Интегрируемость комплексных гамильтоновых систем с неполными потоками в C^2, страница 23

, s2n (ξ), d1 (ξ), . . . , dn+1 (ξds2ξ , |ξ| < ε, íåïðåðûâíî çàâèñÿùèõ îò ξ , èìåþ-ùèõ íà÷àëî è êîíåö â áåñêîíå÷íî óäàëåííûõ òî÷êàõpξ,j (ñì. ñëåäñòâèå 4.2.3 èïîÿñíåíèå 4.2.7),sk (0) = sk , k = 1, . . . , 2n,óäîâëåòâîðÿþùèõ ñîîòíîøåíèÿìdm (0) = dm , m = 1, . .

. , n + 1,è íå ïðîõîäÿùèõ ÷åðåç òî÷êè150(0, ai (ξ)) ∈ Tξ ,i = 1, . . . , 2n + 2,ãäåÎáîçíà÷åíèå 4.2.9sk , dmêàê â óòâåðæäåíèè 4.2.4.((ðàçðåçàíèå ñëîÿ, áëèçêîãî ê íóëåâîìó, ãåîäåçè÷åñêèìè2íà öèëèíäðû)). Ñîãëàñíî ñëåäñòâèþ 4.2.8, ïðè ëþáîì ξ ∈ D0,εèìååì ñëåäó-þùèå ðàçáèåíèÿ ïëîñêîñòè C = Cw è ñëîÿ Tξ = f −1 (ξ), àíàëîãè÷íûå ââåäåííûì â îáîçíà÷åíèè 4.2.5 ïðè ξ = 0:C = S1 (ξ)∪. . .∪Sn (ξ)∪C1 (ξ)∪. . .∪Cn+1 (ξ), Tξ = s1 (ξ)∪.

. .∪s2n (ξ)∪c1 (ξ)∪. . .∪cn+1 (ξ),ãäå Ck (ξ) êîìïîíåíòà ñâÿçíîñòè Cw \(S1 (ξ)∪. . .∪Sn (ξ)), ñîäåðæàùàÿ òî÷êèa2k−1 (ξ), a2k (ξ), 1 ≤ k ≤ n + 1. Ïðè n ≥ 2 ãðàíèöû öèëèíäðè÷åñêèõ îáëàñòåéck (ξ) èìåþò âèä ∂c1 (ξ) = s1 (ξ) ∪ s2 (ξ), ∂ck (ξ) = s2k−3 (ξ) ∪ s2k−2 (ξ) ∪ s2k−1 (ξ) ∪s2k (ξ) ïðè k = 2, .

. . , n, ∂cn (ξ) = s2n−1 (ξ) ∪ s2n (ξ).4.2.3Êîìïëåêñíûå êîîðäèíàòû äåéñòâèå-óãîë è ôóíêöèè ïåðåõîäà. Êîìïëåêñíàÿ òåîðåìà ËèóâèëëÿÎïðåäåëåíèå 4.2.10.ε-îêðåñòíîñòüþk -îéÏóñòü ε > 0 êàê â ñëåäñòâèè 4.2.8.íóëåâîãî ñëîÿ÷åòûðåõìåðíîé×åòûðåõìåðíîéT0 íàçîâåì îáëàñòü Uε (T0 ) :=ε-ðó÷êîé Gε,k íàçîâåì åå ïîäìíîæåñòâîGε,k :=[ck (ξ) ⊂ Uε (T0 ),STξ ⊂ C2 , à|ξ|<ε1 ≤ k ≤ n,|ξ|<εãäå ck (ξ) çàìûêàíèå öèëèíäðè÷åñêîé îáëàñòè ck (ξ) ⊂ Tξ èç îáîçíà÷åíèÿ 4.2.9.Îòìåòèì, ÷òî êîëè÷åñòâî ÷åòûðåõìåðíûõ ε-ðó÷åê Gε,k ðàâíî n + 1 (ñì.îáîçíà÷åíèå 4.2.9), è ðó÷êè ïîêðûâàþò âñþ ÷åòûðåõìåðíóþ ε-îêðåñòíîñòüUε (T0 ) ñëîÿ T0 , ò.å.

Uε (T0 ) =n+1SGε,k .k=1151Òåîðåìà 26((êîìïëåêñíàÿ òåîðåìà Ëèóâèëëÿ äëÿ ãèïåðýëëèïòè÷åñêîãî ãà-ìèëüòîíèàíà ÷åòíîé ñòåïåíè)). Äëÿ C-ãàìèëüòîíîâîé ñèñòåìû (C2 , dz∧dw, f )ñ ôóíêöèåé Ãàìèëüòîíàæåâà ñëîåíèÿf (z, w) = z 2 +P2n+2 (w) è ñîîòâåòñòâóþùåãî ëàãðàí-(ñì. îïðåäåëåíèå 3.1.6),ãäåP2n+2 (w) = (w − a1 ) . . . (w − a2n+2 ),ai ∈ R, i = 1, .

. . , 2n + 2, a1 < a2 < . . . < a2n+2 , n ∈ N,ñóùåñòâóåòε > 0,òàêîå ÷òî âûïîëíåíû ñëåäóþùèå ñâîéñòâà:1) äëÿ ëþáîãîξ ∈ C, |ξ| < ε,ãîìåîìîðôåí ñôåðå ñnñëîéTξ = f −1 (ξ)ÿâëÿåòñÿ íåîñîáûì èðó÷êàìè è äâóìÿ ïðîêîëàìè;2) ëàãðàíæåâî ñëîåíèå â ÷åòûðåõìåðíîéε-îêðåñòíîñòè Uε (T0 )ñëîÿT0T0íàòðèâèàëüíî, ò.å.

ïîñëîéíî ãîìåîìîðôíî ïðÿìîìó ïðîèçâåäåíèþ ñëîÿîòêðûòûé äâóìåðíûé äèñê3) â îêðåñòíîñòè2= {ξ ∈ C | |ξ| < ε};D0,εUε (T0 )ñóùåñòâóþò2n + 2ãîëîìîðôíûõ ôóíêöèéI1 , . . . , In+1 , J1 , . . . , Jn+1 : Uε (T0 ) → C,à äëÿ êàæäîé ÷åòûðåõìåðíîéε-ðó÷êè Gε,k ⊂ Uε (T0 ), k = 1, . . . , n + 1,ñóùå-ñòâóåò ãîëîìîðôíîå âëîæåíèå (çàäàâàåìîå êîìïëåêñíûìè êîîðäèíàòàìèäåéñòâèå-óãîë)(Ik |Gε,k , ϕk mod 2π) : Gε,k ,→ C × (C/2πZ),1 ≤ k ≤ n + 1;ñî ñëåäóþùèìè ñâîéñòâàìè:à) êàæäàÿ ôóíêöèÿåéIk = Ik (f )èIk , Jk : Uε (T0 ) → CJk = Jk (f )îòfÿâëÿåòñÿ ãîëîìîðôíîé ôóíêöè-áåç êðèòè÷åñêèõ òî÷åê, åå ìíîæåñòâîçíà÷åíèée ε,k := Ik (Uε (T0 )) = Ik (Gε,k ),Db ε,k := Jk (Uε (T0 )) = Jk (Gε,k ) ⊂ CD152îòêðûòî âíîñòèCUε (T0 )è ãîìåîìîðôíî îòêðûòîìó êðóãó, îíà âûðàæàåòñÿ â îêðåñò÷åðåç ëþáóþ äðóãóþ òàêóþ ôóíêöèþ ôîðìóëàìèIk = Ik (f (I` )),ãäåf (Ik )èñòâåííî (kf (Jk )Ik = Ik (f (J` )), ôóíêöèè, îáðàòíûå ê ôóíêöèÿìk = 1, .

. . , n + 1ïëåêñíîé êîîðäèíàòû óãîëçàìêíóòîé îáëàñòè(k)(k)A1 (Ik ), . . . , A6 (Ik ) ∈ Ck = n+1(n+1)A3èe ε,kIk ∈ DIk (f )èJk (f )ñîîòâåò-ìíîæåñòâî çíà÷åíèé êîì-ϕk mod 2π|Gε,k ∩Tf (Ik )Wk,Ik ⊂ C,ëîãðàìì ñ âåðøèíàìèïðèJk = Jk (f (J` ));= 1, . . . , n + 1);á) ïðè ëþáûõíàìèJk = Jk (f (I` )),ïîëó÷àåòñÿ èç íåêîòîðîéîãðàíè÷åííîé øåñòèóãîëüíèêîì ñ âåðøè-k = 1(âûðîæäàþùèìñÿ ïðèA11 (I1 ) = A16 (I1 ), A12 (I1 ) = A13 (I1 ), A14 (I1 ), A15 (I1 ),(n+1)â ïàðàëëåëîãðàìì ñ âåðøèíàìèA1(n+1)(In+1 ))(In+1 ) = A4â ïàðàëëå-(n+1)(In+1 ), A5îòâåòñòâóþùèìè ãåîäåçè÷åñêèì(n+1)(In+1 ) = A6(n+1)(In+1 ), A2è(In+1 ),è ñòîðîíàìè, ñî-d1 (f (I1 )), s1 (f (I1 )), s2 (f (I1 )) ⊂ Tf (I1 )ïðèk = 1, dk (f (Ik )), s2k−3 (f (Ik )), s2k−2 (f (Ik )), s2k−1 (f (Ik )), s2k (f (Ik )) ⊂ Tf (Ik )ïðè1 < k < n + 1, dn+1 (f (In+1 )), s2n−1 (f (In+1 )), s2n (f (In+1 ))ïðèk = n + 1,ñëåäóþùèì îáðàçîì: (i) âûêèäûâàíèåì âñåõ âåðøèí (ñîîòâåòñòâóþùèõ áåñêîíå÷íî óäàëåííûì òî÷êàìpf (Ik ),1 , pf (Ik ),2 ),è (ii) îòîæäåñòâëåíèåì (ò.å.ñêëåèâàíèåì) ïðè ïîìîùè ïàðàëëåëüíîãî ïåðåíîñà ëþáîé ïàðû ñòîðîí, îòâå÷àþùèõ îäíîé è òîé æå ãåîäåçè÷åñêîédk (f (Ik ));ïðè÷åì øåñòèóãîëüíèê(ñîîòâåòñòâåííî ïàðàëëåëîãðàìì ïðèk = 1, n + 1)îäíîçíà÷íî çàäàåòñÿñëåäóþùèìè óñëîâèÿìè (ñì.

ðèñ. 4.14,4.15, 4.16 ïðèk = 1 , 1 < k < n + 1,∂Wk,Ik ⊂ Cîáðàçîâàí òðåìÿ ïà-k =n+1•ñîîòâåòñòâåííî):øåñòèóãîëüíèê (èëè ïàðàëëåëîãðàìì)ðàìè ðàâíûõ è ïàðàëëåëüíûõ ñòîðîí, ñîîòâåòñòâóþùèõ ñëåäóþùèì153ãåîäåçè÷åñêèì è ïîëó÷àþùèõñÿ äðóã èç äðóãà ñëåäóþùèìè ñäâèãàìè âïëîñêîñòèC:(k)2π(k)(k)(k)(k)(k)(k)(k)A1 (Ik )A2 (Ik ) = ϕk (dk (ξ)) −→ A5 (Ik )A4 (Ik ) = ϕk (dk (ξ)),(k)(k)−hDk (ξ)ik(k)(k)−hDk (ξ)ikA2 (Ik )A3 (Ik ) = ϕk (s2k−3 (ξ))A3 (Ik )A4 (Ik ) = ϕk (s2k−1 (ξ))ãäå2,ξ := f (Ik ) ∈ D0,εíà âåêòîðδ∈C÷åðåçhDk (f (Ik ))ik := 2π•ïðè ëþáîìk <n+1−→A1 (Ik )A6 (Ik ) = ϕk (s2k−2 (ξ)),A6 (Ik )A5 (Ik ) = ϕk (s2k (ξ)),δ:C→Câ ïëîñêîñòè−→k > 1,k < n+1,îáîçíà÷åí ïàðàëëåëüíûé ïåðåíîñC, s0 (ξ) := s1 (ξ),dJk−1dJkk−1 dJ1(Ik ) −(Ik ) + . .

. + (−1)(Ik ) ,dIkdIkdIkâûïîëíåíî−−−−−−−−−−−→ −−−−−−−−−−−→(k)(k)(k)(k)A3 (Ik )A4 (Ik ) = A6 (Ik )A5 (Ik )dIk+1dIk+2dIn= hSk+1 (f (Ik ))ik := 2π(Ik ) −(Ik ) + . . . + (−1)n−k−1(Ik ) ,dIkdIkdIk•òî÷êà ïåðåñå÷åíèÿ äèàãîíàëåé (âûðîæäàþùåãîñÿ â îòðåçîê ïðèïàðàëëåëîãðàììà(k)(k)â)(k)(îòêóäà òî÷êà ïåðåñå÷åíèÿ äèàãîíàëåé (âûðîæ-äàþùåãîñÿ â îòðåçîê ïðè(k)12 (A3 (Ik )(k)(k)A1 (Ik )A2 (Ik )A3 (Ik )A6 (Ik ) ðàâíà 21 (A1 (Ik )+A3 (Ik )) =0 = ϕk (0, a2k−1 (f (Ik )))ðàâíà(k)k = 1)(k)(k)(k)(k)+ A5 (Ik )) = π = ϕk (0, a2k (f (Ik )));(dz ∧ dw)|Gε,k = dIk ∧ dϕk , k = 1, . . .

, n + 1;ã) ïåðåìåííàÿ äåéñòâèåIk (ξ) =1πaZ2k (ξ)(k)k = n+1) ïàðàëëåëîãðàììà A3 (Ik )A4 (Ik )A5 (Ik )A6 (IIk = Ik (f )è ôóíêöèÿp1ξ − P2n+2 (y)dy, Jk (ξ) =πa2k−1 (ξ)a2k−1Z (ξ)èìåþò âèäpξ − P2n+2 (y)dy,a2k−2 (ξ)154Jk = Jk (f )2ξ ∈ D0,ε,ãäåa0 (ξ) := −∞,è â êà÷åñòâå ôóíêöèè√áåðóòñÿ åå âåòâè, òàêèå ÷òîqq1−P2n+2 ( 2 (a2k−1 + a2k )) > 0 â ïåðâîì ñëó÷àå, è i −P2n+2 ( 12 (a2k−2 + a2k−1 )) <0âî âòîðîì ñëó÷àå;ä) äëÿ ëþáûõ äâóõ ðó÷åêGε,k , Gε,` ,òî æå ñåìåéñòâî ãåîäåçè÷åñêèõsj (ξ),1 ≤ k < n+1 ïåðåñå÷åíèå Gε,k ∩Gε,k+1Gε,k ∩ Gε,k+1 =[ñîäåðæàùèõ â ñâîåé ãðàíèöå îäíî èâûïîëíåíîk = ` ± 1,ïðè÷åì â ñëó÷àåÿâëÿåòñÿ îáúåäèíåíèåì ãåîäåçè÷åñêèõ(s2k−1 (ξ) ∪ s2k (ξ)) =|ξ|<ε[(Prw |Tξ )−1 (Sk (ξ)),|ξ|<εè íà ýòîì ïåðåñå÷åíèè êîìïëåêñíûå êîîðäèíàòû óãîëϕk mod 2π è ϕk+1 mod 2πñâÿçàíû äðóã ñ äðóãîì ôîðìóëàìè:00Ik+1(ξ) ϕk+1 |s2k−1 (ξ) + πJk+1(ξ) = Ik0 (ξ) (ϕk |s2k−1 (ξ) − π),00Ik+1(ξ) ϕk+1 |s2k (ξ) − πJk+1(ξ) = Ik0 (ξ) (ϕk |s2k (ξ) − π);å) óðàâíåíèÿ Ãàìèëüòîíà â êîîðäèíàòàõk = 1, .

. . , n + 1,(Ik , ϕk mod 2π)íà ðó÷êåïðèíèìàþò âèä:I˙k = 0,4) àíòèêàíîíè÷åñêàÿ èíâîëþöèÿùàÿ Ãàìèëüòîíèàíϕ̇k =df (Ik );dIkC2 → C2 , (z, w) 7→ (−z, w),ñîõðàíÿþ-f , ïåðåâîäèò êàæäóþ ÷åòûðåõìåðíóþ ε-ðó÷êó Gε,káÿ, è îãðàíè÷åíèå ýòîé èíâîëþöèè íà ýòó ðó÷êó â êîîðäèíàòàõèìååò âèäGε,k ,â ñå-(Ik , ϕk mod 2π)(Ik , ϕk mod 2π) 7→ (Ik , −ϕk mod 2π), 1 ≤ k ≤ n + 1.Äîêàçàòåëüñòâî.Ïóíêò 1) ñëåäóåò èç ñëåäñòâèÿ 4.1.3. Ïóíêò 2) ñëåäóåò èç÷àñòíîãî ñëó÷àÿ ëåììû 3.3.1: êîãäà íà ñëîå T0 íåò îñîáûõ òî÷åê. Äîêàæåìïóíêò 3).155Ðèñ.

4.14: Ïàðàëëåëîãðàìì Ðèñ.íèê∂W1,I1 ⊂ Cϕ1 ïðè n > 14.15:Øåñòèóãîëü-⊂∂Wk,IkCϕkïðè1<k <n+1Ðèñ. 4.16: Ïàðàëëåëîãðàìì∂Wn+1,In+1 ⊂ Cϕn+1Øàã 1. Îïðåäåëèì ôóíêöèè Ik = Ik (ξ), Jk = Jk (ξ) ôîðìóëàìè ïóíêòà ã).Äîêàæåì ïóíêò à), ò.å. ãîëîìîðôíîñòü ôóíêöèé Ik (ξ), Jk (ξ). Ïî ïîñòðîåíèþçíà÷åíèå ïåðåìåííîé äåéñòâèå Ik çàâèñèò òîëüêî îò ξ . Äàëåå,dIk (ξ)1= dξπaZ2k (ξ)ppd( ξ − P2n+2 (w))dw + a02k (ξ) ξ − P2n+2 (a2k (ξ))dξa2k−1 (ξ)aZ2k (ξ)p1− a02k−1 (ξ) ξ − P2n+2 (a2k−1 (ξ)) =2πdwpa2k−1 (ξ)ξ − P2n+2 (w).Îòñþäà ñëåäóåò, ÷òî ñóùåñòâóåò ε > 0, òàêîå ÷òî ïðè |ξ| < ε ïðîèçâîäíàÿdIk (ξ)/dξ ñóùåñòâóåò, ïîýòîìó Ik = Ik (ξ) ÿâëÿåòñÿ ãîëîìîðôíîé ôóíêöèåé.2Áîëåå òîãî, ïîñêîëüêó dIk (0)/dξ 6= 0, òî ìîæíî ñ÷èòàòü, ÷òî Ik |D0,εÿâëÿåò-2e ε,k :=ñÿ äèôôåîìîðôèçìîì îòêðûòîãî êðóãà D0,εðàäèóñà ε íà îáëàñòü D2Ik (D0,ε).Àíàëîãè÷íîJk0 (ξ)=12πa2k−1R (ξ)√a2k−2 (ξ)dw.ξ−P2n+2 (w)Ïóíêò à) äîêàçàí.Øàã 2.

Äëÿ ëþáîé êóñî÷íî ãëàäêîé îðèåíòèðîâàííîé êðèâîé αξ íà ñëîåTξ îáîçíà÷èìZhαξ i :=∆ξ ,hαξ ik :=αξ156hαξ i,Ik0 (ξ)2ξ ∈ D0,ε,(4.2.1)ãäå ∆ξ ãîëîìîðôíàÿ 1-ôîðìà íà ñëîå Tξ , ñì. îïðåäåëåíèå 3.1.11(À). Òàêêàê ôîðìà ∆ξ çàìêíóòà, òî åå èíòåãðàë hαξ i ïî ëþáîìó îðèåíòèðîâàííîìóöèêëó αξ ⊂ Tξ íå ìåíÿåòñÿ ïðè ëþáûõ ãîìîòîïèÿõ öèêëà â ñëîå. Ðàññìîòðèìíåïðåðûâíûå ïî ξ ñåìåéñòâà îðèåíòèðîâàííûõ öèêëîâαξ,k := (Prw |Tξ )−1 (γa2k−1 (ξ),a2k (ξ) ) ⊂ Tξ ∩ Gε,k ,α̂ξ,k := (Prw |Tξ )−1 (γa2k−2 (ξ),a2k−1 (ξ) ) ⊂ Tξñ òàêîé îðèåíòàöèåé, ÷òî α0,k è α̂0,k ÿâëÿþòñÿ èíòåãðàëüíûìè òðàåêòîðèÿìèïîëåé sgrad C f è i sgrad C f ñîîòâåòñòâåííî, k = 1, . . . , n + 1.