Ergodic Theory - Kozlov - Lectures - MAIN (1114445), страница 4
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. . + pn f (xn )→ f (x)dx, n → ∞.p1 + . . . + pn0×àñòü VIIIËåêöèÿ 3Ðàññìîòðèì óïðàæíåíèå ñ ïðåäûäóùåé ëåêöèè.Óïðàæíåíèå 21 (Ïðèìåð Ïóàíêàðå).I(t) = limt→∞∞XAn sinn=131Äîêàçàòü, ÷òî èíòåãðàët→ ∞,ΛnΛ+1<2Ðàññìîòðèìãäå1<A < Λ.π n−1πΛ< t 6 Λn , n = 1, 2, . . .22n+kíà Λ.0<Ïîäåëèì íåðàâåíñòâà0<tπππ< n+k 6<k+1k2ΛΛ2Λ2ÐàçîáüåìèñõîäíóþñóììóíàäâåI(t)=∞XAj sinj=1n−1∞Xt X n−kttj=Asin+AsinΛjΛj k=0Λkj=1è ðàññìîòðèì ñíà÷àëà ïåðâóþ ïîäñóììón−1Xn−1Aj sinj=1XtAn − Aj6A=.ΛjA−1j=1Ðàññìîòðèì òåïåðü âòîðóþ ñóììó. Äëÿ åå ïðåîáðàçîâàíèÿ èñïîëüçóåìíåðàâåíñòâîsin x >2xπèç êîòîðîãî âèäíî, ÷òîsinÒîãäà∞XtΛn+kAn−k sint>2 t.π Λn+k∞X>An−kΛn+kk=0∞kAn X AAn 1>>Λ k=0 ΛΛ 1−k=0òî åñòü âòîðàÿ ñóììà óâåëè÷èâàåòñÿ ïðèI(t) >AΛ2 t>π Λn+k=An,Λ−An → ∞.Òàêèì îáðàçîì,An (2A − Λ − 1)→ ∞, n → ∞, t → ∞.(Λ − A)(A − 1)Ðàññìîòðèì òåïåðüf (x1 , x2 ) =∞ XAn=1ãäåΛcos 2π(un x1 + vn x2 ),√√( 2 − 1)n = un + vn 2.32Ïîëîæèìx2 = 0, x1 = x.Òîãäàf (x1 , x2 ) = g(x),òî åñòü âìåñòî ôóíêöèèîò äâóõ ïåðåìåííûõ ìû ðàññìîòðèì ôóíêöèþ îò îäíîé ïåðåìåííîég(x) =∞ nXAΛn=1ÏóñòüΛ=√2+1=cos 2πun x.√1 .2−1√1un + vn 2 = nΛÒîãäà√√(− 2 − 1)n = un − 2vn = (−1)n Λn ,òî åñòü(−1)nun =2Ðàññìîòðèì ðÿäG(x) =Óïðàæíåíèå 22.Ôóíêöèÿ1Λ + nΛn∞ nXAn=1Äîêàçàòü, ÷òîΛ= (−1)n Λn .cos πΛn x.g(x) − G(x) ∈ C 1 .G(x) íå èìååò ïðîèçâîäíîé íè â îäíîé òî÷êå ïðè 1 < A < Λ.ż íàçûâàþò ôóíêöèåé Âåéåðøòðàññà.2Ïóñòü òåïåðü ôóíêöèÿ f (x1 , x2 ) ∈ C ,00f (x1 , x2 ) è f (x1 , x2 ) 6= 0.
ÐàññìîòðèìI(τ, x01 , x02 )Zτ=hf i = 0, f (x1 (mod 1), x2 (mod 1)) =f (ω1 t + x01 , ω2 t + x02 )dt,ω1∈ R \ Q.ω20 òàêèõ ïðåäïîëîæåíèÿõI(t) áåñêîíå÷íî ìíîãî ðàç ñêîëü óãîäíî áëèçêîïîäõîäèò ê íóëþ.9.7Äèôôåðåíöèàëüíûå óðàâíåíèÿ íà äâóìåðíîì òîðå ñ èíòåãðàëüíûì èíâàðèàíòîì. íàøåì ñëó÷àå èíòåãðàëüíûì èíâàðèàíòîì áóäåò èíâàðèàíòíàÿ ìåðà.Ïóñòüρ(x, y) > 0 ïëîòíîñòü èíâàðèàíòíîé ìåðû.339.8ÝéëåðÓðàâíåíèå Ëèóâèëëÿẋ = f (x, y)ẏ = g(x, y)div(ρv) = 0â ýòîì ñëó÷àå èìååò âèä∂(ρf ) ∂(ρg)+= 0.∂x∂yÐàññìîòðèì1-ôîðìó−ρgdx + ρf dy = dH, H = H(x, y).Ïîÿñíåíèå:dH ïîëíûé äèôôåðåíöèàë, òàê êàê∂H∂H= −ρg,= ρf,∂x∂yòî åñòüÒåì ñàìûì∂2H∂2H=.∂x∂y∂y∂xHÝéëåð íàçâàë9.9 ïåðâûé èíòåãðàë èñõîäíîé ñèñòåìû óðàâíåíèé.ρèíòåãðèðóþùèì ìíîæèòåëåì.ßêîáè ẋ1 = f1 (x1 , .
. . , xn ),...ẋn = fn (x1 , . . . , xn ).Ïóñòü äëÿ ñèñòåìû âûïîëíåíû ñëåäóþùèå óñëîâèÿ:1. Èìåþòñÿn−2íåçàâèñèìûõ ïåðâûõ èíòåãðàëàG1 (x), . . . , Gn−2 (x), x = (x1 , . . . , xn ).Èç ôóíêöèîíàëüíîé íåçàâèñèìîñòèrankGi , i = 1, nñëåäóåò, ÷òî∂(G1 , . . . , Gn )= n − 2.∂(x1 , . .
. , xn )2. Èìååì èíòåãðàëüíûé èíâàðèàíò ñ ïëîòíîñòüþdiv(ρv) =nX∂(ρfj )j=134∂vj= 0.ρ(x1 , . . . , xn ) > 0:Òîãäà ñèñòåìà èíòåãðèðóåìà â êâàäðàòóðàõ.Ïðèìåð (Óðàâíåíèÿ Ýéëåðà-Ïóàññîíà).I ω̇ + ω × Iω = pr × γγ̇ + ω × γ = 0.Ïåðâûå èíòåãðàëû ñèñòåìû:H=1(Iω, ω) + p(r, γ),2(Iω, γ) = c1 ,(γ, γ) = c2 .Òàê êàê óñëîâèå 2 âûïîëíåíî,ρ ≡ 1,òî äî èíòåãðèðóåìîñòè â êâàäðàòó-ðàõ íå õâàòàåò îäíîãî ïåðâîãî èíòåãðàëà.×àñòü IXÏîñëåäíÿÿ ëåêöèÿM = [0, 1]T x = {2x}.Ðàññìîòðèì ìíîæåñòâî÷òî åñëèx ∈ [0, 1],òîè ïðåîáðàçîâàíèåÏóñòü êðîìå òîãî, çàäàíà ìåðà ËåáåãàïðåîáðàçîâàíèÿAòàêîå,èíâàðèàíòíàÿ îòíîñèòåëüíîT.Îïðåäåëåíèå 5.âàíèÿ. Ïóñòüµ,T :M →MÑîõðàíåíèå ìåðû íå âçàèìíîîäíîçíà÷íîãî ïðåîáðàçî- èçìåðèìîå ìíîæåñòâî.
ÒîãäàµT −1 A = µA,ãäåT −1 A ïîëíûé ïðîîáðàçÓïðàæíåíèå 23.A.Äîêàçàòü òåîðåìû Ïóàíêàðå î âîçâðàùåíèè äëÿ íåâçà-èìíîîäíîçíà÷íûõ îòîáðàæåíèé ñîõðàíÿþùèõ ìåðó.Ðàññìîòðèì ôóíêöèþf : M → R, f ∈ R[0, 1].Ðàññìîòðèì åå ñðåäíååâðåìåííîåf (x) + f (T x) + . . . + f (T n−1 x)= f (x),n→∞nlimÏî ýðãîäè÷åñêîé òåîðåìå Áèðêãîôà-Õèí÷èíàf (x) ∈ R[0, 1], f (T x) = f (x).35Ëþáîåx ∈ [0, 1]ìîæíî ïðåäñòàâèòü â âèäå äâîè÷íîé äðîáèx=ãäån (x)1 (x) 2 (x) 3 (x)+ 2 + 3 + ... + n + ...,2222∀i ∈ Ni (x)Ãðàôèêîì ôóíêöèèÏðèìåíèì êxi (x)Òî åñòü,Tÿâëÿåòñÿ ñòóïåíüêè.ïðåîáðàçîâàíèåTx =01T.2 (x) 3 (x)n (x)+ 2 + . . .
+ n−1 + . . . .222ñäâèãàåò ïîñëåäîâàòåëüíîñòü{n (x)}âïðàâî íà îäèí ýëåìåíò.T : {1 (x), 2 (x), 3 (x), . . .} → {2 (x), 3 (x), 4 (x), . . .}.ÏðåîáðàçîâàíèåTíàçûâàþò ñäâèãîì Áåðíóëëè. Âåëè÷èíàf (x) + f (T x) + . . . + f (T n−1 x)nõàðàêòåðèçóåò ñðåäíåå êîëè÷åñòâî åäèíèö â ðàçëîæåíèè ÷èñëà è ñòðå1ìèòñÿ ïðè n → ∞ (ïîêà íå ÿñíî â êàêîì ñìûñëå) ê .2Ðàññìîòðèì âìåñòî ôóíêöèé i (x) ôóíêöèè ri (x) = 1 − 2i (x), i > 1.Ôóíêöèèri (x)íàçûâþòñÿ ôóíêöèÿìè Ðàäåìàõåðà.Ôóíêöèè Ðàäåìàõåðà îáðàçóþò îðòîíîðìèðîâàííóþ ñèñòåìó ôóíêöèé, âòîì ñìûñëå, ÷òîZ1rk1 (x)rk2 (x) . . .
rks (x)dx = 0,0ïðèk 1 < k 2 < . . . < ks .Ðàññìîòðèì âåëè÷èíór1 (x) + r2 (x) + . . . + rn (x).nÏðèn→∞îíà ñòðåìèòñÿ (îïÿòü æå ïîêà íå ÿñíî â êàêîì ñìûñëå) êÄîêàæåì, ÷òîr1 (x) + r2 (x) + . . . + rn (x)→ 0, n → ∞n360.â ñìûñëå ñðåäíå-êâàäðàòè÷åñêîé ñõîäèìîñòè.Ðàññìîòðèì1Z0Z=01(r1 (x) + r2 (x) + . . .
+ rn (x))2dx =n2r12 (x) + r22 (x) + . . . + rn2 (x) + 2(r1 (x)r2 (x) + . . . + rn−1 (x)rn (x))dx =n2Z 111ndx = → 0, n → ∞.= 2n 0nÒåîðåìà 23. Âñïîìîãàòåëüíàÿòåîðåìà, âûòåêàþùàÿ èç òåîðåìû ËåPRâè. Ïóñòü fn (x) > 0 èÒîãäà ðÿä∞P∞n=110fn (x)dx < ∞.fn (x) < ∞ äëÿ ïî÷òè âñåõ x ∈ [0, 1].n=1Òåîðåìà 24. Ñëåäñòâèå. Äëÿ ïî÷òè âñåõ xfn (x) → 0, n → ∞.Ïîýòîìó èç òîãî, ÷òîäëÿ ïî÷òè âñåõr1 (x) + r2 (x) + .
. . + rn (x)nx ∈ [0, 1],4→ 0, n → ∞ñëåäóåò ÷òîr1 (x) + r2 (x) + . . . + rn (x)→ 0, n → ∞.nÓïðàæíåíèå 24.Êàê ìîæíî óñèëèòü òåîðåìó Áèðêãîôà-Õèí÷èíà èóñèëåííûé çàêîí áîëüøèõ ÷èñåë? êà÷åñòâå ïîäñêàçêè: ìîæíî èñïîëüçîâàòü ñõîäèìîñòü íå ïî ×åçàðå,à ïî Ðèññó.Ðàññìîòðèì ìåòîä Ðèññà, êîòîðûé ñëàáåå ìåòîäà ×åçàðå. Ñóììèðóÿ ýòèììåòîäîì ïîëó÷èì ðåøåíèå çàäà÷è.Òàêèì îáðàçîì, çàäà÷ó ìîæíî ñôîðìóëèðîâàòü ñëåäóþùèì îáðàçîì:ñóùåñòâóåò ëè ìåòîä Ðèññà(R, pn ),êîòîðûé ñëàáåå ìåòîäà ×åçàðå è âf (T n x) → f (R, pn )óñëîâèÿõ ïðèìåíèìîñòè òåîðåìû Áèðêãîôà-Õèí÷èíàäëÿ ïî÷òè âñåõÏðèìåð.nαex. êà÷åñòâå âåñîâûõ êîýôôèöèåíòîâ ìîæíî ðàññìîòðåòü.37pn =Óïðàæíåíèå 25.Èçâåñòíî, ÷òîp1 r1 (x) + p2 r2 (x) + . .
. + pn rn (x)→0p1 + p2 + . . . + pnx.Óêàçàòü òàêèå pn , ÷òî ìåòîä áóäåò êàê ìîæíî áîëåå ñëàáûì.nÅñëè pn = e lnγ n∀γ > 1, òî òàêèå êîýôôèöèåíòû ïîäõîäÿò. ×òînñëó÷àå, êîãäà pn = e ln n ïîêà íåèçâåñòíî.äëÿ ïî÷òè âñåõ38áóäåò â.
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