Đánh giá các phép biến hình á bảo giác lên hình tròn bị cắt theo các cung tròn đồng tâm

gsa a,b,a::/:O. Chungminh:Xemchungminhtrong[4,tr.2I2],ho~c[11,tr.6]- Tlib6d€ naytasuyrabah~quara'tquailtn;mgd6ivoiPBHBGmi€nnhilien: H~ qua3.1(djnhnghiamodunmi@nbjlien) I I I I I Gillsami~nhilienD quacacPBHBGj va1; bienZenhaihlnhvanhkhan H:r<I~<R, HI :rI <IWII<RI. 11 fJH.J<H.TlrNt-HEN TH-U \tIEN 001063 Khi d6 R _RJ---. (3.2)r rJ Tys5naygQiZam6duncilami~n hilienD vadllf/CkYhi~uZamod(D). Chungminh. X6t haiPBHBG 101;-Jmi€n HJ leD H va 1;01-1mi€n H leD HJ' Tli b6 d€ 3.I,tasuYfa (~r~(~'r va (~rs(~l Tlid"R Ro. -= !...- r rJ H~ qua3.2(Hnbbiltbie'ncuamodunmi~nnhilien) Ne'umi~nnhilienA c6cacthankphdnbienkh6ngthaaih6athanhmiltdiim dllf/Cbie'nbaagiacdandi~pZenmi~nhilienB thz mod(A) =mod(B). (3.2.a) Chungminh GQi1 laPBHBGddndi~pmi€nA leDmi€nB.X6thaiPBHBGg mi€nA leDhlnhvanhkhanA' : lj <Isl<RJ va h mi€n Bien hlnhvanhkhan B' : rz<ItI <Rz.Tli h~qua3.1,tasuyfa mod(A) =RJ va mod(B) =Rz. lj rz GQi cp=hot thl cpla PBHBG ddndi~pmi€n A' leDmi€n B'. Khi d6dob6d€ 3.1,tac6:mod(A)= RI =Rz =mod(B) - lj lj H~ qua3.3(Hnbddndi~ucuamodunmi~nnbilien) 12 GiasacacmMnnhilienD vaDI vfJimoduntlldngangR va !!J...,cotinh r rJ chatD c DJ va D ngdncachhaithanhpht1nbiencuaDJ' Khido R~RJ. r 1j (3.3) DiingthucxayrakhivachikhiD ==DI' Chungminh Xet f laPBHBGdondi~pDI len hinh vanhkhan HI : 1j<Iwl<Rl' Khi do mi€n nhi lien D (c D1)quaphepbienhinh f setrdthanhmi€n nhi lien H voi bientrongCI (CJ baaquanhho~ctrimgduongtroll Iwl=1j)va bienngoaiCz (duongtroll Iwl=RJ baaquanhho~ctriingCz). GQiS ladi~ntich(trong)cuat~pmddoCzbaabQcvas Iadi~ntich(ngoai) cuat~pdongdo CJ baabQc. T ,R I' Ad ? H ' h b.:!d); 3 1aco - amo uncua vat eo 0 e . r ~~(~)z. M~tkhac,tacoba'td£ngthuchi~nnhien s ~trR},s ~7rrJz. Dodo (~:r 2 ~2(:)' Tit d6 c6 (3.3) vdi d£ng thlte xay fa (~,)'=~=(~r Theob6d€ 3.1,CJ vaCzphaila duongtroll Iwl=1jvaIwi=RI' tucla D ==DJ8 13 ? , Bo de 3.2 (mdrQngb6'd~3.1choPBHKABG bdiThao) Gidsa w=j(z) Za mQtPBHKABG hinhvanhkhan0 <r <Izl<R <00 Zen mQtmi~nnhilienD khongchaadiim oc>,vdibientrongC]vabienngoaiCzsaD choIzi=R tudnglingvdi Cz. Gri S Zadi~nrich trongcila ti7pmd do Cz baa brc, s Zadi~nrichngoai cila t(ipdongdo C]baabrc. Khi do,taco s~(~)*s. (3.4) Ddngthacxdyra ~ j(z) =a Izli--lz +b vdicachangso'a("*0)vab. Chungminh: (Thao[14],tr.521) R5 rangtantl,limQtPBHBG t=g(w)bie'nmi€n D leumQthlnhvanhkhan 1J< ItI <R] saocho Cz tu'dngungvdi du'ongtroll ItI =Rl' Ap d1;mgbe)d€ 3.1 cho phepbie'nhlnhngu'<;lcg-], ta co s~(~}, (3.5) trongdod~ngthucKayra~ w=g-](t)=a/+bl' vdicaeh~ngs6 a]("*0) va b]. M~t khac phepbie'nhlnh h<;lpt =gf(z) Ia mQtPBHKABG hlnhvanhkhan r <Izi<R leuhlnhv~lllhkhan1J<ItI<R]. £)~ - z - t hK - -(-) 1 gf( -) I' A PBHKABG b' Kl,lt z =-, t =-, ta t ay t =t z =- rz a mQt len r 1J 1J 1<Iz/<R leu 1<ItI<R] , nentheo(2.2)va(2.6)taco: r r] 1 ~? ( R ) K, r1 r 14 1trongdod~ngthucxayrat(z)=elzlTlz,lei=1, hay 1 t =(gofXz)=e-t-lzIK-1z ,lei=1. rK Ke'thQpba"td~ngthucvuaneuvoi (3.5)taduQc(3.4)voi ke'tlu~nv€ kha ~ ? d:t h~ d~ 'ica1 b b .nangxayra angt lic trong 0 a =~, = I rK 3.2CaehamphI}T{p,r,s)vaR{p,r,s) Cachams6tht!c 1=T(p,r,s) r =R(P,I,s) (O~s<r<I), (0~s <t <1), pEN, duQCdinhnghlasaochohlnhvanhkhanr <Izi<1tudngdudngbaagiac voihlnhvanhkhans <Iwl<1bi dt dQcp do;;tn(hlnh3.1) F; ~ {ws ,; 11<1,; t,argw ~ j 2;} (j=O,l,...,p-I). z G) 1 ~ 1 Hlnh3.1:PBHBG r <Izi<lIen s <I~<1bidt dQcp(=2) do~n. Dotinhddndi~ucuamodunmiennhilien(xemh~qua3.3),tacocactinhcha"t saucuahamT(p,r, s) va R(P, I, s): r <T(p,r,s)<I (0~s <r <I), (3.6) 15 Nhocaec6ngthuccua[10,tr.295],[13,tr.l0l-l04], tatlmdu<jcbi~uthuccua R(p,t,s) nhusau: { -ltK'(tP) }R(p,t,O)=exp 2pK(tP) (0<t <1,pEN), (3.14) voi I K(k)= J ~ dx , K'(k) ~ K(.JI- k'),(1- x2)(I- k2x2)0 vavoi 0 <s <t <1,pEN, { -ltK'(U) }R(p,t,s) =exp 2pK(u) , (3.15) voi U=1+h - .Jh(2+h), trongdo: h =(1- k)(l- ak), k =4sPIT [ 1+<pj ] 4, k(1+a) j=l 1+S4PJ-2p a =sn(b+i 2~bIn~,k). b =K(k) , ddaysn(z,k) chisinelipticvoithams6 k. Vi~ctinhtoaDK(tP) va K'(tP) ([13,tr.1l7],[19,tr.177]),cho 1 R(p,t,o)~ 4 P t khit~O (3.16) 16 T(p,r, s()>T(p,r,S2) (0 SI <S2<r <1), (3.7) T(p,r(,s) <T(p,r2's) (0 s <lj <r2<1), (3.8) T(p,r,s) <T(1,r,s) (0 s <r <1,P 2), (3.9) s <R(p,t,s) <t (0 s <t <1), (3.10) R(p,tl,s) <R(p,t2's) (0 S <tl <t2<1), (3.11) R(p,t,sl)<R(P,t,S2) (0 SI<S2<t <1), (3.12) R(p,t,s) >R(1,t,s) (0 s <t <l,p 2), (3.13) va 2tr l-R(p,t,O) ~ 8 2p In p(l- t) khi t ~ 1. (3.17) Nho [13,tr.l02-105],taclingchifabi€u thuccuaT(p,r, s) nhu'sau: 1 [ 4 T(p,r,O)=4PrfI 1+ r4pj ] p j=l 1+r4pj-2p (O<r<I,pEN), (3.18) { a } -tri dx T rs =sex (p, , ) P 2pK(k)1~(l-x')(l-k'x') , 0 <s <r <1,pEN , v~iK(k) nhu'tfen, (3.19) 1- m k(1- h)2 00 [ 1+r4pj ] 4 a =- m= h =4rP k +m' 2h(1-k) , D 1+r4pj-2p . Tli bi€u thuccuaT(p,r,O)tad~dangtha"yding 1 T(p,r,O)<4Pr (0<r <I,pEN) . (3.20) Vi v~y,nho(3.6)va(3.7)c6 1 r <T(p,r,s) <4Pr. (3.21) Tli d6surfa Vi v~ynho(3.10)va(3.12)c6 4 Pt <R(p,t,s) <t. (3.24) Tli d6surfa 17 limT(p,r,s) =r (0 s <r <1). (3.22) p-+oo Mt khactU(3.20)c6 1-- R(p,t,O)>4 p t (0<t <I, pEN). (3.23) lim R(p, I,s) = I p-+oo (0 ::;s < I < 1). (3.25) Hannua,tanh~nduQctit(3.16)va(3.17) T(p,r,O)~ 4P r (3.26) khi r ~ 0, va { 2 } 8 -1f 1- T(p,r,0)~ p exp 2p(1- r) (3.27) khi r ~ 1. 3.3 Caeb6d@khae B6 d~3.3(mdrQngmQtba'tdiingthat GrotzschbdiThao) GidsaD la hlnhvanhkhan R <Izi<1 Irit pn (p E N,n E N u {o})nhal cdl nlimIren caeduiJngIron donglam 0 saDchoD Irung vdi chfnhno blJi phepquay ,27r 1- Z = e P z, f la PBHKABG miin D len miin E nlim trong m(it phdng phac 0 <Iwl<1 saDcho duiJngtron Izi =R tUdngang bien trongC] gidi h(;mblJi mQt t(ipdongchaa gdc tQadQ,duilngtron Izi =1 tUdngang bienngoai C2.Hdn nila, ,27r 1- gid saE trungvdi chfnhnoblJi phepquayW = e P w. Khi do M, ,; T(p,R~,m,) (3.28) vdi m1=min{lwllw E C1} M1 =max {IwlIw E C1} (~0), vaT(p,r,s) lahamph{/-du(lcdinhnghiaIrongphdn3.2. 18 IDdng thuGxdy ra ~ fez) =fo(z)=ah(t), lal=1,t =blzlK-1z, Ibl=1, h za I I PBHBG hinhvimhkhanRK <ItI<1 Zenmi~nnhj lienP saDcho ItI=RK tUdng ringvlii c,~ {wiIWI= m,}u{wm, ,; 1»1'; M"argw ~ j 2; }, j =O,I,...,p-1 va ItI=1 tUdngringvlii Cz={wJlwI=I}. Chungminh:Bfftd~ngthuc(3.28)vdiK =1vaC2 la duongtroll Iwl=1lamQt d.;mgkhaccuabfftd~ngthucGrotzsch[6],tr. 372khongtrinhbaychungminh. Ngo Thu Luong[11],tr.I8 dachungminhd mi bfftd~ngthuc(3.28)chotru'ong hQpda lieUcua Grotzschva trlnhbayh;li[11,tr.33]stfma rQngcuaThao[14], tr.63thanhb6d€ 3.3. B6d~3.4 Gid saD ZahinhvanhkhanQ <Izl<R trit pn (p E N,n E N u {v})nhat cdtndmtrencacduongtrimdangtam0 saDchoD trungvlii chinhnobaiphep .2" quayZ =e'f;z, f ZaPBHKABGmi~nD Zenmi~nE ndmtrongmijtphdngphac 0 <Iwl<00saDcho duilngtrim Izl=Q tUClngringbien trongC] gilii h~nmQtttJ-p dongchaagoctQadQ,duilngtrim Izl=R tUdngringbienngoaiC2.Hdnnaa,gid ,z"1- sit'E trungwJi chinhnobaiphepquayW =e P w.Khido > mIl ] mz - Q K ml T[P.(R) 'M, (3.29) vlii mj=min{lwllwECj}, j =12, , 19 M2 =max{IWIIwE C2} va T(p,r,s) Zahamph,!dl1(Jcdinhnghfatrongphdn3.2. 1 Dllng thac xdy ra <;:::)fez) =fo(z)=aH(t), lal=1,t =blz/K-lz, Ibl=1, H Za 1 1 1 PBHBG hlnh vanh khan QK < It1< RK Zenmiin nhi lien P' saD cho It1= RK tl1dng ilng v{ti c, ~{wll1<1~M,}u {w In, "11<1"M"argw ~j 2; }. I j=O,l,...,p-l va Itl=QK tl1angilngv{ti C] ={wllw1=m1}. Chung minh: XemchUngminhtrong[20,tr.16],ho~c[11,tr.35]- B6 d~3.5("D~oham"cuahamngu'fjcho PBHKABG) V{ticac kY hi~udl1avao11m,!c2.3,gid sa w=fez) ZaPBHKABG cua miin chaa z = 0 v{tif(O) =0 va m'(O,f) > O.Dt;itg =1-1, ta co: 1 m'(O,f) =M* (O,g)-K, (3.30) 1 M' (0,f) =m* (0,gfK . (3.31) Chungminh: LffyR >0 diibe,d~tCR={z:Izi=R}vac~=/(CR),r6rang t6nt~iWI E C~va Zl E CR saocho m(R,f) =Iwl! =!f(ZI)!=r (r>O). D~tLr ={w:Iwl=r}vaL; =g(Lr) Vi L; n~mtrongIzl:::;R, taco M(r,g)=lg(WIX=lzJ/=R. Dodo,tum'(O,/»O,taco 20 , ' (0 f) _ I' m(R,f) _ I' r _ I ' [ M(r,g) ] -K - M * (0 ) -~ m, -1m , -1m ,-1m K - ,gK,R~O - r~O M( ) - r~O r RK r,g K Tu'dngtv, U(y R >0 diibe,d~tCR={z: Izl=R} va c~=f(CR),rarang t6ntC;liWzE C~va ZzE CR saocho M(R,f)=lwzl=lf(zz~=r (r>O), B~tLr ={w: Iwl=r} va L; =g (Lr)' Vi Izi~R niimtrongL;, taco m(r,g)=Ig(wz)1=Izzi=R, Do do,taco I M '(Of) _ I' M(R,f) _ 1' r _ I ' [ m(r,g) ] -K - * (0 )-~., - 1m , - 1m , - 1m K - m ,g KR~O - r~O ( ) - r~O r RK m r,g K H~ qua3.4 Cho K=l, ta co m'(O,f)=If'(O~va M*(o,g)=lg'(o~,phl1angtrinh (3.30)triJ thanh1f'(0~=Ig'(ofl, ._.