) MO?RHO HDG@JIGTABHTb B BHne (XXY,iX V), T]H€ /¿X V ~ Ø, a B HOJIHOïi rpynue [«] BEOpaTh odpasyiomme (S,,Q,), Helicr- Bylomme cormacuo (1.3) § 1 O6parHo, ecam 3ananE(S,, O,), TO MOAHO NOCTPOUTS asreOpy N= W*(A, S,,Z), tuna IT, rge A = LO(X XY, eX v), aBaTreM NOCKOIbKY Q,€ V[S,], ro QO, O0Ip6/GIđ6T 8BTOMODÙHSM N, a O7IHOBATGJIbHO MO?KHO
IOCTDOHTE (baRTOD M = W*(N, QO, , Z) [loHnrno, wro M ~ M, vak Kak TIOTORN Becos (cm [5], [6]) gua Mu M cosnanaior
Bec p ua M, KOTODBIf MWHyIMpyerca Mepolt wxXv Ha XXY aABuAeTCA WakyHapHpim, ak KaK B cusmy (1.4) §1 Sp(log4,)n[—5, 6] = 0, rge 4, — Mo- HÿJHPHHHH 0116parop 71H M , 0TB6WAIOIHHH Beoy p OOparHo, B cuy pesyib- ratos 5.2 [8] u padorst [6] qua BCHKOrTO MakyHapHoro Beca p Ha M MO?RHO CHHTATb, aro M umeer GTDVRTYDY M, Torma N= M, ˆ
J[H yjanbHelitero momesHo samerurn, wro ecumn Z(N) ~ L°(X, H), Ay = L°(Y,v), a N= W(A,, S, Z), To =Z()đN, \T.â ĐJICM©HTAMH n © N CILyKAaT USMEPUMELe OlleparopHute Mota x > n(x), CO 8Haq6HHRMH B aKrope N JIEMMA 3.1 Hycmp « € Aut M, moda =- W,(M), p € R, u a(N) =N, moz0a cywmecmeyem t € Int N maxoti, umo cyscenue B = t-te na Z(N), yeump N, mpusu- đUbH0, T.© ỞÌ xu = 1d, 4w PỈạ = \e V.du(x), 20e x > V, — uamepumoe none
x '
asmomoppiussoe na X co snauenunmu ¢ Aut N, 2de N= W*(Ay, S,Z), npures ecan moda = W,, mo OV.) =p daa ng XEX
Hokasameascmeo Vlonomum ama upocrorn, wo mode = id Tak Kak,
^ bì Aw A
Trang 1820 LB AL PO/TO.IEH ce Ha N, a£s — ye1opHoe 02RH/AHH© M na N flostoeary p(xÚn)):= pm), rae m € ẤT, A đ —- TO3ITHBHBIHI ÕDATHAMEHÍ OHEDATOP, HPHCO€/HH€HHBHI Kk Z(Đ) C;1e1oBaTe.TbHO, (Dp-# : Dp), = đ
lÏЩ7102107RHM, WTO M Nelictpyer »& upocrpanctse Hf 3 rưibỗđep- ToBOM lpocrpancrze L°(R, H), snekTop-ynHIuii € == (€(7)) Ha R co 3nage- HHữMU B ÏŸ paẴ€CMOTDHIAM asedpy M, Tutia I, , TYacIbHVIO k M, KOTOPAH HODOz-
ena onepatopamu (em [415]): ; (XMM) = of (EN x EM, (22) := éŒ — s) t,5ER Torga 2 € Aut paciinpAeTcH FO 44 ¢€ Aut My: aa(m(x)) = a(x(x)) WA ve M ag(4,) = HZ 8 ER
\lyern K ROMMYTATILBHAA Woyadredpa M, „II0D07RTOHHäH 01I6)aToDaMH ZÊY), re xe ZN) I/,.s€R Anrẽpa K MozxeT 0BITb pea:zIH30bBana RaR /®(XxR, 421), Ip)H46M eC21L @(X, 0) 6 (XS R, 1 X00) 1 VJIOBRZIETBODR©T VeloBnnM (1.2) To n(pye Z(M,) la OID€/I6GHHW %y, €/I6/W€T, WTO ag(K) = K Teuepb §1, mockOcIbRY +4(n(Z(Đ)))= : r(s(Œt N))) = - (Z(N)), TO, TIPUMeHHH Te aie CO- OOpakeHuA, Kak MW Upli POKABATeIBCTBE ;IâATMDL 1.2 Đẽ, HOZIYHHM, MTO
ˆ
cymectrByer / € Int ă, ;IH ROTODOTO t7lz za id fq
IPEZVIO/RBHIIE 3.2 /emb epWnna G = (S,, Q_)u meen no rung 9éemeone Ha (XXY,Hx<v) llpeƠno.to2icuM, mo x€ Aut M, a(N) = Uu (mod a)(x u) - - (ax, p) 20e 4,0 == Ox, u p=: Const (ca nped 2.2 SH, To2da cyruecmeyem 6€ Tnt M makott, umo cyoicenne t~'x Ha Z(N) ~ 1 x 8 onpedeasem (01041000032 % na (X, 10) Rpome moeo, cyupecmeyent nasepiimoe node X —> VỆ UEMOMOPPUI.MO6 us Aut NV, npiuuew @(V2) == p —log (data, x) duty), umo een n= {r(x) € N, 20e n(x) € N O29 wa x EX, mo a(n): {VA Ny}
(cp.e meop 2.3 §1)
SAMEGAHIE 3.3 1) V2 — Í-ROHIR¿I ;I€HCTBHH 2, Ha (X u) co Bnade- HHINHMI B Aut N
2) ŒaKT, HA/IOTHHHBI II€/IORGHHRB 3.2, CHPĐAB@/LINIB Oe3 wpe io- NOMCHNA O HI@PUIOTHWHOCTI nNGHCTBHN G = (S, Ĩ,) OH OCHOBAH Ha pesyebta- Tax [22] 1 goRasblBaeTCA TOUHO Take, Kak Il Wpequomenne 3.2
ˆ
Z[0N03@110.1bC100 Cor JIACHO Teopeme 2.3 §Ì cvmteerpver /@6€.fT{Ớ] ¢ mod B —= mod # =: (⁄;X, ứ 2) BH/1A
Trang 19ABTOAIOPG®II3MBI HHbERTIBHBIX PAKTOPOB THOA TI, 21
re x > W, — ri mote BTOMOPÙIH3MOB (Y, v), TIPHWeM từ, Ee V[S] TUIH II.B X H $(2) = — log(d(s,x)/du(x)) Apromopiam B pacmupđeron no agrowopdusawa Đarropa M rakoro, WTO BN) =: Nun= {n(x)} —> B(n) = = (W(œï}'(x))}, rue n= {n(x)} € Â, a x W,— uamMepumoe HO2I6 ABTO- moppusmos W, npndew (W,) = p —log(duœ;x)/du(x))
HomowemM y = #øÿƑ~1, rorya mody = modz-modfØ~!—=id, m" y() == = a-B-(N) = N Cormacno tmemme 3.1 cyijecrpyer te Int M TAKOH, WTO Yam = id, wot rtyls =\ ® V.du(x), rae x > V, — memepimoe nose
x
aBToMopaMop N, wpuyem O(V,) = J aaa o.B x € X Ho rorya t-te =: t-1y-B
oOsanaer UCKOMBIMH CBOLICTBAMIT ⁄
3 .XBPFOMODP@M3MBbI HIUBBRPHBHBHX DARTOPOB M THILA TO, CG T(M) = (t:o,¢IntM) 40
3 §3 õyneT n0Ea8aHo, wTo ©G7IH y aRTopa Ä nuBapuanr 7(M) 40 u (p — iIAWyHAapHHH sec Ha #ƒ (T.© Ì HBZIHGTGH H8071HpOBA8HHOII TO4ROR ŠSp 4), TO BOHRHH # 3 Aut Mƒ MO?RHO IIDHBECTH BHGHIHHM CONpAKeHMeM K HEKOTOPOMy K“aHonneo£oMV BHNY (6M (2.3 § 1) cam mea, B € Aut M, p,(a) = p,(8) = 0u moda == y-modf-y7!, rye y € C(W) (t.e yW, == Wy, t € R), ro x w B BHemHe coupsuKeHbt B Aut M
1 \launem c onpeemenuit
OUPEDENEHME 1.1 Tycrs M ~ daxrop tuna Ill) c T(M) 4 0 Bec @ nua M dyneM HasErBaTh nepuoduueckum c nepuodom T #0, ecam of = id, HO or, # id, ne N,v > 2, 0ymGHHe @ Hà Ä, (H€HTPAJIH3ATOD @) GGTb T.HODM 11071yROHBHHBUI ©7161 r Hà Äf„, @(1) = co
EMMA 1.2 Ecau M — Giaxmop muna Ill), ROmopoeo @ — nepuo-
Ouueckuli cec ¢ nepuodom T, mo cywmecmeyem yuumapnorii onepamop U6 M, o6.1adaywit ceolticmeamu AdU € AutM,, npuvem U u M, nopostdaiom M; 6ouee mozeo, to ACU < dt, 20e t ~ m.HopM cacd na M, , NOMODbLỦ 86/89- CINCH CYACEHUCM (0 Hasee, U = » u,, 20e u, — %*GCHLHUHG8 uszomempus uz M
n>0 ,
+ —Ìnt
Makan, UMNO OF (Uy) = 2 Mạ, 1/ ROHLODOB Q, = HH HD, t= HIỀH, € Zo, uenmpy M, , NPUACM OG =? PnPm =O, M#n, US p= 3 q,,, logd = 2n/T
Tak Kak nepMoRMuecKHi Bec ~ ABNAeTCH JIAHRVHADHHM, TO nemma 1.2
Trang 2022 B XH, POZIONEBIT
OIIPETRESIBHHE 1.3 llepnonwecKnii Beo @ Hà ă © IepHOJOM 7 Ha30BeM O0WJHHblMựM â8600  nepuodom T, ecum BM cyulecrByer VHHTApHHH OHe- parop U, raRol, aro of(U,) = HU, , AdU,€ AutM,, a M nopompen M, u U, (cp ¢ oupeyed 4.3.1 [1] ma HHÍ,-ageopos),
CÿHICTBOB8HH© ÕO0ỐH@HHOTO G7167A @ c nepuojom T cueqyet, Hanpa- Mẹp, Ha reopeMEI 2.1 § 1 (em nakze [15])
1IPEZ12107KEHHE 1.4 lĨycmo M — 0awmop, ưố.4a9ai0uli 861/469 0Ố0ỐM{@H- Homan G8601 p,, i= 1,2, ¢ nepuodom T Toeda cyryecmeyem uucao a > 0 u yuumapuoli onepamop ue M maxue, umo ¢(x) = ap,(Adu(x)) On xe M,
¬ 9 ọ *
ayge › > 1 > 2 1 Qe :— r a
Aorasamessemeo.» Tar Rak 6,2 -= Oph = id, To (Do, : DQy)y =: ai, rye ạ€ TT lĨyoTrp >Ơ TaROBO, T0 27 =a Tloxamem, WTO (x) = ss ap,(Adu(x)) qa xe M, Ổ3aMeTHM, wfto ecm @¿(x) = fo,(Adu(x)), To
#ạ = (Độ; : DQ, )r == Dee : DP) (D1, : Der)e =* iT, » ? iv
== BT (u*op(u)) = BY,
ufi-+ a, rye Pp > 0, o,,,(:) =: @(Adu-) lycra P = M@F,, rae Fy L-haxrop NMonomm YC YY x,j,@e;;) = 29%) + G(x.) aun x= YO x,,@e,; rae
ij-s1,2 i,j=1,2
¢jj -~ MATpHIMHEIG GNHHII(EI yin #;, TOT/tA (em 1.2.2 [8]) Y — T.HopM mosty- KonequErii Bec Ha P Mockoapnry o¥(1 @ ey») = (Dee : DO) 7-(De, : Dagy)p@eny =
; ? : +
== Op" Bey, == 1@eq,, a Øz/ =: 1d, j r= Ì, 2, OOTJIACHO ID€NHO¿ZIO?R@HHO, TO co}: id, Ho Torga WY — uepitoquyecKHit Bec c uepHogom T (cm ompeje- ceniue 1.1) I1 GOT¿IACHO semme 1.2 cymjectByer yHuTapHtit oneparop U co CBOLCTBAMIL, ©DG4I10218HHBIMI B (ÌODXIY-THDOBRG WemMbL Tak kak Ad Ve Aut P,,
a Un P, wopoayaor P, to Ad U apronHweoRm ;teliorByer Ha Z„:› H€HTp© Đụ
lÏo:roznHm /¡ := Xe, ¡ =: L, 2, noraa ƒ/¡ € Pạ„ depes Z(ƒ,) 0ố03HaWHAM Z,, — HocH- TozIb /; TaR RaE Ad apro/tt4eoku neiicrpyer Ha Z„, to ƯZ(ƒq)U~'(1—Z(f)) %0 C71610BATG/IBHO, GVII@CTBV€T HO/MHĐO€RTOPĐ ? < ƒ¡ H8 Ä„ @€¿¡ H WAGTHWHAH 1I8OM€TpHH „€ P (GCM (ĐODAMV/IHpOBKV zI6MMHI Í.2) TAKH©, ato 30H,
< I~ Z(f,) Tak kak Bec (0¡ HBIHGT0H OỐOỐIH@HHBIM GJI610M Ha Ä/, TO B M cyulecTBye? VHHTRDHBII OGI6pATOD ;; CO CBOHÏOTBAMH, IHDĐWHCZIOHHBIMH B oupegenenun 1.3 Ho rorga e(zIM 0; -= U;,,@e;;, TO WA UpoeRtopa đ =: 0Ề"pUỶ S < fy (us My @e) BEUIOMHeHO ©OOTHOIIGHH© 01g07”, € Ï — Z( fi), upruem unvy € Py Orcioya cuexyer, uto Z(f,) = 7 AHA/IOTHSHO /ORA3BIBA©TOH, WTO Z( fo) == I Cneqosatenpno, ƒ¡ H ƒ¿ ORBHBAJICHTHBI OTHOCHT©ZIbHO ý ; IO2TOMV Đ „ CYIICTBYT WAGTHHHaAR H3OMGTDHH 0, TâKAH, %TO ĐỶ0 = f,, vv* == f, Tax Kak (1 @®@e,)v = 2(1 @éo9), TO cymectByer we M TARO©, YO v= Aes
Trang 21ABTOMOP@IU?MDIE HHbĐBRTHBHBIX Œ®ARKTOPOB THHA Il, 23
u™u == uu* = Ï MO#{HO HOKA88Tb, WTO OH€DATOP 0 ABIIAeTCH MCKOMBIM (CM
4.3.2 [8)) Z2
2 IÏepeHH€M K HMAVHGHIIQO BHGIHHH€ GOOHDH?RGHHBIX BTOMODÙHSMOB TEOPEMA 2.1 llyeme M — wHsekmueHuli agmop mwna TỰ c T(M)+#0 Licau x, B € Aut M, p,(%)—=p,()=0 mo u 6H€MUI€ COND4200€HbI Tn02Ư6 lì 90/1bR0 rmo¿Ða, ko¿Ða moda u mod conpsocens 6 C(W(M)), m.e cywecmeyem y € C(W) makoti, umo
(2.1) moda = y-mod 8-y~1,
200 C(W) — yenmpaauzamop nomoxa W(M) (cm Beedenue )
3AMEYAHUE 2.2 Caryuaii mod « = mod f = W,(M) yme pacemorpen B § 2, IOOTOMY TOCTATOYHO PAGOMOTPGTbE BAPHAHTH, Rorda moda, mod ¢ W(M) Ilyets @ —- o6o00nenHerii cnex Ha Mc nepuopom T € T(M) (em oupenenenue 1.3) Cor.tacHo npenaonmenuio 2.2 §1 mpocrpanctsBo (X,, uy) B KOTOpOM jeit- crpyet moTok W(M) umeer Buy Xy = {(x, u) | xeX,Q%< u<@(x)}, re o(x) == 2n/T, a fly == XM, TOrAA
(mod ar) (x, u) = (a,x, H + Pa): (mod 8G, u) = (Bix, u + Đạ)
rye %,, 8, € Aut(X, 2), 8 Ø„, 0, — ROHOTAHTBI, H© HID€BOCXO/RHie @ Tenepb yeuosue (2.1) o3Hawaer, To cyinectByert y, € Aut (X, p) raxott, aro a, =7,-8,-yr4,
a Pz = Pg =" Pp, Mpuyem y,O = Qy,
Ormerum Take, ITO B cusy TeopemEL 3.1 §1, (reopempt 2.3 B paccma- TPHBA6MOM ©1yae) Amt y, © C(W) cymecrByer ÿ € Aut Mƒ, npHdeM mod ÿ == ?ị, IIODTOMWV /ORA38T€IbGTBO T€OD€MBI 2.Í GBOHHTCH K ĐAGOMOTP©HHIO 8BTOMOD- usmos « u B, y KoTOpEIX moda = mod ổ
[PET JIOMEHME 2.3 Tycme M, a, B — maxue aec, Kar uo meopeme 2.1, amoda = mod B anepuoduxen no modys0 (W(M),t € R),7.e mod a" ¢(W,, 7 R) Osan VnE Zn #0 To2da a u B eneune conpasicends Kak asmomopgiuasno M Hoxazameavcmeo Momuo upennonarata, avo M umeer Buy M, xan Bu.3 § 2, rae (S,, @,) saxanbl 1o @opmysam (1.3) $4, IpHweM g(x) = y = Const {om (1.4) $4) Torga makyHapHnit Bec p, Korophit uu_yuupyerca Mepoit uxv Ha XXY, apuserca oOo6ujeHHEIM ceqom Ha M B cmbicae onpefenenua 1.3 HooroMy BBHNY npetnomenun 1.4 VMHO7RAH, ©071H Heobxonumo, & u B Hi BHYTĐCHHH€ ABTOMODÙHSMBI, MO?RHO €wnrarb, ato a(N) = B(N) = N, rye Nes M, (om a 3 §2)
Trang 2224 LB AL ĐO7IO7I1E11
.ÀHa/IOTHWHO, HHIYHHpVy€T Hà (X,£) apToMopnsM B,, B,Q == Of, u onpexeiset 1-komma x > V8 , npidem O(V8) =~ py — log (du(B,x)/du(y))
Tak Kak moda = mod, ro (GM 3awewaHne 2.2)
% = B, >
o(V2) = (V4), waa mB x
;la:z1ee, oốpaTumeom K (opMyale (1.3) §1 Ouesuguo, U,, xe X, ecTh {-Kouka xXelicTenn Q Ha (X,) co sHayeHnamn B #{S]}] llonsrHo, aro CL xe X, onpeyeaser l-RonnHdl UỦ, xe€ XY, n6HCTBHH (| co 8HA4GHHHMI B Aut N, npiuem (U2) = » — log (du(Qx)/du(x)), rae o(x) -: g == Const 1H 1B xe Xx,
Paccmotpum jeiicrsua a, u QO Ha (X, u) Wockoapbry 4,Q ==: Ở2;, TƠ TOẠI CaMBIM MbL 3a7aeM Bprojurgseckoe 1elorBne rpynnm Z° Ha (X,1) B cray [3] OTO /IĐÏGTBHG HB:IHGTCH âHHDOKCHMATHBHO ROHOSHBM TAEIM OỐP830A1, HO đ6liorBHO PDVHHB Z* wa (X, #) MOMHO OTIpeecIUT 8THDORGHMATHBHO ROHĐW- Hbưti rpynnownx ¥ = X¥xZ? (14] Ho rorxa nonunap (U2, V2) 1 (UP un VỆ) OIID@/I6:THOT Õ0OD6:I€RCRH TOMOMOD(ĐHSMBI 0; H 0; COOTBGTCTBÊHHO PPYHHOH;1A GpAutN
eIEMMA 2.4 (A Roun, B Rpurep, u ap.) Mycme G — annporcu.sa- HHHGHO ROH€MHĐLĂ UsMepusl epynnoud, G — nowweRad epynnd, Py N Py — bope.weenue comomoppusns G 6 G makue, umo
0 = 0z(mod H)
20e H — HOD.MWd.tbHŒq Ốop€.leoeka% nodepynna G, a A ee samoranue ¢ G Toeda cymcemeyem Oopereccnue wsmepu.swie omobpancenua h:G—oH u P:X:=#° + H manue, no
ply) == Aly) Prop) Psy) "1 op EY,
2de rus — 16006 1L npacoe omodpaacenue G na X coomsememeoenno (TOR888AT€7IbGTBO HPHBG/I€HO, HanpuMep, B [23].)
B wamem ecayyae G=:AutN, H=-IntN u A=:IntN Tax Kar (V2) == OVE), ®(UỆ) = ®(U#) n¿tä n.B x, a N — anIDORCBMATHBHO ROH€HHHÍI H,„-a£Top, TO n3 [Í, c:16A©TBH€ Ố] BBIT€RAGT, WTO Ø„=z Øg (modIntN) B CM:IY JI€MMBI 2.4 CVHI@CTBVET HS3M©@PpHMO€ H0¿I8 x > P,, re P, € Int N, raxoe, T0
(2.2) UỆ == PgyUfs‡yPy! ANA 11B x€ X,
Trang 23ABTOMOPORSMDE DITLERTHBHEIX DAKRTOPOB THILA IT, 2%
rye x —> ấy, Í = Ì, 2, — M8M@PHMO6 HO¿I6 8BTOMOPpH3MOB us Int N Teneps 13
a ˆ
(2.2) cnenyer, aro P = (P,) onpeneaaet astomoppusm M, a (2.3) nor/a 03Ha-
4A@T BHGHIHIOIO GOHDØZ“6HHOGTb ở 1ï Ổ Z
3 LepelineM « paccmorpenmio x € Aut M, y Koropbix mod 2” = WM) JWI HEKOTOPHX f€ RuneN, p,(«) = 0 Haumenpiee rakoe m Ha80BeM epHoONoM moda w o603HauuM epea p(moda) = q Tak Kak moda(x, u) = (a,x, u + p,) (cM mnpe;uI 2.2 § 1), To mod a? = W, o3Hauaet, To and0 of = id u p,q = 4, au6o0 ef == 0", gEN, mE Zu gp, — om =t
Haiteli I(6@JtblO Ốy/1€T 1ORA8AT€JIbOTBO G716YIOHISTO HĐ€/JI07R6HHN:
HPBH7IO/REHHE 3.1 Hựemo M, xu — mANkH€ 2€, KkúW 6 me0p€MUe
2.1, modx = modf u p(modx) = p(modf)=q #0 Tozda x u GH€UN€ conpsacenn
3AMEUAHME 3.2 Yconosne mod af = mod ff = W,, te R, MousHo 3aMeHHTh yeuosHem moda? = mod? = id JleiicrsureabHo, mycth modat=W,, a N — runepOuunruait I, daxrop, t — r.Hopm cueq Ha N, p(t) — onHo- napaMerpHueckad rpynma aBTomopdusmosp N TaEKaW, aro t-p(t) = e't Pac- CMOTDHM aBToMOpÙH3M 2@p(—t/¢) daxropa M@N, usomopduoro M Torxa mod(z@ ø(—//4))” = mod «?.W_, = id Teneps ecan y = «@p(—t/q)@ plt/q) €
c AutM@N@N, T0 # H ÿ BH€HI€ COIDØ/RGHBI, HTO G219/W@T M3 T@OD€MDI Í H eenerepua 6 [†], nooRozbxy id@p(—t/q)@ p(t/q) € Int M BBY Teopembt 1.1 § 2 Ifooromy Bompoc o BHemmHeH COHDW?40HHOCTH 2 H j6 Aută GBONHTCH R BOIPOOV O BHGIIHeÏH GOIpfØ?⁄“€HHOCTH Z@j(—//4) @p(—(jg), upwaem
mod(a@ p(—t/q))* : = id Z
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MW oaHaJIOTHIHble COOTHOMIeCHUA BepHBl MILA Oy
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©yitecrpyeT ÿ€ Aut#, (F¿) = Fạ, Yea = 1d TARKOH, WTO G=y-0,-y*+s, seIntF,, ‘f.€ MOMKHO IIDG/ITOZIAPATE, YTO BLIMOUHEHO (3.6)
llepelneM K NOKA8AT€JIbCTBY cymecrBoBanua P € Int N, yaosuersopa- youjero (3.5), T.e KR 3AB@DHIHHIO /0E838T€ebcrna IIpenozgeHnn 3.1 Boc- TI0JIb3V@MGR M€TOJOM, Ip€NIO?R€HHEfM B § 2.5 [23]
B cmy sawegaHun 3.Á MOZIHO GHHTATb, iro uMeet MecTo (3.6) Uepes AuON oGosHayum mMHomecTBo map (6,u), rme 6 € AutN, we U(N), a U(N) — TPVHHA BGEX YHHTAPHBIX OI€DATODOB B Đ, TAKHX, WTO
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llo nopo7y noRasarezscrpa (cm Hemmy 2.5.6 [23])
Tenepb, KEaAK H HDM /OREA3AT€JIbCTB€ HĐGJHJ07R€HHH 2.3, DACCMOTDHM IHIDOKROHMATIBHO KOH©HHEIHI TDYHHOHN 'f, HOPO?RH€HHHIH pelicrBMem aBTO- Mopjwawos Q@ nø, na (X, ø) Torxa 1-gonmimr {Uƒ, V?} n {Uÿ, VỆ} sanaror SopezeBckne TOMOMODHWB8MBIL Ø„ H Ø0; ©OOTBĐTOTBEHHO TpyHHOMnaA Y B AutN B cmay (3.1) 218 FOMOMOP(MSMBI HpHHA/167£AT rpyHnne Aut?W, paocMorpeHHofl spre OJIA6 TOPO, HOGEOJbRV Œ®(WZ) = đ(VỆ) HH I.B x, TO BBHAY JIMMH 3.5
Pa = Pg (mod Int°N)
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B oH POCIOUIEIL
4TO {LTA IB Vv € X
P.UP — UPP,s}, P,.VE== VPPs},
HIPHMGM HOCROEIbRV P,€ In, ro PV0,P;107'c IntN 71H TB x TaRHM 0- õpaso, ; 1n Ð =: (P,) nbnnozIneno (3.5), a aHawnr # n ổ pHeInHe conpwzReHm GB CHITEPATYPA 1 Connes, A., Outer conjugacy classes of automorphisms of factors, Ann Sci Ecole Norm Sup., 8(1975), 383—420 2 Connes, A.,On the classifications of von Neumann algebras and their automorphisms, Sypo- sia Mathematica, XX(1976), 435—478
3 Connes, A.; KRIEGER, W., Measure space automorphisms, the normalizers of their full groups
and approximate finiteness, J Functional Analysis, 241977), 336—352
4 Araki, H.; Woops, E.J., A classifications of factors, Publ Res Inst Math Sci., 3(1968), $1~—130
Connes, A., On hyperfinite factors of type III, and Krieger’s factors, J Functional Analysis,
18(1975), 318 327
6 KRIEGER, W A., On ergodic flows and isomorphism of factors, Math Ani., 223(1975), 19 70
7 Connes, A.; TAKESAKI, M., The flow of weights on factors of type IIIf, Téhoku Math J.,
29(1977), 473 575
8 Connes, A., Une classification des facteurs de type HI, Aan Sci Ecole Norm Sup., 2(1973),
133-252
9 Hamacui, T.; Oka, Y.; Osikawa, M., Flows associated with ergodic non-singular transfor- mations groups, Publ Res Inst Math Sci., 1101975), 31 —50
10 POZIO/LEIL B H., O crpyErype aareốp on HelmaHa, I1BỌCTBCHHIBX K aciredpam, NOCTPOCHHbIM HO /IHHAMHHOCRHM CHCTEMAM, (1//0E1(60110/1Đ001Š 41011183 t6 ©°0
ta
npu.to2reentas, 9: 3(1975), 87-88
11 POXZLHH, B Á., H23ØPpAaHHb BOIPOCBI MGTDHHGCHỌÏ T€ODHH /UHHAMHHOCRHX CH€TOM,
Veneru dam nauz#, 4: 3 (1949), 57—128.,
12 Dye, H.A., On groups of measure preserving transformations I, Amer J Math., 81(1959), 119-153,
13 Connes, A A classification of injective factors, Ann of Math., 104(1976), 73 115
14 FELDMAN, J.; HAHN, P.; Moore, C.C., Orbit structure and countable sections for actions of continuous group, Adv in Math , 28(1978), 186 -230
15 TAKEFSAKI, M., Duality for crossed product and structure of von Neumann algebras of type IH, Acta Math., 131(1973), 249—310
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19 PO/IO/USH, B H., O aBTOMOpÙH2MAX HAbeRTIBHEIX dakropon tutta III), Upe- pH? 4-80, @GPHHT AH VYGCP, Xapbkon, 1980, 1—32
20 Connes, A.; TaKesaki, M., Flot des poids sur les facteurs de type If], C.R Acad Sci Paris,
Sér, A, 3781974), 937-940
21 Hamacut, H., The normalizer group of an ergodic automorphism of type III and the commu- tant of an ergodic flow, J Funcrional Analysis, 40(1981), 387 —403
22 BezuGLyt, S.I.; Gotopets, V Ya., Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type LIL full groups, J Functional Analysis, 60(1 985), 341-369
23 Jones, V F R.; TAKesaki, M., Actions of compact abelian groups on semifinite injective factor, preprint
V YA GOLODETS
Institute for Low Temperature Phivsics and Engineering,
Ukr SSR Academy of Sciences,
47, Lenin Avenue, Kharkov, 310164,
U.S.S.R
Received August 12, 1982; revised July 9 1984