O(CEEI-OAOI(*IEE (INOEOOO IECUeEEO OAIIA?AOO?

IAOe(IIAEUeII( AEAAeAI(( IAOE OE?A(IE

NIOAO Ieaenaiae? Ia?eiae/

OAeE 517 + 519.46

AI?IENEIAOEAIA O?AICEOEAI(NOUe

A?OII IA?AOAI?AIUe

A A?AIAeE*I(E OAI?((

01.01.01 — Iaoaiaoe/iee aiae(c

Aaoi?aoa?ao aeena?oaoe(( ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o(ceei-iaoaiaoe/ieo iaoe

Oa?e(a 1999

Aeena?oaoe((th ( ?oeiien

?iaioa aeeiiaia a O(ceei-oaoi(/iiio (inoeooo( iecueeeo oaiia?aoo?
iaoe(iiaeueii( Aeaaeai(( iaoe Oe?a(ie

Iaoeiaee ea?(aiee – aeieoi? o(ceei-iaoaiaoe/ieo iaoe

AIEIAeAOeUe Aaeaioei sseiae/

(O(ceei-oaoi(/iee (inoeooo iecueeeo oaiia?aoo? IAI Oe?a(ie,

i?ia(aeiee iaoeiaee ni(a?ia(oiee)

Io(oe(ei( iiiiaioe:

aeieoi? o(ceei-iaoaiaoe/ieo iaoe, i?ioani? Naiieeaiei TH?(e
Noaoaii-ae/, i?ia(aeiee iaoeiaee ni(a?ia(oiee (inoeoooo iaoaiaoeee IAI
Oe?a(ie, Ee(a;

eaiaeeaeao o(ceei-iaoaiaoe/ieo iaoe N(iaeueueeia Na?a(e Aeieo?iae/,
noa?-oee iaoeiaee ni(a?ia(oiee O(ceei-oaoi(/iiai (inoeoooo iecueeeo
oaiia?aoo? IAI Oe?a(ie, Oa?e(a.

I?ia(aeia onoaiiaa: Oa?e(anueeee aea?aeaaiee oi(aa?neoao,
iaoai(ei-iaoaia-oe/iee oaeoeueoao, Oa?e(a.

Caoeno a(aeaoaeaoueny “ 4 ” /a?aiy 1999 ?ieo i 1500 aiaeei(
ia can(aeaii( niaoe(ae(ciaaii( a/aii( ?aaee Ae 64.175.01 o
O(ceei-oaoi(/iiio (inoeooo( iecueeeo oaiia?aoo? IAI Oe?a(ie, Oa?e(a,
i?.Eai(ia, 47.

C aeena?oaoe((th iiaeia iciaeiieoenue o a(ae(ioaoe( O(ceei-oaoi(/iiai
(inoeoo-oo iecueeeo oaiia?aoo? IAI Oe?a(ie, Oa?e(a, i?.Eai(ia, 47.

Aaoi?aoa?ao ?ic(neaiee “ 30 ” eaioiy 1999 ?ieo.

A/aiee nae?aoa?

niaoe(ae(ciaaii( a/aii( ?aaee ________ Eioey?ia A.I.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei(noue oaie. A?aiaee/ia oai?(y c(yaeeany ia ii/aoeo 30-o ?ie(a
iaoi-ai noi?(//y; aiia ii/aeanue c ?iaio Aeae. oii Iieiaia, yeee
caeiaany aeine(aeaeai-iyie iaoaiaoe/ieo iiaeaeae eaaioiai( oa eeane/ii(
iaoai(e, ( c oiai /ano ?icae-oie a?aiaee/ii( oai?(( caaaeaee eoia
ia?aeaeueieie oeyoaie c ?icaeoeii oai?(( iia-?aoi?ieo aeaaa?.

Iiiyooy ai?ieneiaoeaii( o?aiceoeaiino( c(yaeeiny a 1985 ?ioe( a ?iaio(
A.Eiiia oa A.Aeae.Aoaena o ca(yceo c i?iaeaiith eeaneo(eaoe(( oaeoi?(a,
ye( ( iane(i-/aieie oaici?ieie aeiaooeaie oaeoi?(a oeio I; aeyaeeiny, ui
ai?ieneiaoeaii ne(i/ai( oaeoi?e ?iciaaeathoueny o oaeee iane(i/aiee
aeiaooie, yeui e o(eueee yeui eiai iio(e aaa caaeiaieueiy( oiia(, yeo
aaoi?e iacaaee “ai?ieneiaoeaiith o?aice-oeai(noth” (AO). Ie
i?iaeiaaeo(ii aeine(aeaeaiiy oeueiai iiiyooy c oi/ee ci?o a?ai-aee/ii(
oai?((.

Iaea(eueo o?aaeeoe(eia /anoeia a?aiaee/ii( oai?(( – oea aeine(aeaeaiiy
iiiaeeii-eeo ia?aoai?aiue oa oa/(e, ye( caa?(aathoue i(?o. Aea oe(
iaiaaeaiiy aeaaii aaea i(eiio ia caeathoueny i?e?iaeieie, ( oiio /anoi
iinoa( i?iaeaia c(ynoaaoe, /e iiaeia oca-aaeueieoe oa /e (ioa iiiyooy
aai ?acoeueoao aeey a(eueo caaaeueii( neooaoe((, aea, ii-ia?oa, ae((
aeia(eueia a?oia, a ia o(eueee Z aai R, (, ii-ae?oaa, oey ae(( caa?(aa(
o(eueee eean i(?e, a ia naio i(?o. Iaoo ?iaioo, e?(i (ioiai, iiaeia
?icaeyaeaoe ye ?iaioo a oeeo aeaio iai?yieao.

Ca(ycie c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. A(eueoa /anoeia oe(((
?i-aioe aeeiiaia aaoi?ii i(ae /an iaa/aiiy a anie?aioo?( Iaoaiaoe/iiai
a(aeae(eaiiy OO(IO AI (iio(i IAI) Oe?a(ie i(ae ea?(aieoeoaii
A.ss.Aieiaeoey. ?iaioo aeeiia-ii a iaaeao oaiaoe/iiai ieaio OO(IO IAI
Oe?a(ie ca oaiith 1.4.10.22.6 “Aeaaa-?a(/i( oa aaiiao?e/i( iaoiaee a
oai?(( iia?aoi?(a oa oai?(( ae(iai(/ieo nenoai” (? aea?aeaaii(
?a(no?aoe(( 0196V002943).

Iaoa ( caaea/( aeine(aeaeaiiy: 1) ciaeoe yeiaea(eueoa aeinoaoi(o oiia,
ye( a ca-aacia/oaaee ai?ieneiaoeaio o?aiceoeai(noue; 2) ciaeoe oi/iee
iien AO ae(e aoaeue-yeeo a?oi, a ia o(eueee R, iiaith neaaei(
aea(aaeaioiino(, i?iaeaeo-iaeiiao?(a oa i?iaeaeo-eioeeee(a.
3) iiaoaeoaaoe oe(eaa( i?eeeaaee ae(( a?oie, ui ( i?yiei aeiaooeii aeaio
a?oi, ( c(noaaeoe (o c a(aeiia(aeieie ae(yie a?oi-iiiaeiee(a.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao(a. On( ?acoeueoaoe aeena?oaoe((
( iiae-ie. O ae?oaiio ?icae(e( aia?oa ei?aeoii aecia/aii iiiyooy
eoiaaeiiai ?aiao iaeei (funny rank one; rang un gauche) aeey ae(e
aeia(eueieo a?oi, ye( ia caa?(aathoue i(?o, ( aeiaaaeaii, ui eoiaaeiee
?aia iaeei ia caeaaeeoue a(ae aeai?a i(?e a iaaeao oiai ae naiiai eeano
oa ui c eoiaaeiiai ?aiaa iaeei aeo(ea( AO. Aeiaaaeaii, ui o?aiceoeai(
ae(( oa ae(( c aeene?aoiei niaeo?ii ( AO, a o aeiaaeeo ae(e ?ica(ycieo
a?oi aiie, e?(i oiai, iathoue eoiaaeiee ?aia iaeei. Oia?oa aeiaaaeaia
oaea oai?aia i?i (iaeoeiaai( ae((: iaoae H – caieiaia ii?iaeueia
i(aea?oia a?oie G. sseui ae(y a?oie H ia eaaa-aiaiio i?inoi?( oa
i?e?iaeia ae(y a?oie G ia G/H iaeaea( iathoue eoiaaeiee ?aia iaeei, oi e
(iaeoeiaaia ae(y a?oie G oaae ia( eoiaaeiee ?aia iaeei. Iiae(aio (aea
i?in-o(oo) oai?aio aeiaaaeaii oaeiae aeey AO ae(e, ( iaa(oue aac
i?eiouaiiy i?i ii?iaeue-i(noue i(aea?oie H. A aeey ae(e oe(eeii
iaca(ycieo a?oi iaa(oue ia o?aaa i?eioneaoe ?ic-a(yci(noue, uia aeiaanoe
eoiaaeiee ?aia iaeei.

O o?aoueiio ?icae(e( aia?oa aeiaaaeaii aeaa aaaeeeaeo e?eoa?(( ?iciaaeo
a ian-e(i/aiee i?yiee aeiaooie ia?e, yea neeaaea(oueny c ae(( oa
eioeeeeo. Aia?oa aeiaaaeaii, ui iiaea(eia ae(y Iaee(, iiaoaeiaaia ca
aiaiaaaeueiith a?oiith ia?aoai?aiue oeio II aai III G oa 1-eioeeeeii
(((, aea ( – eioeeee ?aaeiia-I(eiaeeia, a ( – aeia(eueiee 1-eioeeee c(
cia/aiiyie a a?oi( A, ( AO oiae( e o(eueee oiae(, eiee ia?a (G, ((() (
neaa-ei aea(aaeaioiith aei ia?e, ui neeaaea(oueny c i?iaeaeo-iaeiiao?o
oa i?iaeaeo-eioeee-eo. Ca(aene anoaiiaeaii, ui aoaeue-yeo ae(th Iaee(
a?oie A iiaea aooe ?aae(ciaaii ye ae(th Iaee(, iiaoaeiaaio ca ae((th
aeia(eueiiai caaeaiiai oeio oa ca i?iaeaeo-eioeeeeii. A yeui aeaia AO
ae(y c naiiai ii/aoeo aoea cia?aaeaia ye ae(y Iaee(, anioe(eiaaia aei
ae(( oeio II oa aei eioeeeeo, oi oeae eioeeee aeyaey(oueny ia o(eueee
neaaei aea(aaeaio-iei, aea iaa(oue eiaiiieia(/iei aei aeayeiai
(-i?iaeaeo-eioeeeeo. An( oe( ?acoeueoaoe ni?aaaaeeea( aeey a?oi A, ye(
caaeiaieueiythoue oaeei o?ueii oiiaai: (1) A – aiaiaaaeue-ia e.e.n.
a?oia; (2) A i(noeoue c/eneaiio u(eueio aiaiaaaeueio i(aea?oio; (3)
eioeeee ( ((Z1((, G; A) ia( aooe oaeei, uia log (((((, g)) aoa
eia?aieoeath, aea ( – iiaeo-ey?ia ooieoe(y a?oie A.

E?(i oiai, ii?(aiyii iiaea(eio ae(th Iaee(, iiaoaeiaaio ca ae((th oeio
III oa iiaea(eiei eioeeeeii (((, c aeaiia ae(yie Iaee(, iiaoaeiaaieie ca
o((th ae naiith ae(-(th oa ca eioeeeeaie ( oa ( ie?aii, oa aeiaaaeaii,
ui c AO ia?oi( ae(( aeo(ea( AO aeaio (ioeo. Cai?ioia oaa?aeaeaiiy (
oeaiei, ( iaaaaeaii a(aeiia(aeiee i?eeeaae.

O /aoaa?oiio ?icae(e( aia?oa iiaoaeiaaii oaeiae oe(eaaee i?eeeaae
noeoiii( ae(( ( aeaio a?oi G = Z2 ( Z2 ( … oa Z, yea ia ( (ciii?oiith
aei aeiaooeo a(aeiia(aeieo ae(e a?oi G oa Z; a naia, ae(( G oa Z ie?aii
ia a?aiaee/i(; ae(( Z ia (( a?aiaee/ieo eii-iiiaioao ( (ciii?oieie oiae(
e o(eueee oiae(, eiee oe( eiiiiiaioe ia?aaiaeyoueny iaeia a iaeio ae((th
G; oa oeaio?ae(caoi? C((G(Z) aeyaey(oueny oaeei, ui C((G(Z) / ((G(Z) (
Z2. Oeae i?eeeaae iiaoaeiaaii ia aac( i?eeeaaeo Aa((aa a?aiaee/iiai
ia?aoai?aiiy c eaaaaianueeith eiiiiiaioith ia?ii( e?aoiino( o niaeo?(.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao(a. ?acoeueoaoe aeena?oaoe((
iiaeooue aooe aeei?enoai( aeey iiaeaeueoeo aeine(aeaeaiue ai?ieneiaoeaii
o?aiceoeaieo ae(e oa iia(ycaieo c ieie ieoaiue a?aiaee/ii( oai?((. Aei
/enea a(o/eciyieo onoaiia, ui ii-aeooue aooe caoe(eaaeaieie
?acoeueoaoaie oe((( aeena?oaoe((, aoiaeyoue O(ceei-oaoi(/iee (inoeooo
IAI Oe?a(ie (Oa?e(a), (inoeooo iaoaiaoeee IAI Oe?a(ie (Ee(a),
Oa?-e(anueeee aea?aeaaiee oi(aa?neoao oa (i.

Ai?iaaoe(y ?acoeueoao(a aeena?oaoe((. ?acoeueoaoe aeena?oaoe((
aeiiia(aeaeenue oa ia-aiai?thaaeenue ia nai(ia?( c oai?(( iia?aoi?ieo
aeaaa? oa a?aiaee/ii( oai?(( ((inoe-ooo iecueeeo oaiia?aoo?, Oa?e(a;
ea?(aiee nai(ia?o A.ss.Aieiaeaoeue), ia nai(ia?( c oai?(( iia?aoi?ieo
aeaaa? (Eiee(ae aea O?ain, Ia?eae; ea?(aiee nai(ia?o AE.Neai-aeae(n), ia
nai(ia?( c a?aiaee/ii( oai?(( (Oi(aa?neoao Ia?eae-6; ea?(aiee nai(ia?o
AE.-I.Ooaaii).

Ioae(eaoe((. Aeea. ia?ae(e ioae(eaoe(e aaoi?a ca oaiith aeena?oaoe((
iai?ee(ioe( oeueiai aaoi?aoa?aoo.

Iniaenoee aianie caeiaoaa/a. Oa, ui aeeeaaea(oueny a ?icae(eao 2.1,
2.2, 2.4, 3.1, 3.4.1 oa 3.5, c?iaeaia aaoi?ii iaeiiiniaiai. Oa, ui
aeeeaaea(oueny a ?icae(eao 2.3, 3.2, 3.3, 3.4.2, 3.6 oa 4, c?iaeaia
aaoi?ii o ni(a?ia(oieoeoa( c A.ss.Aieiaeoeai i?e ?(aiiio aeeaae( noi?(i.

Ia((i oa no?oeoo?a. Aeena?oaoe(y neeaaea(oueny c anooio, /ioe?ueio
aeaa, aeniia-e(a oa nieneo aeei?enoaieo aeaea?ae, ui i(noeoue 35
iaeiaioaaiue. Ianya aeena?oaoe(( 104 noi?(iee. Ianya nieneo aeei?enoaieo
aeaea?ae 4 noi?(iee.

Oi/o ue?i iiaeyeoaaoe Aaeaioeia sseiae/a Aieiaeoey oa AEi?aea
Neaiaeae(na ca aeiiiiiao a ?iaio( iaae aeena?oaoe((th.

INIIAIEE CI(NO ?IAIOE

Ia?oee ?icae(e. Iiia?aaei( a(aeiiino(

Ooo iaaaaeo?ii aeaye( iaeaaaeeea(o( iiiyooy a?aiaee/ii( oai?((, na?aae
yeeo ai-?ieneiaoeaia o?aiceoeai(noue:

Aecia/aiiy 1.1. Eaaeooue, ui ae?y a?oie G ia eaaaaiaiio i?inoi?? (w, B,
m) ai-?ieneiaoeaii o?aiceoeaia (AO), yeui aeey aoaeue-yeiai e>0 oa
aeia?eueiiai ne?i/aii-ai iaai?a ooieoe?e f1, f2,…, fN I L1+(w, m)
ciaeaeooueny ooieoe?y f I L1+(w, m), aeaiaioe gj I G oa eiao?oe??ioe l i
j ? 0 (ooo i=1,…,n; j=1,…,Ni), oae?, ui

.

Ae?oaee ?icae(e. Aeinoaoi? oiiae aeey ai?ieneiaoeaii? o?aiceoeaiino?

oa eoiaaeiee ?aia iaeei

Aecia/aiiy 2.5. Ae?y a a?oie G ia eaaaaiaiio i?inoi?? (W, B, m) ia?
eoiaaeiee ?aia iaeei, yeui aeey aoaeue-yeeo A1, A2, …, An I B, m(Ai) < Y, ? aeia?eueieo e > 0, s > 0 ciaeaeooueny E0 I B e gik I G, rik I R+
(ooo i=1, …, n; ; k=1,…, Ni) oae?, ui iiiaeeie gikE0 aai ia
ia?aoeiathoueny, aai ni?aiaaeathoue, ?

N?iaenoai iiiaeei {gik E0 : i=1,2,.., n; k=1,…, Ni } aoaea
iacaaoeny noaeii (aaaeath), a E0 – eiai aacith.

I/aaeaeii, ui c eoiaaeiiai ?aiaa iaeei aeo?ea? a?aiaee/i?noue.

Oaa?aeaeaiiy 2.1.2. Eoiaaeiee ?aia iaeei ia caeaaeeoue a?ae aeai?a i??e
a a?aieoeyo iaeiiai eeano aea?aaeaioieo i??.

Oaa?aeaeaiiy 2.1.3. Ae?y a?oie G ia eaaaaiaiio i?inoi?? (W, B, m), yea
ia? eo-iaaeiee ?aia iaeei, ? AO.

Eaiia 2.1.4. Iaoae G1 oa G2 aoaeooue aea? a?oie aaoiii?oeci?a, ui
ae?thoue ia a?aeiia?aeieo eaaaaiaeo i?inoi?ao (w1, B1, m1) oa (w2, B2,
m2 ). ?icaeyiaii ?oi?e i?y-iee aeiaooie, naaoi ae?th G1 ? G2 ia (w1 ?
w2, B1 ?B2, m1 ? m2).

(1) yeui iaeaea? aeai? ae?? iathoue eoiaaeiee ?aia iaeei, oi ? i?yiee
aeiaooie oaae.

(2) sseui iaeaea? aeai? ae?? AO, oi ? i?yiee aeiaooie oaae.

Eaiia 2.1.5. Iaoae a?oia aaoiii?oeci?a G ae?? ia eaaaaiaiio i?inoi??
(w, B, m), H – oiiieia?/ia a?oia, oa j: H ® G – iaia?a?aiee aiiiii?o?ci
a?oi c u?eueiei ia?acii. ?icaeyiaii ae?th H ia (w, B, m), caaeaio oae:
h * w = j(h)* w.

(1) sseui aeaia ae?y G ia? eoiaaeiee ?aia iaeei, oi e iiaoaeiaaia ae?y
H oaae.

(2) sseui aeaia ae?y G ? AO, oi e iiaoaeiaaia ae?y H oaae.

Aeae( iaaaaeo(ii ca(ycie i(ae aeene?aoiei niaeo?ii oa o?aiceoeai(noth,
a ii-o(i a(aeiio oai?aio i?i ai?ieneiaoe(th aeia(eueieo e.e.n. a?oi
a?oiaie E(.

Aecia/aiiy 2.6. Iaoae e.e.n. a?oia G ae?? ia i?inoi?? Eaaaaa (w, B, m).
Iii-aeeia EIB iacaa?oueny G-iaeaea caieiaiith, yeui aeey aeayei? (a oiio
? aeey aoaeue-yei?) ooiaeaiaioaeueii? iine?aeiaiino? Un ieie?a iaeeieoe?
e I G ia?ii m(UnE \ E) ® 0. Iiiaeeia EIB ia-caa?oueny G-iaeaea
a?aee?eoith, yeui w\E ? G-iaeaea caieiaiith. Iiiaeeie, ui aiaeii/an
iaeaea a?aee?eo? ? iaeaea caieiai?, iacaathoueny iiiaeeiaie c ioeueiaith
G-a?aieoeath.

Aeiaiaeeii (nioaaiiy oaeeo iiiaeei, a oaeiae iiaeeea(noue ai?ieneioaaoe
aei-a(eueio aei(?io iiiaeeio ca aeiiiiiaith iiiaeei c ioeueiaith
G-a?aieoeath.

Oai?aia 2.2.1. Iaoae G – e.e.n. a?oia, H — ?? caieiaia i?aea?oia. Ae?y
i?aaeie cno-aaie a?oie G ia i?inoi?? i?aaeo noi?aeieo eean?a a?aeiinii H
? AO.

C?icoi?ei, ui oaea ae oai?aia ni?aaaaeeeaa ? aeey e?aeo cnoa?a ia
i?inoi?? e?-aeo noi?aeieo eean?a. Iaaaaea?ii, ui oaeoi?-ae?y AO ae??
naia ? AO. Oiio o?aaa o?eue-ee aeiaanoe oai?aio aeey aeiaaeeo a?eueii?
o?aiceoeaii? ae??, oiaoi aeey ae?? G i?aaeie cnoaaie ia nia?. Iaoiae
aeiaaaeaiiy – iiaoaeiaa ai?ieneiaoeaii? iaeeieoe? a L1(G).

Aeniiaie 2.2.2. Ae?y e.e.n. a?oie, yea ia? aeene?aoiee niaeo?, ? AO.

Ie?aii ?icaeyaea(ii aaaeaa( a?oie oa aeiaiaeeii aea( oai?aie:

Oai?aia 2.3.7. A?aiaee/ia ae?y aaaeaai? e.e.n. a?oie, yea ia?
aeene?aoiee niaeo?, ia? eoiaaeiee ?aia iaeei.

Oai?aia 2.3.8. Aoaeue-yea o?aiceoeaia ae?y aaaeaai? e.e.n. a?oie ia
eaaaaiaiio i?inoi?? ia? eoiaaeiee ?aia iaeei.

Oaia? ia?aoiaeeii aei (iaeoeiaaieo ae(e oa oi?ioeth(ii ?acoeueoaoe:

Oai?aia 2.4.1. Iaoae G – e.e.n. a?oia, H – ?? caieiaia ii?iaeueia
i?aea?oia, H ae?? ia eaaaaiaiio i?inoi?? (w, B, m). sseui ae?y H ia?
eoiaaeiee ?aia iaeei, oa ae?y G cnoaaie ia i?inoi?? e?aeo noi?aeieo
eean?a X=G/H oaeiae ia? eoiaaeiee ?aia iaeei, oi e ?iaeoeiaaia ae?y
a?oie G oaeiae ia? eoiaaeiee ?aia iaeei.

Oai?aia 2.4.2. Iaoae ciiao G – e.e.n. a?oia, H – ?? caieiaia i?aea?oia,
H ae?? ia eaaaaiaiio i?inoi?? (w, B, m). I?eionoeii, ui oey ae?y ? AO
(i?e?iaeia ae?y G ia X = G/H caaaeaee ? AO aiane?aeie oai?aie 2.2.2).
Oiae? ?iaeoeiaaia ae?y a?oie G oaae ? AO.

Iiaeaeueoo /anoeio ae?oaiai ?icae(eo caeia( aeiaaaeaiiy oeeo oai?ai.

Aeniiaie 2.4.6. On? o?aiceoeai? ae?? e.e.n. ?ica’ycieo a?oi iathoue
eoiaaeiee ?aia iaeei.

Aeniiaie 2.4.7. Ae?? e.e.n. ?ica’ycieo a?oi, ui iathoue aeene?aoiee
niaeo?, ia-thoue ? eoiaaeiee ?aia iaeei.

Aeniiaie 2.4.8. sseui aaoiii?o?ci ia? eoiaaeiee ?aia iaeei, oi e
niaoe?aeueia oa/?y, iiaoaeiaaia i?ae iino?eiith noaeaaith ooieoe??th,
oaeiae ia? eoiaaeiee ?aia iaeei.

Oaa?aeaeaiiy 2.4.9. An? o?aiceoeai? ae?? oe?eeii iaca’ycieo e.e.n. a?oi
iathoue eo-iaaeiee ?aia iaeei.

I?eeeaae 2.4.10. Iaoae SL(n, Zp) – a?oia an?o iao?eoeue c aecia/ieeii,
yeee aei??aith? 1, iaae e?eueoeai p-aaee/ieo oe?eeo /enae. Oiae? SL(n,
Zp) – oe?eeii iaca’ycia eiiiaeoia a?oia, a SL(n, Z) – ?? u?eueia
i?aea?oia. Ae?y SL(n, Z) ia SL(n, Zp) e?aeie cnoaaie ia? eoiaaeiee ?aia
iaeei ? ? AO, oae naii ye ? ae?y SL(n, Zp) ia nia?.

O?ao(e ?icae(e. I?iaeaeo-eioeeeee oa ai?ieneiaoeaia o?aiceoeai?noue

mn. Ia?anoaaeaiiy ln aei? ia wn oae: ln(k) = k+1 mod(Rn). Oei ln
ii?iaeaeothoue c/eneaiio aieueio a?oio ia?aoai?aiue Gpr i?inoi?a wpr,
yea caaoueny i?iaeaeo-iaeiiao?ii (aiae aiaeienueeiai «product» –
aei-aooie).

Aecia/aiiy 3.1. Iaoae apr — oaeee 1-eioeeee wpr ? Gpr ® A ci cia/aiiyie
a aaaeaaie a?oii A, ui apr(w, lnl) (0 F l < Rn) caeaaeeoue oieueee aiae n-oi? eii?aeeiaoe oi/ee w = (w1, w2, ..., wn, ...) I wpr. Eioeeee apr oaeiai aeaeo iacaa?oueny i?iaeaeo-eioeeeeii. Iaoae (w, B, m) – i?inoi? Eaaaaa, a G – a?aiaee/ia c/eneaiia aieueia a?oia ia?aoai?aiue oeueiai i?inoi?a. Iaoae a : w ? G ® A – 1-eioeeee. sseui inio? aei(?ia aiaeia?aaeaiiy q: (w, m) ® (wpr, mpr), m°q = mpr, yea caa?iaa? ii?o, ia?aaiaeeoue i?aeoe a i?aeoe i i?e oeueiio q [G] q-1 = [Gpr], oa a (q-1 w, g)= apr (w, qgq-1), aea g I [G], w I wpr, oi eioeeee a iacaa?oueny ieae/a q-i?iaeaeo eioeeeeii. Oaa?aeaeaiiy 3.1.1. Iaoae c/eneaiia aiaiaaaeueia a?oia G ae?? a?aiaee/ii ia eaaa-aiaiio i?inoi?? (w, B, m). I?eionoeii, ui a – eioeeee aeey oe??? ae?? c? cia/aiiyie a aaaeaa?e e.e.n. a?oi? A. Ia?a (G, a) ? neaaei aea?aaeaioiith aei ia?e, ui neeaaea?oueny c i?iaeaeo-iaeiiao?a oa i?iaeaeo-eioeeeea, yeui e o?eueee yeui aeey aoaeue-yeeo e, q, s > 0 oa
aeia?eueiiai ne?i/aiiai iaai?a /anoeiaeo ia?aoai?aiue g1, g2,…, gn I
[G]m* ciae-aeooueny ne?i/aia i??a P ~ m, eioeeee b, eiaiiieia?/iee a,
ooieoe?y f, ui nie?oa? b c a, oa i?inoa aaaea z c iino?eieie (b,
P)-ia?ao?aeieie cia/aiiyie, oae?, ui Dom gi, Im gi Im,e B(z);

m(w I Dom gi ( supp(z): giw I Orbz(w)) > (1 — e)* m(Dom gi);

(ooo E = (i (Dom gi ( Im gi));

m(w I supp(z) ( E : dist(eA, f(w)) > s) < q * m(supp(z) ( E). A?aei?oeii, ui ia?a (G, r) (aea r – eioeeee ?aaeiia-I?eiaeeia aeey i??e m), oaae aiaeii/an aeyaey?oueny neaaei aea?aaeaioiith aei ia?e, ui neeaaea?oueny c i?iaeaeo-iaeiiao?a oa eiai (i?iaeaeo-)eioeeeea ?aaeiia-I?eiaeeia aeey i?iaeaeo-i??e. Oaa?aeaeaiiy 3.1.2. Iaoae G – c/eneaiia a?aiaee/ia aiaiaaaeueia a?oia iaae?iae-aeaieo ia?aoai?aiue oeio III ia eaaaaiaiio i?inoi?? (w, B, m), a a – eioeeee aeey ia? c? cia/aiiyie a aaaeaa?e e.e.n. a?oi? A. Ia?a (G, a) ? neaaei aea?aaeaioiith aei ia?e, ui neeaaea?oueny c i?iaeaeo-iaeiiao?a oa i?iaeaeo-eioeeeea, yeui ni?aaaaeeeaa oaea oii-aa: c iino?eieie (P, b)-ia?ao?aeieie cia/aiiyie (ooo P – aeayea ne?i/aia i??a, aea?aaeaioia aei m, a b – aeayeee eioeeee, eiaiiieia?/iee a), aeey aoaeue-yeeo e > 0 oa s > 0 ciaeaeooueny ne?i/aia i??a Q ~ m, eioeeee g ~
a, i?in-oa aaaea x c iino?eieie (Q, g) — ia?ao?aeieie cia/aiiyie, yea ?
iiae??aiaiiyi aeaii? e?aoii? aaaee, oa ooieoe?y f, g ~f b, oae?, ui

Oey oiiaa ? ia o?eueee aeinoaoiueith, aea e iaiao?aeiith; iiaa/eii oea
ieae/a.

Iaoae c/eneaiia aiaiaaaeueia a?oia oeio III G a?eueii ae?? ia
eaaaaiaiio i?inoi?? (w, B, m), a r aoaea eioeeeeii ?aaeiia-I?eiaeeia
oe??? ae??, oa a – aeia?eueiei eioeeeeii c? cia/aiiyie a aaaeaa?e a?oi?
A. Ia?a (r, a) ?icaeyaea?oueny ye iiaea?eiee eioeeee. Aecia/eii
i?ino(?-aeiaooie w?A?R c oaeith i??ith: dn(w,a,u) = dm(w) * da *
exp(u)du; ooo w I w, a I A, u I R. I?e?iaei? i?iaeoe?? c (w , B, m) ia
w, A oa R iicia/eii a?aeiia?aeii pw, pA oa pR. O??eoeth (w, a, u)
iicia/eii aoeaith z. ?icaey-iaii iaae?nio ae?th a?oie G ia oeueiio
i?inoi??:

(w, a, u) = (g w, a * a(w, g), u + log (dm°g/dm)(w))

?acii c oaeeie ae?yie a?oi A oa R (t I R, b I A):

Tt(w, a, u) = (w, a, u+t), Vb(w, a, u) = (w, b * a, u).

oa Vb caa??aathoue i??o n, a Tt – i?. Oe? o?e ae?? eiiooothoue iaeia c
?ioith.

-?iaa?eaioieo iiiaeei. Iaoae p – i?e?iaeia i?iaeoe?y c w ? A ? R ia X.
Iaa?aii aeia?eueio neaia-ne?i/aio i??o m ia X, aea?aa-eaioio aei n0 °
p-1, aea n0 – aoaeue-yea ne?i/aia i??a, aea?aaeaioia aei n. Oiae?
iiaeaii iaienaoe oaeee ?iceeaae:

-?iaa?eaioio i??o, yea i?e iaeaea an?o x I X caaei-aieueiy? oiia?
n({(w, a, u) I w ? A ? R : p(w, a, u) ( x} | x) = 0.

Aecia/aiiy 3.3. ?icaeyiaii oaeoi?-ae??

Ft(p(w, a, u)) = p(Tt(w, a, u)) oa

Wb(p(w, a, u)) = p(Vb(w, a, u))

ia X a?oi, a?aeiia?aeii, R oa A. Noeoiia ae?y (Ft, Wb) aoaea iacaaoeny
iiaea?eiith ae??th Iaee?.

* Td * Vb : g I G, d I D, b I B }.

Oaa?aeaeaiiy 3.2.1. Ia?a (G, (r, a)) (aea G oeio III) ? neaaei
aea?aaeaioiith aei ia?e (G, (r1, a1)), aea r1 oa a1 – oae? eioeeeee:

, Vb, Tt) = b-1,

? B ? D; A).

I?ae /an aeiaaaeaiiy aiaeii/an aeyaey?oueny, ui a?oie ia?aoai?aiue G oa
G ? i?a?oaeueii aea?aaeaioieie ? ui ia?e (G, a) oa (G, a1) ? neaaei
aea?aaeaioieie.

Aecia/aiiy 3.5. Iaoae h I [G]n*, a E I Dom h – aei??ia iiiaeeia. fh
(x) oa fE(x) aoaeooue iaa?ae’?iieie ooieoe?yie I L1(X, m), oaeeie, ui
fh(x) = n(Im h | x); fE (x) = n(E| x).

C?icoi?ei, ui fE = fid |E, fh = fIm h, || fE ||1 = n(E).

Eaiia 3.2.2. Iaoae e > 0, E – aei??ia iiiaeeia a w ? A ? R, oa f I
L1(X, m)+ oae?, ui || f – fE ||1 < e. Oiae? ?niothoue aei??ia iiiaeeia E1 I w ? A ? R oa a?aeia?a-aeaiiy h I [G]n* c E ia E1, oae?, ui fE1 = f, || n(h () – n(* ( E) || F || f - fE || + e < 2 e, oa n(z I E : a1(z, h) ( eA) < 2 e. I(ney aeiaaaeaiiy ua e(eueeio aeiiii(aeieo eaii iiaeia ia?aeoe aei iniiaii( oai?aie: Oai?aia 3.3.1. Iaoae G – aiaiaaaeueia a?aiaee/ia a?oia iaae?iaeaeaieo ia?aoai-?aiue oeio III ia (w, B, m), oa iaoae a – 1-eioeeee aeey oe??? ae?? c? cia/aiiyie a aaaeaa?e e.e.n. a?oi? A, a r – eioeeee ?aaeiia-I?eiaeeia. Ia?a (G, (r, a)) ? neaaei aea?aaeaioiith aei ia?e, ui neeaaea?oueny c i?iaeaeo-iaeiiao?a oa i?iaeaeo-eioeeeea, yeui e o?eueee yeui iiaea?eia ae?y Iaee? (Ft, Wb) ? AO. Eaaei aeiaanoe, ui oey oiiaa ? aeinoaoiueith: oea aeo?ea? c oai?aie 2.2.1 oa c oiai oaeoa, ui oaeoi?-ae?y AO ae?? naia ? AO. Iao?ea?aeueia /anoeia oai?aie – oea aeiaanoe, ui c ai?ieneiaoeaii? o?aiceoeaiino? iiaea?eii? ae?? Iaee? aeo?ea?, ui aeaia ia?a ? neaaei aea?aaeaioiith aei i?iaeaeo-iaeiiao?a c i?iaeaeo-eioeeeeii. Caaaeyee oaa?aeaeaiith 3.2.1, cai?noue aeaii? ia?e (G, (r, a)) ?icaeyaeaoeiaii ia?o (G, (r1, a1)). Nei?enoa?iiny oaa?aeaeaiiyi 3.1.2. Iaoae Z = (zi - aeia?eueia e?aoia aaaea aeey G c iino?eieie (P, b)-ia?ao?aeieie cia/aiiyie, aea P ~ n, b ~h a1. Iaoa iaoa – iiaoaeoaaoe x, Q, g ye a oaa?aeaeaii? 3.1.2. A?cueiaii aeia?eueiee iiaa?o er(i) a eiaei?e zi. ?icaeyiaii iiiaeeio E = (er(i). Eaiia 3.3.2. Ii oiaeo iiaeaeueoiai aeiaaaeaiiy oai?aie 3.3.1 ie ia?ii i?aai i?e-ioneaoe, ui P = n ia E ? ui bE = (a1)E. Oaia? aeieno(ii AO-oiiao aeey ae(( G, ei?enoo(iinue eaiiith 3.2.2. oa (ioe-ie aeiiii(aeieie eaiiaie oa yaii aoaeo(ii iio?(ai( x, Q, g. Aeiaaeie oeio II ?icaeyaea(oueny iiae(aieie iaoiaeaie, aea aeiaaaeaiiy a?oioo-(oueny ia aeei?enoaii( oaa?aeaeaiiy 3.1.1 cai(noue 3.1.2 ( i?iaiaeeoueny aac eaiie 3.2.2. Oai?aia 3.4.1. Iaoae G – c/eneaiia aiaiaaaeueiy a?oia ia?aoai?aiue oeio II, ui ae?? ia eaaaaiaiio i?inoi?? (w, B, m), oa a : w ? G ® A – 1-eioeeee aeey oe??? ae?? c? cia/aiiyie a aaaeaa?e a?oi? A. Ia?a (G, a) ? noaa?eueii neaaei aea?aaeaioiith aei ia?e, ui neeaaea?oueny c i?iaeaeo-iaeiiao?a oa i?iaeaeo-eioeeeea, yeui e o?eueee yeui ani-oe?eiaaia ae?y ? AO. Aeniiaie 3.4.3. Eiaeio AO ae?th aaaeaai? a?oie iiaea aooe ?aae?ciaaii ye ae(th Iaee(, anioe(eiaaio aei i?iaeaeo-eioeeeeo oa aei ae(( aoaeue-yeiai caaeaiiai oeio. Oai?aia 3.4.5. AO ae?y aaaeaai? a?oie, ui ? ae??th Iaee?, iiaoaeiaaiith ca ae??th oe-io II oa eioeeeeii, ? aiaeii/an ? ia?acii q-i?iaeaeo-eioeeeea, eiaiiieia?/iiai aei ii-/aoeiaiai. O aeiaaeeo iaaaaeaai( a?oie A o?aaa ci(ieoe aecia/aiiy i?iaeaeo-eioeeeeo: Aecia/aiiy 3.7. a: W ? G ® A iacaa?oueny p-eioeeeeii, yeui aeeiiai? oae? o?noue oiia: (1) aeey eiaeiiai j I N ?nio? ?icaeooy {Ekj, 0 F k < pj }, naaoi Ek1j ( Ek2j = ( i?e k1 ? k2, oa (kEk = W; (2) oa ?niothoue ia?aoai?aiiy Tj oeio I, ye? ia?aeeaaea-thoue iiiaeeie Ekj, naaoi Tjl * Ekj = Ejk+l (mod pj), i?e/iio (3) Tjl * Eki = Eki aeey an?o l, ye o?eueee i ? j, oa (4) a?oia, ui ?? ii?iaeaeathoue an? {Tj }j=1Y, ni?aiaaea? c [G]; (5) dm ° Tj / dm iino?eia ia eiaei?e Ekj; (6) oa ua a(w, Tj) ? a?aeia?aaeaiiyi c w a A, aei??iei a?aeiinii neaia-aeaaa?e, yeo ii?iaeaeathoue an? {Ekj : k=0,..., pj-1; j=1,...,l }. Oiae?, c?iaeaoe a?aeiia?aei? ci?ie a aeayeeo aeieacao, aaea?oueny ocaaaeueieoe ?acoeueoaoe /ioe?ueio iiia?aaei?o ?icae?e?a aeey aeiaaeeo iaaaaeaai? a?oie A. Aea A ia iiaea aooe aeia?eueiith: (1) A ia? aooe aiaiaaaeueiith e.e.n. a?oiith; (2) A ia? aeeth/aoe a naaa c/eneaiio u?eueio aiaiaaaeueio i?aea?oio B; (3) eioeeee a I Z1(w, G; A) ia? aooe oaeei, uia log D(a(w, g)) aoa eia?aieoeath, aea D – iiaeoey?ia ooieoe?y a?oie A (oea o?ea?aeueiei /eiii ni?aaaaeeeaa, yeui A oi?iiaeoey?ia). Uia c(noaaeoe iiaea(eio ae(th Iaee( c aeaiia i?inoeie, ?icaeyiaii eioeeeee a oa r ie?aii. Aaaaeaii oae? ae??: g(w, a) = (g w, a * a(w, g)), Vb(w, a) = (w, a * b). Aiie ae?thoue ia i?inoi?? w ? A; iicia/eii /a?ac Wb oaeoi?-ae?th ae?? Vb ia a?aiaee/ieo eii-iiiaioao ae?? g. E??i oiai, aaaaeaii ae?? g(w, u) = (g w, a + log (dm °g)/(dm)), Tt(w, u) = (w, u + t) ia i?inoi?? w ? R, oa iicia/eii /a?ac Ft oaeoi?-ae?th ae?? Tt ia a?aiaee/ieo eiiii-iaioao ae?? g. Oaa?aeaeaiiy 3.6.1. sseui iiaea?eia ae?y Iaee? (Ft, Wb) ? AO, oi e iaeaea? i?ino? ae?? Iaee? Ft oa Wb oaae ? AO. I?eeeaae 3.6.2. Iaoae w = { 0, 1}Z, m - i??a-aeiaooie, m = ( mi, mi(0) = mi(1) = 1/2. Iaoae q – ia?aoai?aiiy Aa?ioee?. ?icaeyiaii i?ino?? X = w ? w c i??ith m = m?m ? oae? aeaa ia?aoai?aiiy, ui caa??aaoeiooue i??o: q1 = q ? q, q2 = id ? q. Iaoae (Y, n) – eaaaaia i?ino?? c neaia-ne?i/aiith i??ith n, S – aeia?eueia a?aiaee/ia ia?a-oai?aiiy ia Y, ui caa??aa? n, oa iaoae u1, u2 – aeaa aaoiii?o?cie, ui eiiooothoue iaeei c ?ioei oa caaeiaieueiythoue oiia?: n ° u1 = exp(t1) * n, n ° u2 = exp(t2) * n, aea t1, t2 – aeaa ?aoe?iiaeueii-ianoi??ieo /enea. Iaoae, e??i oiai, u1, u2 iaeaaeaoue aei ii?ia-e?caoi?a [S]. ?icaeyiaii eaaaaia i?ino?? (X ? Y, m ? n) oa iiaoaeo?ii oae? o?e ae??: Q1(x, y) = (q1 x, u1 y), Q2 (x, y) = (q2 x, u2 y), S0 (x,y) = (x, S y). Aiie eiiooothoue oa ii?iaeaeathoue iiaio a?oio, yeo ie iicia/eii G. (oiy ni?eueia ae?y ? ae??th oeio III1 oa i?a?oaeueii aea?aaeaioia aei ae?? Z. Aecia/eii oaeee eioeeee a I Z1(X ? Y, G; Z): a(x, y; Q1) = 0, a(x, y; Q2) = 1, a(x, y; S0) = 0. Ae?y Iaee?, iiaoaeiaaia ca G oa a, o?ea?aeueia e oiio AO. Ae?y Iaee?, iiaoaei-aaia ca G oa eioeeeeii ?aaeiia-I?eiaeeia, oaae o?ea?aeueia e oiio AO. Aea iiaea?eia ae?y Iaee? ia? oaeoi?-ae?th aeiaeaoiuei? aio?ii?? e oiio ia iiaea aooe AO. *aoaa?oee ?icae(e. I?i aeanoeaino? iaeii? noeoiii a?aiaee/ii? ae?? i?yiiai aeiaooeo aeaio a?oi i?inoi?o X, ui caa-??aa? i??o, oae, uia: a) xn ® e, aea e – ?icaeooy ia oi/ee; b) qn o aea?eeiaiio caieno ia? aeaeyae w0 w1 w2 ...wn, n>0, aea w0 =
1;

;

d) A1(n+1) ( A1+qn(n+1) = A1(n), n? 0.

.

Ieae/a aaaaea?ii iine?aeiai?noue w aaea ia?aiith, ? i?eoiio oae, uia
aeeiio-aaeenue oae? aea? oiiae: ii-ia?oa, ?nio? iane?i/aii aaaaoi wk,
??aieo 1; ii-ae?oaa,

A(aeiii, ui aeaa ?icoe?aiiy T, iiaoaeiaai? ca eioeeeeaie a1 (x) oa
a2(x), iao-?e/ii ia?ciii?oi?, yeui a?aeiia?aei? iine?aeiaiino? {en1}n=1Y
oa {en2}n=1Y a?ae??ciy-thoueny a c/eneaiiiio /ene? i?noeue. Aoaeaii
eacaoe, ui aeaa eioeeeee a1 (x) oa a2 (x) iaeaea iaeiaeia?, yeui
iine?aeiaiino? {en1}n=1Y oa {en2}n=1Y a?ae??ciythoueny eeoa a
ne?i/aiiio /ene? i?noeue.

Eaiia 4.2.1. ?icoe?aiiy T c iaeaea iaeiaeiaeie eioeeeeaie iao?e/ii
?ci-ii?oi?.

x) = (a2(x)/a1(x)) b(x): aei-aiaeeii (nioaaiiy oi/ii aeaio oaeeo b, ui
a(ae?(ciythoueny ciaeii, oa ia/eneth(ii (o-i( cia/aiiy.

x, a{en}(x) y, {en}). Oey ae?y ?iciaaea?oueny ia a?aiaee/i? eiiiiiaioe
aeaeyaeo X ?Z2 ? {en}. Ieae/a ie ?o oa a?aeiia?aei? iaiaaeaiiy T ia ieo
iicia/a?ii /a?ac X{en} oa T{en}.

Oaia? aecia/eii ae?th a?oie G ia oiio ae i?inoi??. Aeey an?o k I N0
aecia/e-ii ia?aoai?aiiy pk i?inoi?o X :

pk (x,y, {en}n=0Y) = (x, bk(x)y, {d(k-n) en} n=0Y).

G aoaea a?oiith, yeo ii?iaeaeathoue an? oe? ia?aoai?aiiy; aiia aaaeaaa.
Aa/eii, ui T eiiooo? c pk. Ia/eneeii oaia? oeaio?ae?caoi? noeoiii? ae??
T oa pk. Iaoae V – aaoiii?o?ci i?inoi?o X, ui eiiooo? c T oa pk.
Iicia/eii s (x, y, z) = (x, –y, z).

pk, aea p, q I Z oa ia caeaaeaoue a?ae {en}.

Aeniiaie 4.3.3. Ae?y a a?oie Z?G ia ?ciii?oia i?yiiio aeiaooeo ae?e Z
oa G, ine?eueee Ca(G) = Z2N, Ca(Z) = Z ? Z2, aea Ca(G ? Z) ia
?ciii?oiee Ca(Z) ? Ca(G).

AENIIAEE

1. Iiiyooy eoiaaeiiai ?aiaa iaeei iiaea aooe ei?aeoii aecia/aiei aeey
ae(e aoaeue-yeeo e.e.n. a?oi, ui ia caa?(aathoue i(?o, i?e/iio eoiaaeiee
?aia iaeei caa?(aa-(oueny, eiee i(?o cai(iththoue ia aea(aaeaioio aei
ia(.

2. Ae(y e.e.n. a?oie ia eaaaaiaiio i?inoi?(, yea ia( eoiaaeiee ?aia
iaeei, ( AO.

3. sseui e.e.n. a?oia G ae(( ia eaaaaiaiio i?inoi?(, oi iiaeeeai
aecia/eoe eeane G-iaeaea a(aee?eoeo oa G-iaeaea caieiaieo i(aeiiiaeei
oeueiai i?inoi?a, i?e-/iio aoaeue-yea aei(?ia iiiaeeia iiaea aooe
ai?ieneiiaaia ca aeiiiiiaith iii-aeei, ui ( aiaeii/an iaeaea a(aee?eoeie
oa iaeaea caieiaieie, c aeia(eueiith oi/i(n-oth o ?icoi(ii( i(?e
neiao?e/ii( ?(cieoe(.

4. O?aiceoeai( ae(( e.e.n. a?oi ( AO; ae(( e.e.n. a?oi, ui iathoue
aeene?aoiee niaeo?, oaeiae ( AO.

5. (ia?oa oai?aia i?i (iaeoeiaai( ae(() Iaoae G – e.e.n. a?oia, H – ((
caieiaia ii?iaeueia i(aea?oia, oa iaoae H ae(( ia eaaaaiaiio i?inoi?(
((, B, (). sseui ae(y H ia( eoiaaeiee ?aia iaeei, oa ae(y G cnoaaie ia
e(aiio iaeii?iaeiiio i?inoi?( X=G/H oaeiae ia( eoiaaeiee ?aia iaeei, oi
e (iaeoeiaaia ae(y a?oie G oaeiae ia( eoiaaeiee ?aia iaeei.

sseui X ( aeene?aoiei, oi ia o?aaa i?eioneaoe, uia H aoea ii?iaeueiith,
(, a(eueo oiai, o oeueiio aeiaaeeo cai?ioia oaa?aeaeaiiy aei iaoi(
oai?aie oaeiae ni?a-aaaeeeaa.

6. (ae?oaa oai?aia i?i (iaeoeiaai( ae(() Iaoae G – e.e.n. a?oia, H – ((
caieiaia i(aea?oia, oa H ae(( ia eaaaaiaiio i?inoi?( ((, B, (). sseui
ae(y H ( AO, oi e (iaeo-eiaaia ae(y a?oie G oaeiae ( AO.

7. O?aiceoeai( ae(( e.e.n. ?ica(ycieo a?oi iathoue eoiaaeiee ?aia
iaeei; ae(( ?ic-a(ycieo a?oi, ui iathoue aeene?aoiee niaeo?, oaeiae
iathoue eoiaaeiee ?aia iaeei.

8. O?aiceoeai( ae(( oe(eeii iaca(ycieo e.e.n. a?oi iathoue eoiaaeiee
?aia iaeei.

9. Aeiaaaeaii aeaa e?eoa?(( ?iciaaeo ia?e, ui neeaaea(oueny c ae(( oa
eioeeeeo, a iane(i/aiee i?yiee aeiaooie. Aeea. oi?ioethaaiiy oeeo
e?eoa?((a o ?icae(e( 3.1.

10. Iiaea(eia ae(y Iaee(, iiaoaeiaaia ca aiaiaaaeueiith a?oiith
ia?aoai?aiue oeio III G oa eioeeeeii (((, aea ( – eioeeee
?aaeiia-I(eiaeeia, a ( – aeia(eueiee 1-ei-oeeee c( cia/aiiyie a aaaeaa(e
a?oi( A, ( AO oiae( e o(eueee oiae(, eiee ia?a (G, ((, ()) ( neaaei
aea(aaeaioiith aei ia?e, ui neeaaea(oueny c i?iaeaeo-iaeiiao?o oa
i?iaeaeo-eioeeeeo.

11. Ae(y Iaee(, iiaoaeiaaia ca aiaiaaaeueiith a?oiith ia?aoai?aiue oeio
II G oa eioeeeeii (, c( cia/aiiyie a aaaeaa(e a?oi( A, ( AO oiae( e
o(eueee oiae(, eiee ia?a (G, () ( noaa(eueii neaaei aea(aaeaioiith aei
ia?e, ui neeaaea(oueny c i?iaeaeo-iaei-iao?o oa i?iaeaeo-eioeeeeo.

12. Aoaeue-yea ae(y aaaeaai( a?oie A iiaea aooe i?aaenoaaeaiith ye ae(y
Iaee(, iiaoaeiaaia ca ae((th aeia(eueiiai caaeaiiai oeio oa
i?iaeaeo-eioeeeeii.

13. Iiia?aaei( oaa?aeaeaiiy iiaea aooe i(aeneeaia: o ?ac(, eiee aeaia
AO ae(y ao-ea c naiiai ii/aoeo i?aaenoaaeaia ye ae(y Iaee(, anioe(eiaaia
aei ae(( oeio II oa ei-oeeeeo, oeae eioeeee aoaea ia o(eueee neaaei
aea(aaeaioiei, aea e eiaiiieia(/iei aei aeayeiai (-i?iaeaeo-eioeeeeo.

14. An( iaaaaeai( aeua (ii. 9 – 13 ) ?acoeueoaoe ni?aaaaeeea( ia
o(eueee aeey ae-iaaeeo aaaeaai( a?oie A, aea e a iaaaaaoi a(eueo
caaaeuei(e neooaoe((. A?oia A ioneoue o(eueee caaeiaieueiyoe oaeei
o?ueii oiiaai: (1) A – aiaiaaaeueia e.e.n. a?oia; (2) A i(noeoue
c/eneaiio u(eueio aiaiaaaeueio i(aea?oio; (3) aeaiee eioeeee ( ( Z1((,
G; A) ia( aooe oaeei, uia log (((((, g)) aoa eia?aieoeath, aea ( –
iiaeoey?ia ooieoe(y a?o-ie A.

15. Ie ii?(aiyee iiaea(eio ae(th Iaee(, iiaoaeiaaio ca ae((th oeio III
oa ca ii-aea(eiei eioeeeeii ((, (), c aeaiia i?inoeie ae(yie Iaee(,
iiaoaeiaaieie ca o((th ae naiith ae((th oa ca eioeeeeaie ( oa ( ie?aii,
oa aeiaaee, ui c ai?ieneiaoeaii( o?ai-ceoeaiino( ia?oi( ae(( aeo(ea(
ai?ieneiaoeaia o?aiceoeai(noue aeaio (ioeo. Cai?io-ia oaa?aeaeaiiy (
oeaiei, ( ie iaaaee a(aeiia(aeiee i?eeeaae.

16. Iiaoaeiaaia noeoiia ae?y a aeaio a?oi G= Z2 A Z2 A… oa Z, ui ia (
(ciii?oiith aei i?yiiai aeiaooeo a(aeiia(aeieo ae(e a?oi G oa Z , ai ia(
oae( aean-oeaino(:

– yeui Ca (G ? Z) – oeaio?ae?caoi? ae?? a, oi Ca(G ?Z) / a(G ? Z) ( Z2;

– yeui z oa h – aea? aeia?euei? a?aiaee/i? eiiiiiaioe ae?? a({e} ? Z),
oi ae?? a({e}? Z)|z oa a({e}? Z)|h iao?e/ii ?ciii?oi? oiae? e o?eueee
oiae?, eiee z = a(g, 0)h aeey aea-yeiai g I G;

– ae?? eiaeii? c a?oi G oa Z ia a?aiaee/i?.

IOAE(EAOe((

Golodets V.Ya., Sokhet A.M. Ergodic actions of an Abelian group with
discrete spectrum, and approximate transitivity. J.Sov.Math. 52, No.6 ,
p. 3530–3533 (1990); translated from Teor.Funkts., Funks.Anal.Prilozh.,
51, p.117–122 (1989).

Golodets V.Ya., Sokhet A.M. On properties of jointly ergodic action of
the direct product of two groups. Ukr.Math.J. 43, No.5, p.635–639
(1991); translated from Ukr. Mat.Zh. 43, No.5, 684–688 (1991).

Golodets V.Ya., Sokhet A.M. Transitive actions of solvable groups and
approximate transitivity. Preprint of Institute for Low Temperature
Physics and Engineering, Kharkov, 1989, No.19. – 26 p.

Golodets V.Ya., Sokhet A.M. A representation of approximately
transitive group actions as a product cocycle range. Preprint of
Institute for Low Temperature Physics and Engineering, Kharkov, 1991,
No.2. – 12 p.

Golodets V.Ya., Sokhet A.M. Measure-preserving approximately transitive
actions and spectrum. Preprint of Institute for Low Temperature Physics
and Engineering, Kharkov, 1991, No.11. –16 p.

Nioao A.I. I iaeioi?uo naienoaao aii?ieneiaoeaii o?aiceoeaiuo
aeaeno-aee. XII iao/.-oaoi. eiioa?aioeey iieiaeuo enneaaeiaaoaeae OOEIO
AI ONN?: oacenu aeieeaaeia. Oa?ueeia, 1991. n.85–86.

Aieiaeaoe A.ss., Nioao A.I. I niaeo?aeueiuo naienoaao aii?ieneiaoeaii
o?aiceoeaiuo aeaenoaee. XVI Ananithciay oeiea ii oai?ee iia?aoi?ia a
ooieoee-iiaeueiuo i?ino?ainoaao. Oacenu aeieeaaeia. Ieaeiee Iiaai?iae,
1991. N.54.

Golodets V.Ya., Sokhet A.M. Cocycles of type III transformation group
and AT property for the double Mackey action. Preprint of the Erwin
Shroedinger International Institute for Mathematical Phisics, 1994, ESI
97. – 39 p.

Golodets V.Ya., Sokhet A.M. Product cocycles and the approximate
transitivity. Mathematical Physics, Analysis and Geometry, 1, No.4,
p.331–365 (1998).

Sokhet A.M. Funny rank one and the approximate transitivity for induced
actions, Monatshefte fuer Matematik, 1998, to appear.

Sokhet A.M. Transitive actions have funny rank one. Matematicheskaya
fizika, analiz, geometriya, 6, No.1/2, p.124–129 (1999).

Nio(o I.I. Ai?ieneiaoeaii o?aiceoeai( a?oie ia?aoai?aiue a a?aiaee/i(e
oai?((. – ?oeiien. Aeena?oaoe(y ia caeiaoooy iaoeiaiai nooiaiy
eaiaeeaeaoa o(ceei-iaoaiaoe/ieo iaoe ca niaoe(aeuei(noth 01.01.01. –
iaoaiaoe/iee aiae(c. – O(ceei-oaoi(/iee (inoeooo iecueeeo oaiia?aoo? IAI
Oe?a(ie, Oa?e(a, 1999.

Aeena?oaoe(th i?enay/aii aea/aiith ai?ieneiaoeaii o?aiceoeaieo (AO)
ae(e. Aecia/aii aeanoea(noue, yeo eee/ooue eoiaaeiei ?aiaii iaeei, oa
aeiaaaeaii (( iaca-eaaei(noue a(ae aeai?o i(?e a iaaeao iaeiiai eeano, a
oaeiae ui c ia( aeo(ea( AO. Aeiaaaeaii oai?aio i?i (iaeoeiaai( ae((:
iaoae H – caieiaia ii?iaeueia i(aea?oia ei-eaeueii eiiiaeoii(
naia?aaaeueii( a?oie G. sseui i?e?iaeia ae(y G ia G/H oa aei-a(eueia
ae(y H ia eaaaaiaiio i?inoi?( iathoue eoiaaeiee ?aia iaeei, oi e
(iaeoeiaaia ae(y a?oie G oaae ia( eoiaaeiee ?aia iaeei. sseui G
?ica(ycia, ia?oa oiiaa caaaeaee aeeiiaia. Iiae(aia oai?aia ni?aaaaeeeaa
e aeey AO ae(e, iaa(oue aac i?eiouaiiy i?i ii?iaeuei(noue i(aea?oie H.
Iaea?aeaii e(eueea oe(eaaeo aeniiae(a. Aeiaaaeaii e(eueea e?eoa?((a
ai?ieneiaoeaii( o?aiceoeaiino(. Iiaea(eiee eioeeee aeey a?aiaee/ii( ae((
oeio II /e III, ui neeaaea(oueny c eioeeeeo ?aaeiia-I(eiaeeia oa
aeia(eueiiai eioeeeeo, ( neaaei aea(aaeaioiei aei i?iaeaeo-eioeeeea aeey
i?iaeaeo-iaeiiao?a oiae( e o(eueee oiae(, eiee anioe(eiaaia iiaea(eia
ae(y Iaee( ( AO. O aeiaaeeo oeio II iiaeia ciaeoe iaa(oue
i?iaeaeo-eioeeee, eiaiiieia(/iee aei ii/aoeiaiai. Iaaaaeaii i?eeeaae, ui
aeiaiaeeoue iaiao(aei(noue ?icaeyaeaiiy naia iiaea(eieo ae(e Iaee(.
Iaaaaeaii i?eeeaae noeoiii( ae(( aeaio a?oi G oa Z, ia (ciii?oiiai aei
aeiaooeo a(aeiia(aeieo ae(e a?oi G oa Z; a(i ia( e(eueea ianiiae(aaieo
aeanoeainoae.

Eeth/ia( neiaa: a?aiaee/ia oai?(y, ai?ieneiaoeaia o?aiceoeai(noue,
eoiaae-iee ?aia iaeei, (iaeoeiaaia ae(y, ae(y Iaee(, i?iaeaeo-eioeeee.

Nioao A.I. Aii?ieneiaoeaii o?aiceoeaiua a?oiiu i?aia?aciaaiee a
y?ai-aee/aneie oai?ee. – ?oeiienue. Aeenna?oaoeey ia nieneaiea o/?iie
noaiaie eaiaeeaeaoa oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe
01.01.01 – iaoaiaoe/aneee aiaeec. – Oeceei-oaoie/aneee einoeooo ieceeo
oaiia?aoo? IAI Oe?aeiu, Oa?ueeia, 1999.

Aeenna?oaoeey iinayuaia aii?ieneiaoeaii o?aiceoeaiui (AO) aeaenoaeyi.
Aaiaeeony iiiyoea no?aiiiai ?aiaa iaeei e aeieacuaaaony aai
iacaaeneiinoue io au-ai?a ia?u a i?aaeaeao iaeiiai eeanna, a oaeaea /oi
ec iaai neaaeoao AO. Aeieacu-aaaony oai?aia ia eiaeooee?iaaiiuo
aeaenoaeyo: ionoue H – caieiooay ii?iaeueiay iiaea?oiia eieaeueii
eiiiaeoiie naia?aaaeueiie a?oiiu G. Anee anoanoaaiiia aeae-noaea G ia
G/H e i?iecaieueiia aeaenoaea H ia i?ino?ainoaa Eaaaaa iaa eiatho
no?aiiue ?aia iaeei, oi oi aea aa?ii e aeey eiaeooee?iaaiiiai aeaenoaey
G. Anee G ?ac?aoeia, ia?aia oneiaea anaaaea aa?ii. Aiaeiae/iay oai?aia
ni?aaaaeeeaa e aeey AO aeaenoaee, e aeaaea aac i?aaeiieiaeaiey i
ii?iaeueiinoe H. Iieo/ai ?yae eioa?an-iuo neaaenoaee. Aeieacaii
ianeieueei e?eoa?eaa AO-naienoaa. Aeaieiie eioeeee aeey y?aiaee/aneiai
aeaenoaey oeia II eee III, ninoiyuee ec eioeeeea ?aaeiia-Ieeiaeeia e
i?iecaieueiiai eioeeeea, neaai yeaeaaeaioai i?iaeaeo-eioeeeeo aeey
i?iaeaeo-iaei-iao?a, anee e oieueei anee annioeee?iaaiiia aeaieiia
aeaenoaea Iaeee – AO. A neo/aa oeia II iiaeii aeaaea iaeoe
i?iaeaeo-eioeeee, eiaiiieiae/iue aeaiiiio. I?eaaaeai i?eia?,
iieacuaathuee iaiaoiaeeiinoue ?anniio?aiey aeaieiuo aeaeno-aee Iaeee.
Iino?iai i?eia? niaianoiiai aeaenoaey a?oii G e Z, ia eciii?oiiai
i?iecaaaeaieth niioaaonoaothueo aeaenoaee a?oii G e Z e iaeaaeathuaai
iaeioi-?uie iaiaeeaeaiiuie naienoaaie.

Eeth/aaua neiaa: y?aiaee/aneay oai?ey, aii?ieneiaoeaiay
o?aiceoeaiinoue, no?aiiue ?aia iaeei, eiaeooee?iaaiiia aeaenoaea,
aeaenoaea Iaeee, i?iaeaeo-eioeeee.

Sokhet A.M. Approximately transitive transformation groups in the
ergodic theory. – Manuscript. Thesis for a candidate’s degree in
Physics and Mathematics by speciality 01.01.01 – mathematical analysis.
– Institute for Low Temperature Physics & Engi-neering, Kharkov, 1999.

The class of approximately transitive (AT) actions was introduced by
A.Connes and E.J.Woods in connection with the characterization problem
for the factors which are infinite tensor products of type I factors.
These actions have turned to be very inte-resting from the ergodic
theory point of view, and a lot of papers were devoted to study-ing of
these actions.

A lot of problems arose in connection with the approximate
transitivity, and some of them are already solved. The first natural
group of questions is how the property of approximate transitivity (AT
property) is related with other properties studied in the ergodic
theory. Some sufficient conditions for the approximate transitivity are
presented here. Another group of questions was to connect the study of
AT actions with the classification problem of 1-cocycles of ergodic
actions. A description of AT actions constructed as Mackey actions in
terms of product cocycles and the weak equivalence relation is obtained
in this thesis.

In fact, a lot of results concerning the AT property were primarily
obtained for the most natural particular cases of a single automorphism
and a flow, i.e. for the actions of Z and R. The author’s purpose was to
understand what do these results mean for the general case of an action
of an arbitrary locally compact separable group and to prove their
appropriate analogues.

This paper consists of four chapters.

Chapter 1 presents some preliminaries: we recall some definitions used
in the ergodic theory, introduce the notion of the approximate
transitivity and discuss some its reformulations, and then pass to the
definitions of a cocycle, a Mackey action and of the weak equivalence.

Chapter 2 contains some theorems providing sufficient conditions for
the approximate transitivity. We start from the definitions of rank and
funny rank; we define funny rank one for a non measure-preserving action
of an arbitrary locally compact separable (l.c.s.) group. We prove that
the funny rank one is a sufficient condition for the AT property. We
show that any transitive action of a l.c.s. group is AT, as well as any
action having discrete spectrum. Then, we pass to Abelian group actions
and prove that their transitive action are not only AT, but even have
funny rank one. We extend then these results from Abelian l.c.s. groups
to a more general class of solvable groups. To do this, we prove (and
this is the main result in this chapter) funny rank one for an induced
action: namely, if H is a closed normal subgroup of a l.c.s. group G,
and both an (arbitrary) action of H on some Lebesgue space and the
natural G-action on G/H have funny rank one, then the induced G-action
will also have it. A similar (but much simpler) result is proved for the
AT property also. We obtain some important corollaries of our result;
among them, there is an interesting one concering to totally
disconnected group actions; namely, their transitive actions have funny
rank one, and to prove this, we need not assume the solvability.

Chapter 3 contains some theorem providing criterions of the AT property
in the terms of Mackey actions and product cocycles. We introduce
product cocycles, and then prove two technical criterions of the product
property. Then, we consider so-called double Mackey actions, and deal
with type II and type III actions separately. The principal results of
this chapter state that (1) the Mackey action constructed by an amenable
type III transformation group G and a 1-cocycle (((, where ( is the
Radon-Nikodym cocycle while ( is an arbitrary 1-cocycle with values in a
l.c.s. group A, is AT if and only if the pair (G, ((,()) is weakly
equivalent to a product odometer supplied with a product cocycle; (2)
the Mackey action constructed by an amenable type II trans-formation
group G and a 1-cocycle ( with values in a l.c.s. group A is AT if and
only if the pair (G, () is stably weakly equivalent to a product
odometer supplied with a product cocycle; (3) any AT action of a l.c.s.
group A can be represented as a Mackey action constructed by an action
of any pre-given type and a product cocycle; (4) the latter statement
can be strengthened: in the case when the given AT action from the very
beginning was a range of a type II action and a cocycle, then this
cocycle turns out to be not only weak equivalent but even cohomologous
to a (-product cocycle; (5) the Mackey action constructed by a type III
action and ((,() is compared with two Mackey actions constructed by (
and ( separately, and it is shown that the AT property for the first one
implies the AT property for two others, while the converse statement
turns out to be false by constructing an appropriate example.

Chapter 4 presents another interesting example of a joint action ( of
two groups G = Z2 ( Z2 ( … and Z that is not isomorphic to the product
of the corresponding G and Z-actions and possesses with some interesting
properties: the actions of Z and G separately are not ergodic, the
actions of Z on its ergodic components are isomorphic if and only if
these components are mapped to one another by the action of G, and the
centralizer Ca (G ? Z) is such that Ca(G?Z) / a(G ? Z) ( Z2.

Key words: ergodic theory, approximate transitivity, funny rank one,
induced action, Mackey action, product cocycle.

A(aeiia(aeaeueiee ca aeione – aeieoi? o(c.-iao. iaoe A.I.Eioey?ia

I(aeienaii aei ae?oeo ___________________. Oiiai.-ae?oe.a?e. 2. Cai.
__________________

Oe?aae 100 aec. ?ioai?(io OO(IO IAI Oe?a(ie, i.Oa?e(a.

Aaceiooiaii

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