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DISTRIBUTION OF LENGTH SPECTRUM OF CIRCLES ON A
also call the image LSpec(M) = L(Cir(M)) ? R in the real line the length [5] and [6]) On a complex projective space CPn(c) of holo-.
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The Second Stage of the Stress-Strain State of Reinforced Concrete
According to the definition of tangential stresses in any circle. 1? is found by the formula: 1. M. M t t r r j j. I. I bcir.
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T.Adachi
NagoyaMath.J.
Vol.153(1999),119{140
DISTRIBUTIONOFLENGTHSPECTRUMOFCIRCLES
ONACOMPLEXHYPERBOLICSPACE
TOSHIAKIADACHI1
x1.Introduction denedKahlermagneticKahlermagnetic
spaceareconjugatetothegeodesic ow,henceareofAnosovtype.Thisre- theactionoftheisometrygroup.Asmoothcurve
:R!MonacompleteRiemannianmanifoldM r trt_ (t)=2_ (t);ReceivedFebruary20,1997.
RevisedJanuary7,1998.
119120T.ADACHI
withrespecttothe withthecomplexstructureJ:Foracircle wedeneitscomplextorsion byh_ ;Jrt_ i=krt_ k.Thisdoesnotdependontandsatisesjj1.ItKahlergeometryofthebasemanifold.
Wecallacircle
closedifthereexistsaconstantTwith (T)= (0);_ (T)=_ (0);rt_ (T)=rt_ (0): (t+T)= (t)forevery periodof andisdenotedbylength( ).Weputlength( )=1foran opencircle noncompacttype.Herewecalltwocircles 1and2arecongruentifthere
existanisometry'andaconstantt0with2(t)='
1(t+t0)forevery
lengtharecongruenteachother? samelength?L:Cir(M)!R[f1gdenedbyL([
])=length( ).Sometimeswe spectrumofcirclesonM.Onanon- atcomplexspaceform,twocirclesare anon-LENGTHSPECTRUMOFCIRCLES121
andthatofgivencomplextorsion. ectonthestructure curvature>p cisbounded.Insection2,wecomparebyamapof (>p sectionalcurvaturec, turebytheirlengthspectrum. withlengthisnotboundedwithrespectto. ects wehavethefollowing: geodesiccurvature, ofgeodesiccurvatureifandonlyif5p c 4.122T.ADACHI
Theunboundedpropertyalsore
ectsonthestructureofthefulllength showinsection4thefollowingresult.Theorem3.
cideswiththehalfline(0;1). eachpoint2R,andismonotoneincreasing.3)Therearenocircles
onCHn(c)ofcomplextorsion(02isclosedwith
primeperiod4 p42c.4)When>()and6=1wedenotebya;,b;,d;(a; d ;)thesolutionsforthecubicequation c 3(42c)+2p
c=0: Acircle
d ;=a;isrational.Itsprimeperiodis length( )=4 pcL:C:M:1b;a;;1d;a; geodesiccurvaturehenceisthesetf[ ;]j01g.ByFact1,wend p function:[0;1)!Rby ()=8 :0;if03p3c;ifp
c 2pc, 3(42c)+2p
c=0:Acircle
d ;=a;isrational.Itsprimeperiodis length( )=4 pcL:C:M:1b;a;;1d;a; geodesiccurvaturehenceisthesetf[ ;]j01g.ByFact1,wend p function:[0;1)!Rby ()=8 :0;if03p3c;ifp
1;if>p
c.124T.ADACHI
<()isbounded.WeputM=f[ ;]j0<()g,anddenea mapofnormalization :M!Mp c=Cirpc(CHn(c))nf[ pc;1]g by ;])=h p c;3p3c (42c)3=2i L([ ;])=CL([ ;]2M withC=q 3c42c.Whenp
c2< Proof.Considerthecubicequationc3(42c)+2p
c=0.By putting=Cwenditisequivalenttoc33c+2p cC3=0. Since0
pcC3=3p3c(42c)3=2<1,thismeansthat a p c;=Ca;;bpc;=Cb;;dpc;=Cd;; with=3p (M[f[ isessentiallyequivalenttoLp cifpc 2< c 2<1,2 2;1([ 1;])=8
121([
1;]);0<(1);
h 2;11(2)
1(1)(1)i
;(1)1; wegetthefollowing: LENGTHSPECTRUMOFCIRCLES125
1)thelengthspectrumL(p
c 2 2)when>p
(402+c)3=2(42+c)3. FollowingLemma1weset
F =8 :n ;0]j>p c 2o ;if=0, n ;]j3p 3c(42c)3=2=;0<<1o
;if0<<1, n ;1]j>p co ;if=1. BCir(CHn(c))=f[
;]j>();01g =0==1 =p c 2=p c=K F ?Hn(c)) duality :Cirp c(CHn(c))nf[ pc;1]g!Cirp2c 4(CPn(c))nf[
p 2c 4;1]g by([ p c;])=[ p2c 4;].Hereweusethesamenotation[
;]forrepre- andcomplextorsion. 126T.ADACHI
morphicsectionalcurvaturec,acircle ofgeodesiccurvaturep 2c=4and
equation 33
2+p 2 2c=0: Thecircle
isrational.Itsprimeperiodis length( )=4 pcL:C:M:1BA;1DA 2)Thecircle
form =(p;q)=q(9p2q2)(3p2+q2)3=2 itsprimeperiodis length( )=8 :4 3pcp2(3p2+q2);ifpqiseven,
2 3pcp2(3p2+q2);ifpqisodd.
lowing: Lemma2.Themapofdualitysatises
L([ p c;])=1p2L([ pc;]) forevery(0<1). theproblemsonthemultiplicityofL. LENGTHSPECTRUMOFCIRCLES127
quenceofLemma2,Facts1and2wehave LSpecp
c(CHn(c)) 4 p3c 43pcp3p2+q2
p>q,pqiseven, pandqaremutuallyprime) 2 3pcp3p2+q2
p>q,pqisodd, pandqaremutuallyprime) 4 p3c;43pcp7;43pcp13;43pcp19;43pcp21; 4 4 3pcp61;43pcp67;43pcp73;43pcp78;43pcp79;
4 geodesiccurvaturep spectrum.Forexample,4 3pcp91isthesmallestdoublelengthspectrum
forcirclesofgeodesiccurvaturep c.Thisspectrumisthelengthofcircles ofcomplextorsion(11;1)=135 91p91and(5;4)=83691p91withthegeodesic
curvaturep thelengthspectrumLp cwithlengthisnot c())=1(seefor ofgrowth,wealsoobtain lim !1(log)](L1p c())=0forevery>0: Wenowconsiderforgeneral.Whenp
c 2,sinceeverycircleof
128T.ADACHI
set.For>p c imageis p c;] 0<3p 3c() (42c)3=2 c 2LSpec(CHn(c)) 4 p42c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) andwhen>pc, LSpec(CHn(c))
=4 p42c;2p2c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) where(>1)denotesthenumberwith3p3c (42c)3=2=921(32+1)3=2.Wecan Proposition2.When0p
c 2,thelengthspectrumofcircles
2, thefollowingpropertieshold: 2)ThebottomofthesetLSpec(CHn(c))is4
p42c,whichisthelength withrespectto;limsup!1](L1())=1. order;lim!1(log)](L1())=0forevery>0. LENGTHSPECTRUMOFCIRCLES129
example,when=p lengthofcirclesofcomplextorsion(3;1)=10 7p7.Herewecompare2p2c,
thelengthofaholomorphiccircle,and4q 7 3(42c),whichcorrespondsto
3p3c(3;1).Theformeris
smallerthanthelatterifandonlyif>5p c 4.Sinceismonotoneincreas-
ingwithrespectto,and5p c 4=3,wegetthat0<(42c)3=23p3c(3;1)<1
if p c 2<<5p c torsion).Thereforewendthefollowing: Proposition3.
2)If5p
c phiccircles.Ifp c 2<<5p c 4,thesecondlengthspectrumis4q
7 3(42c),
21p21c.
c 8(3n1)(3n+
withrespectton. Proposition4.
2)Ifp c 2< 10c 20),thethirdlengthspectrumis4q
13 3(42c),
3p3c(3;1)=
10(42c)3=2
quotesdbs_dbs25.pdfusesText_31
Proof.Considerthecubicequationc3(42c)+2p
c=0.By putting=Cwenditisequivalenttoc33c+2p cC3=0.Since0
pcC3=3p3c(42c)3=2<1,thismeansthat a p c;=Ca;;bpc;=Cb;;dpc;=Cd;; with=3p (M[f[ isessentiallyequivalenttoLp cifpc2< c 2<1,2 2;1([ 1;])=8
121([
1;]);0<(1);
h 2;11(2)
1(1)(1)i
;(1)1; wegetthefollowing: LENGTHSPECTRUMOFCIRCLES125
1)thelengthspectrumL(p
c 2 2)when>p
(402+c)3=2(42+c)3. FollowingLemma1weset
F =8 :n ;0]j>p c 2o ;if=0, n ;]j3p 3c(42c)3=2=;0<<1o
;if0<<1, n ;1]j>p co ;if=1. BCir(CHn(c))=f[
;]j>();01g =0==1 =p c 2=p c=K F ?Hn(c)) duality :Cirp c(CHn(c))nf[ pc;1]g!Cirp2c 4(CPn(c))nf[
p 2c 4;1]g by([ p c;])=[ p2c 4;].Hereweusethesamenotation[
;]forrepre- andcomplextorsion. 126T.ADACHI
morphicsectionalcurvaturec,acircle ofgeodesiccurvaturep 2c=4and
equation 33
2+p 2 2c=0: Thecircle
isrational.Itsprimeperiodis length( )=4 pcL:C:M:1BA;1DA 2)Thecircle
form =(p;q)=q(9p2q2)(3p2+q2)3=2 itsprimeperiodis length( )=8 :4 3pcp2(3p2+q2);ifpqiseven,
2 3pcp2(3p2+q2);ifpqisodd.
lowing: Lemma2.Themapofdualitysatises
L([ p c;])=1p2L([ pc;]) forevery(0<1). theproblemsonthemultiplicityofL. LENGTHSPECTRUMOFCIRCLES127
quenceofLemma2,Facts1and2wehave LSpecp
c(CHn(c)) 4 p3c 43pcp3p2+q2
p>q,pqiseven, pandqaremutuallyprime) 2 3pcp3p2+q2
p>q,pqisodd, pandqaremutuallyprime) 4 p3c;43pcp7;43pcp13;43pcp19;43pcp21; 4 4 3pcp61;43pcp67;43pcp73;43pcp78;43pcp79;
4 geodesiccurvaturep spectrum.Forexample,4 3pcp91isthesmallestdoublelengthspectrum
forcirclesofgeodesiccurvaturep c.Thisspectrumisthelengthofcircles ofcomplextorsion(11;1)=135 91p91and(5;4)=83691p91withthegeodesic
curvaturep thelengthspectrumLp cwithlengthisnot c())=1(seefor ofgrowth,wealsoobtain lim !1(log)](L1p c())=0forevery>0: Wenowconsiderforgeneral.Whenp
c 2,sinceeverycircleof
128T.ADACHI
set.For>p c imageis p c;] 0<3p 3c() (42c)3=2 c 2LSpec(CHn(c)) 4 p42c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) andwhen>pc, LSpec(CHn(c))
=4 p42c;2p2c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) where(>1)denotesthenumberwith3p3c (42c)3=2=921(32+1)3=2.Wecan Proposition2.When0p
c 2,thelengthspectrumofcircles
2, thefollowingpropertieshold: 2)ThebottomofthesetLSpec(CHn(c))is4
p42c,whichisthelength withrespectto;limsup!1](L1())=1. order;lim!1(log)](L1())=0forevery>0. LENGTHSPECTRUMOFCIRCLES129
example,when=p lengthofcirclesofcomplextorsion(3;1)=10 7p7.Herewecompare2p2c,
thelengthofaholomorphiccircle,and4q 7 3(42c),whichcorrespondsto
3p3c(3;1).Theformeris
smallerthanthelatterifandonlyif>5p c 4.Sinceismonotoneincreas-
ingwithrespectto,and5p c 4=3,wegetthat0<(42c)3=23p3c(3;1)<1
if p c 2<<5p c torsion).Thereforewendthefollowing: Proposition3.
2)If5p
c phiccircles.Ifp c 2<<5p c 4,thesecondlengthspectrumis4q
7 3(42c),
21p21c.
c 8(3n1)(3n+
withrespectton. Proposition4.
2)Ifp c 2< 10c 20),thethirdlengthspectrumis4q
13 3(42c),
3p3c(3;1)=
10(42c)3=2
quotesdbs_dbs25.pdfusesText_31
2<1,2 2;1([ 1;])=8
121([
1;]);0<(1);
h 2;11(2)
1(1)(1)i
;(1)1; wegetthefollowing: LENGTHSPECTRUMOFCIRCLES125
1)thelengthspectrumL(p
c 2 2)when>p
(402+c)3=2(42+c)3. FollowingLemma1weset
F =8 :n ;0]j>p c 2o ;if=0, n ;]j3p 3c(42c)3=2=;0<<1o
;if0<<1, n ;1]j>p co ;if=1. BCir(CHn(c))=f[
;]j>();01g =0==1 =p c 2=p c=K F ?Hn(c)) duality :Cirp c(CHn(c))nf[ pc;1]g!Cirp2c 4(CPn(c))nf[
p 2c 4;1]g by([ p c;])=[ p2c 4;].Hereweusethesamenotation[
;]forrepre- andcomplextorsion. 126T.ADACHI
morphicsectionalcurvaturec,acircle ofgeodesiccurvaturep 2c=4and
equation 33
2+p 2 2c=0: Thecircle
isrational.Itsprimeperiodis length( )=4 pcL:C:M:1BA;1DA 2)Thecircle
form =(p;q)=q(9p2q2)(3p2+q2)3=2 itsprimeperiodis length( )=8 :4 3pcp2(3p2+q2);ifpqiseven,
2 3pcp2(3p2+q2);ifpqisodd.
lowing: Lemma2.Themapofdualitysatises
L([ p c;])=1p2L([ pc;]) forevery(0<1). theproblemsonthemultiplicityofL. LENGTHSPECTRUMOFCIRCLES127
quenceofLemma2,Facts1and2wehave LSpecp
c(CHn(c)) 4 p3c 43pcp3p2+q2
p>q,pqiseven, pandqaremutuallyprime) 2 3pcp3p2+q2
p>q,pqisodd, pandqaremutuallyprime) 4 p3c;43pcp7;43pcp13;43pcp19;43pcp21; 4 4 3pcp61;43pcp67;43pcp73;43pcp78;43pcp79;
4 geodesiccurvaturep spectrum.Forexample,4 3pcp91isthesmallestdoublelengthspectrum
forcirclesofgeodesiccurvaturep c.Thisspectrumisthelengthofcircles ofcomplextorsion(11;1)=135 91p91and(5;4)=83691p91withthegeodesic
curvaturep thelengthspectrumLp cwithlengthisnot c())=1(seefor ofgrowth,wealsoobtain lim !1(log)](L1p c())=0forevery>0: Wenowconsiderforgeneral.Whenp
c 2,sinceeverycircleof
128T.ADACHI
set.For>p c imageis p c;] 0<3p 3c() (42c)3=2 c 2LSpec(CHn(c)) 4 p42c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) andwhen>pc, LSpec(CHn(c))
=4 p42c;2p2c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) where(>1)denotesthenumberwith3p3c (42c)3=2=921(32+1)3=2.Wecan Proposition2.When0p
c 2,thelengthspectrumofcircles
2, thefollowingpropertieshold: 2)ThebottomofthesetLSpec(CHn(c))is4
p42c,whichisthelength withrespectto;limsup!1](L1())=1. order;lim!1(log)](L1())=0forevery>0. LENGTHSPECTRUMOFCIRCLES129
example,when=p lengthofcirclesofcomplextorsion(3;1)=10 7p7.Herewecompare2p2c,
thelengthofaholomorphiccircle,and4q 7 3(42c),whichcorrespondsto
3p3c(3;1).Theformeris
smallerthanthelatterifandonlyif>5p c 4.Sinceismonotoneincreas-
ingwithrespectto,and5p c 4=3,wegetthat0<(42c)3=23p3c(3;1)<1
if p c 2<<5p c torsion).Thereforewendthefollowing: Proposition3.
2)If5p
c phiccircles.Ifp c 2<<5p c 4,thesecondlengthspectrumis4q
7 3(42c),
21p21c.
c 8(3n1)(3n+
withrespectton. Proposition4.
2)Ifp c 2< 10c 20),thethirdlengthspectrumis4q
13 3(42c),
3p3c(3;1)=
10(42c)3=2
quotesdbs_dbs25.pdfusesText_31
1;])=8
121([1;]);0<(1);
h2;11(2)
1(1)(1)i
;(1)1; wegetthefollowing:LENGTHSPECTRUMOFCIRCLES125
1)thelengthspectrumL(p
c2 2)when>p
(402+c)3=2(42+c)3. FollowingLemma1weset
F =8 :n ;0]j>p c 2o ;if=0, n ;]j3p 3c(42c)3=2=;0<<1o
;if0<<1, n ;1]j>p co ;if=1. BCir(CHn(c))=f[
;]j>();01g =0==1 =p c 2=p c=K F ?Hn(c)) duality :Cirp c(CHn(c))nf[ pc;1]g!Cirp2c 4(CPn(c))nf[
p 2c 4;1]g by([ p c;])=[ p2c 4;].Hereweusethesamenotation[
;]forrepre- andcomplextorsion. 126T.ADACHI
morphicsectionalcurvaturec,acircle ofgeodesiccurvaturep 2c=4and
equation 33
2+p 2 2c=0: Thecircle
isrational.Itsprimeperiodis length( )=4 pcL:C:M:1BA;1DA 2)Thecircle
form =(p;q)=q(9p2q2)(3p2+q2)3=2 itsprimeperiodis length( )=8 :4 3pcp2(3p2+q2);ifpqiseven,
2 3pcp2(3p2+q2);ifpqisodd.
lowing: Lemma2.Themapofdualitysatises
L([ p c;])=1p2L([ pc;]) forevery(0<1). theproblemsonthemultiplicityofL. LENGTHSPECTRUMOFCIRCLES127
quenceofLemma2,Facts1and2wehave LSpecp
c(CHn(c)) 4 p3c 43pcp3p2+q2
p>q,pqiseven, pandqaremutuallyprime) 2 3pcp3p2+q2
p>q,pqisodd, pandqaremutuallyprime) 4 p3c;43pcp7;43pcp13;43pcp19;43pcp21; 4 4 3pcp61;43pcp67;43pcp73;43pcp78;43pcp79;
4 geodesiccurvaturep spectrum.Forexample,4 3pcp91isthesmallestdoublelengthspectrum
forcirclesofgeodesiccurvaturep c.Thisspectrumisthelengthofcircles ofcomplextorsion(11;1)=135 91p91and(5;4)=83691p91withthegeodesic
curvaturep thelengthspectrumLp cwithlengthisnot c())=1(seefor ofgrowth,wealsoobtain lim !1(log)](L1p c())=0forevery>0: Wenowconsiderforgeneral.Whenp
c 2,sinceeverycircleof
128T.ADACHI
set.For>p c imageis p c;] 0<3p 3c() (42c)3=2 c 2LSpec(CHn(c)) 4 p42c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) andwhen>pc, LSpec(CHn(c))
=4 p42c;2p2c 4s 3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime) 2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) where(>1)denotesthenumberwith3p3c (42c)3=2=921(32+1)3=2.Wecan Proposition2.When0p
c 2,thelengthspectrumofcircles
2, thefollowingpropertieshold: 2)ThebottomofthesetLSpec(CHn(c))is4
p42c,whichisthelength withrespectto;limsup!1](L1())=1. order;lim!1(log)](L1())=0forevery>0. LENGTHSPECTRUMOFCIRCLES129
example,when=p lengthofcirclesofcomplextorsion(3;1)=10 7p7.Herewecompare2p2c,
thelengthofaholomorphiccircle,and4q 7 3(42c),whichcorrespondsto
3p3c(3;1).Theformeris
smallerthanthelatterifandonlyif>5p c 4.Sinceismonotoneincreas-
ingwithrespectto,and5p c 4=3,wegetthat0<(42c)3=23p3c(3;1)<1
if p c 2<<5p c torsion).Thereforewendthefollowing: Proposition3.
2)If5p
c phiccircles.Ifp c 2<<5p c 4,thesecondlengthspectrumis4q
7 3(42c),
21p21c.
c 8(3n1)(3n+
withrespectton. Proposition4.
2)Ifp c 2< 10c 20),thethirdlengthspectrumis4q
13 3(42c),
3p3c(3;1)=
10(42c)3=2
quotesdbs_dbs25.pdfusesText_31
2)when>p
(402+c)3=2(42+c)3.FollowingLemma1weset
F =8 :n ;0]j>p c 2o ;if=0, n ;]j3p3c(42c)3=2=;0<<1o
;if0<<1, n ;1]j>p co ;if=1.BCir(CHn(c))=f[
;]j>();01g =0==1 =p c 2=p c=K F ?Hn(c)) duality :Cirp c(CHn(c))nf[ pc;1]g!Cirp2c4(CPn(c))nf[
p 2c 4;1]g by([ p c;])=[ p2c4;].Hereweusethesamenotation[
;]forrepre- andcomplextorsion.126T.ADACHI
morphicsectionalcurvaturec,acircle ofgeodesiccurvaturep2c=4and
equation 332+p 2 2c=0:
Thecircle
isrational.Itsprimeperiodis length( )=4 pcL:C:M:1BA;1DA2)Thecircle
form =(p;q)=q(9p2q2)(3p2+q2)3=2 itsprimeperiodis length( )=8 :43pcp2(3p2+q2);ifpqiseven,
23pcp2(3p2+q2);ifpqisodd.
lowing:Lemma2.Themapofdualitysatises
L([ p c;])=1p2L([ pc;]) forevery(0<1). theproblemsonthemultiplicityofL.LENGTHSPECTRUMOFCIRCLES127
quenceofLemma2,Facts1and2wehaveLSpecp
c(CHn(c)) 4 p3c43pcp3p2+q2
p>q,pqiseven, pandqaremutuallyprime) 23pcp3p2+q2
p>q,pqisodd, pandqaremutuallyprime) 4 p3c;43pcp7;43pcp13;43pcp19;43pcp21; 4 43pcp61;43pcp67;43pcp73;43pcp78;43pcp79;
4 geodesiccurvaturep spectrum.Forexample,43pcp91isthesmallestdoublelengthspectrum
forcirclesofgeodesiccurvaturep c.Thisspectrumisthelengthofcircles ofcomplextorsion(11;1)=13591p91and(5;4)=83691p91withthegeodesic
curvaturep thelengthspectrumLp cwithlengthisnot c())=1(seefor ofgrowth,wealsoobtain lim !1(log)](L1p c())=0forevery>0:Wenowconsiderforgeneral.Whenp
c2,sinceeverycircleof
128T.ADACHI
set.For>p c imageis p c;] 0<3p 3c() (42c)3=2 c 23p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime)2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) andwhen>pc,LSpec(CHn(c))
=4 p42c;2p2c 4s3p2+q2
3(42c)
p>q,pqiseven, pandqaremutuallyprime)2s3p2+q2
3(42c)
p>q,pqisodd, pandqaremutuallyprime) where(>1)denotesthenumberwith3p3c (42c)3=2=921(32+1)3=2.WecanProposition2.When0p
c2,thelengthspectrumofcircles
2, thefollowingpropertieshold:2)ThebottomofthesetLSpec(CHn(c))is4
p42c,whichisthelength withrespectto;limsup!1](L1())=1. order;lim!1(log)](L1())=0forevery>0.LENGTHSPECTRUMOFCIRCLES129
example,when=p lengthofcirclesofcomplextorsion(3;1)=107p7.Herewecompare2p2c,
thelengthofaholomorphiccircle,and4q 73(42c),whichcorrespondsto
3p3c(3;1).Theformeris
smallerthanthelatterifandonlyif>5p c4.Sinceismonotoneincreas-
ingwithrespectto,and5p c4=3,wegetthat0<(42c)3=23p3c(3;1)<1
if p c 2<<5p c torsion).Thereforewendthefollowing:Proposition3.
2)If5p
c phiccircles.Ifp c 2<<5p c4,thesecondlengthspectrumis4q
73(42c),
21p21c.
c8(3n1)(3n+
withrespectton.Proposition4.
2)Ifp c2< 10c 20),thethirdlengthspectrumis4q
13 3(42c),
3p3c(3;1)=
10(42c)3=2
quotesdbs_dbs25.pdfusesText_31
20),thethirdlengthspectrumis4q
133(42c),
3p3c(3;1)=
10(42c)3=2
quotesdbs_dbs25.pdfusesText_31
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