a Xi :1205.0923 1 [ma h.DS] 4 May 2012
Global pe iodici y condi ions o maps and ecu ences
ia No mal Fo ms
Anna Cima(1), A mengol Gasull(1) and V´ıc o Ma˜nosa (2)
(1) Dep . de Ma em`a iques, Facul a de Ci`encies,
Uni e si a Au `onoma de Ba celona,
08193 Bella e a, Ba celona, Spain
[email p o ec ed]b.ca , ga[email p o ec ed]
(2) Dep . de Ma em`a ica Aplicada III (MA3),
Con ol, Dynamics and Applica ions G oup (CoDALab)
Uni e si a Poli `ecnica de Ca alunya (UPC)
Colom 1, 08222 Te assa, Spain
ic o [email protected]
May 7, 2012
Abs ac
We ace he p oblem o cha ac e izing he pe iodic cases in pa ame ic amilies o
a ional di eomo phisms o Kk, whe e Kis Ro C, ha ing a ixed poin . Ou app oach
elies on he No mal Fo m Theo y, o ob ain necessa y condi ions o he exis ence
o a o mal linea iza ion o he map, and on he in oduc ion o a sui able a ional
pa ame iza ion o he pa ame e s o he amily. Using hese ools we can ind a ini e
se o alues p o which he map can be p-pe iodic, educing he p oblem o inding he
pa ame e s o which he pe iodic cases appea o simple compu a ions. We apply ou
esul s o se e al wo and h ee dimensional classes o polynomial o a ional maps. In
pa icula we ind he global pe iodic cases o se e al Lyness ype ecu ences.
2000 Ma hema ics Subjec Classi ica ion: 37G05, 39A11, 39A20, 37C05
Keywo ds: Pe iodic maps; Linea iza ion; No mal Fo ms; Ra ional pa ame iza ions; Glob-
ally pe iodic ecu ences; Lyness ecu ences.
1
1 In oduc ion
A map Fsuch ha Fp(x)≡x, o some p∈Nand o all x o which Fpis well de ined,
will be called a pe iodic map. I pis he smalles posi i e in ege wi h his p ope y, hen
Fis called p-pe iodic. In his pape we ea he p oblem o cha ac e izing he p-pe iodic
cases in pa ame ic amilies o a ional maps o Kk, whe e Kis Ro C, ha ing a ixed poin .
When Fis a p-pe iodic di e en iable map ha ing a ixed poin , x0, i is well-known ha
(DF (x0))p= Id. In ac his is a simple consequence o he chain ule. As we will see in
P oposi ion 10, m=pis he smalles posi i e in ege numbe such ha (DF(x0))m= Id.
This simple esul allows o ea in a easy way he pe iodici y p oblem when a alue p
such ha (DF(x0))p= Id is known. Fo ins ance i Fhas a ixed poin x0such ha
(DF (x0))2= Id hen i Fis p-pe iodic hen pmus be 2,and no p= 2m, m ∈Nas we
could hink in p inciple, and hen we simply ha e o check whe he F2= Id o no .
In gene al, gi en a pa ame ic amily o maps Fa,a∈Km, he mos di icul p oblem
o inding he pe iodic maps is o de e mine which a e he possible alues psuch ha he e
exis s some asuch ha Fais p-pe iodic. The ools ha we will in oduce in his pape
will allow o ind a ini e se o possible alues o p o which he map can be p-pe iodic,
con e ing he p oblem o inding hese alues o ain o a compu a ional p oblem.
P oposi ion 10 as well as ou app oach o he cha ac e iza ion o p-pe iodic maps ia
No mal Fo m Theo y a e based on he Mon gome y-Bochne Theo em, see [23]. I will be
ecalled and p o ed in Sec ion 2. In a ew wo ds i says ha any p-pe iodic, C1-map wi h
a ixed poin is locally conjuga ed wi h he linea map L(x) = DF(x0)x, and so locally
linea izable. No ice ha he di e en iabili y condi ion is necessa y since i is well known
ha he e a e pe iodic in olu ions (i.e. F2= Id) gi en by homemo phisms wi h ixed poin s
which a e no linea izable, see [8].
Hence any p-pe iodic case in a gi en amily wi h ixed poin s can be locally linea ized.
Thus, he applica ion o a sui able No mal Fo m algo i hm, will gi e necessa y condi ions
o he exis ence o he linea iza ion. As we will see, hese condi ions a e some imes also
su icien .
We ema k ha his app oach does no co e he p oblem in i s ull gene ali y, because
he e a e pe iodic di eomo phisms wi hou ixed poin s in Rkwi h k≥7, see [18, 20].
I is well-known ha he No mal Fo m algo i hms o en lead o e y complica ed ex-
p essions which a e di icul o handle when dealing wi h he gi en pa ame e s o he map.
Some imes, hese obs uc ions can be signi ica i ely so ened by in oducing new pa ame-
e s a ionally depending on he old ones, and such ha he coo dina es o he ixed poin s
as well as he eigen alues o he jacobian ma ix a hese ixed poin s, depend a ionally
on hese new pa ame e s. This is he second main cha ac e is ic o ou app oach, when
2
dealing wi h conc e e applica ions.
The No mal Fo m Theo y is b ie ly ecalled in Sec ion 3. In Sec ion 4 we ob ain some
esul s o plana maps in he case ha he linea pa o Fa he ixed poin is gi en by a
ma ix diag(α, β) wi h αβ = 1, o diag(α, 1). As i s applica ions o he me hod, we ge :
Theo em 1. Conside a smoo h complex map o he o m
F(x, y) = 
αx +X
i+j≥2
i,jxiyj,1
αy+X
i+j≥2
gi,jxiyj
,(1)
whe e αis a p imi i e p- oo o uni y, p≥5. Then he condi ions P1(F) = P2(F) =
P3(F) = 0 a e necessa y o F o be p-pe iodic, whe e
P1(F) := ( 2,1+ 1,1g1,1)α4− 1,1(2 2,0−g1,1)α3+ (2g2,0 0,2− 1,1 2,0+ 1,1g1,1)α2
−( 2,1+ 1,1 2,0)α+ 1,1 2,0,
P2(F) :=g0,2g1,1α4−(g1,2+g0,2g1,1)α3+ ( 1,1g1,1+ 2g2,0 0,2−g0,2g1,1)α2
+g1,1(−2g0,2+ 1,1)α+ 1,1g1,1+g1,2,
and P3(F)is gi en in Appendix A.
In ac , condi ions P1(F) = 0 and P2(F) = 0 also wo k o p= 4.
Theo em 2. Conside a smoo h complex map o he o m
F(x, y) = 
αx +X
i+j≥2
i,jxiyj, y +X
i+j≥2
gi,jxiyj
,(2)
whe e αis a p imi i e p- oo o uni y. Then he ollowing a e necessa y condi ions o F
o be p-pe iodic:
P1(F) := 1,1= 0,
P2(F) :=g0,2= 0,
P3(F) := 1,2α−2 2,0 0,2+ 2 0,2g1,1− 1,2= 0,
P4(F) :=g0,3α−g0,3− 0,2g1,1= 0,
P5(F) := 1,3α2+ (−2 2,0 0,3+ 3 0,3g1,1+ 2g1,2 0,2−2 2,1 0,2−2 1,3)α+ 1,3+ 2 2,0 0,3
+ 2 2,1 0,2−4g2,0 2
0,2−2g1,2 0,2−3 0,3g1,1= 0,
P6(F) :=g0,4α2−( 0,3g1,1+ 2g0,4+g1,2 0,2)α+g2,0 2
0,2+ 0,3g1,1+g0,4+g1,2 0,2= 0.
In his las case, and in con as wi h he one ea ed in Theo em 1, i is no di icul
o ob ain addi ional pe iodici y condi ions. Two mo e pe iodici y condi ions a e gi en in
Appendix B.
3
The abo e esul s a e applied in se e al con ex s. The i s applica ion is o polynomial
maps. Pe iodic polynomial maps a e no o ious examples o in e ible polynomial ones,
which, in u n, a e he ocus o many deep open p oblems like he Jacobian conjec u e,
o he linea iza ion conjec u e. This second conjec u e says ha i F:Cn→Cnis a p-
pe iodic polynomial map, hen he e exis s a polynomial au omo phism ϕ(i.e. an in e ible
polynomial map wi h polynomial in e se) such ha ϕ◦F◦ϕ−1is a linea map. This
conjec u e is ue o n= 2 and as a as we know i is open o n≥3, see [15, Chaps. 8
and 9] and [21].
In Sec ion 5 we cha ac e ize he p-pe iodic maps in a amily o iangula maps, see
Theo em 15, and we gi e a simple and sel -con ained p oo o he linea iza ion conjec u e
o his case. As an applica ion o his esul and Theo em 1 we p o e:
P oposi ion 3. Conside a complex polynomial map
F(x, y) = 
αx +
3
X
i+j=2
i,jxiyj, y/α +
3
X
i+j=2
gi,jxiyj
,(3)
The map is p-pe iodic i and only i αis a p imi i e p- oo o he uni y, and i holds one o
he ollowing condi ions
(i) p= 1 and F(x, y) = (x, y);
(ii) p= 2,4and F(x, y) = (αx + 0,2y2, y/α)o F(x, y) = (αx, y/α +g2,0x2);
(iii) p= 3, and F(x, y) = (αx + 0,3y3, y/α)o F(x, y) = (αx, y/α +g3,0x3);
(i ) p≥5and F(x, y) = (αx+ 0,2y2+ 0,3y3, y/α)o F(x, y) = (αx, y/α+g2,0x2+g3,0x3).
Simila ly, as an applica ion o Theo em 2, we p o e:
P oposi ion 4. The only p-pe iodic cases in he amily o complex maps
F(x, y) = αx +bx2+cxy +dy2
1 + m(x2+y2),y+ x2+sxy + y2
1 + m(x2+y2),
a e, ei he F(x, y) = (x, y)when α= 1, o he ones gi en he polynomial maps F(x, y) =
(αx +dy2, y)o F(x, y) = (αx, y + x2)when αa p imi i e p- oo o he uni y wi h p > 1
and dand a bi a y complex numbe s.
In all he es o examples, gi en in Sec ions 6 and 7, he maps a e he ones associa ed
o some ecu ences. Recall ha gi en a ecu ence, au onomous o no , i is said ha
i is globally p-pe iodic i o all ini ial condi ions o which he sequence is well-de ined i
gi es ise o a p-pe iodic sequence and pis he smalles posi i e in ege numbe wi h his
4
p ope y. We will ace his ques ion s udying an associa ed map F. Wi h his poin o iew,
he ecu ence will be globally pe iodic i and only i he map Fis pe iodic.
The s udy o he global pe iodici y in di e ence equa ions is nowadays he subjec o
an ac i e esea ch, see o ins ance [1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 22, 25, 26], and
e e ences he ein and se e al echniques ha e been used o app oach he p oblem. To he
bes o ou knowledge, his is he i s ime ha he No mal Fo m Theo y is used in his
se ing. As a second applica ion o Theo em 1, we classi y he globally pe iodic second
o de Lyness ecu ences, eob aining he esul s in [13] o his case:
P oposi ion 5. The only globally pe iodic Lyness ecu ences xn+2 =a+xn+1
xn
wi h a∈C,
a e he 5-pe iodic case wi h a= 1; and he 6-pe iodic case wi h a= 0.
Also as a di ec consequence o Theo em 1 we ge nex esul o some Gumo ski-Mi a-
ype ecu ences [17],
P oposi ion 6. The e a e no globally pe iodic cases in he amily o Gumo ski-Mi a ecu -
ences
xn+2 =−xn+xn+1
b+x2
n+1
, b ∈C.
One o he main applica ions in his se ing conce ns he 2-pe iodic Lyness ecu ence
xn+2 =an+xn+1
xn
,whe e an=(a o n= 2ℓ+ 1,
b o n= 2ℓ, (4)
and a, b ∈C. In Sec ion 6.3 we sol e he global pe iodici y p oblem o i by s udying he
amily o maps
Fb,a(x, y) = a+y
x,a+bx +y
xy ,
which as we will see desc ibes he beha io o (4).
Theo em 7. The only globally pe iodic ecu ences in (4) a e:
(i) The cases a=b= 0 (6-pe iodic) and a=b= 1 (5-pe iodic).
(ii) The cases a= (−1±i√3)/2and b=a= 1/a,10-pe iodic.
No ice ha he cases gi en in (i) co espond o he well-known au onomous globally
pe iodic Lyness ecu ences also appea ing in P oposi ion 5.
Finally, o show an applica ion in K3we ind he globally pe iodic hi d o de Lyness
ecu ences, eob aining again he esul in [13]:
P oposi ion 8. The only globally pe iodic hi d-o de Lyness ecu ence
xn+3 =a+xn+1 +xn+2
xn
, a ∈C,
co esponds o a= 1 and is 8-pe iodic.
5

2 Some consequences o he Mon gome y-Bochne Theo em
The nex e sion o Mon gome y-Bochne Theo em is a simpli ied one, adap ed o ou
in e es s. The gene al one applies in a much mo e gene al con ex , see [23].
Theo em 9 (Mon gome y-Bochne ). Le F:U → U be a p-pe iodic C1-di eomo -
phism, whe e Uis an open se o Kk. Le x0∈ U be a ixed poin o F. Then, he e exis s a
neighbou hood o x0whe e Fis conjuga ed wi h he linea map L(x) = DF(x0)x. Mo eo e
he linea iza ion is gi en by he local di eomo phism
ψ(x) = 1
p
p−1
X
i=0
(DF (x0))−iFi(x).
P oo . Since Fis p-pe iodic (DF(x0))p= Id. So (de (DF(x0))p= 1 and DF(x0) is in-
e ible. Conside ψas in he s a emen . By he in e se unc ion heo em i is clea ha
he map ψis a local di eomophism because Dψ(x0) = Id. Mo eo e , using again he
p-pe iodici y o Fwe ge ha ψ(F(x)) = L(ψ(x)), as we wan ed o p o e.
As we ha e seen in he p oo o he abo e heo em, i Fis a p-pe iodic di e en iable
map wi h a ixed poin x0, hen (DF(x0))p= Id. Nex esul ela es pwi h he minimum
posi i e msuch ha (DF(x0))m= Id.
P oposi ion 10. Le Fbe a di e en iable map ha ing a ixed poin x0. Assume ha Fis
p-pe iodic and le mbe he minimum posi i e msuch ha (DF(x0))m= Id. Then p=m.
P oo . By using he Mon gome y-Bochne Theo em we know ha Fis C1-conjuga ed o
L(x) = DF(x0)xin a neighbo hood o x0. Thus F=ψ−1◦L◦ψ, o some C1di eomo phism
ψ. Since Lm= Id i and only i Fm=ψ−1◦Lm◦ψ=ψ−1◦ψ= Id, he esul ollows.
Co olla y 11. Le Fa(x) = Lx+G(x,a),wi h x∈ U ⊂ Knand a∈Km,a smoo h amily
o maps such ha G(0,a)≡DxG(0,a)≡0 o all a∈Km. Assume ha pis he minimum
posi i e in ege numbe such ha Lp= Id. Then i Fais pe iodic o some a∈K hen i is
p-pe iodic, i.e. Fp
a= Id .
In pa icula no e ha i L= Id hen he only pe iodic case is Fa(x) = xand when
L2= Id he pe iodici y condi ions a e gi en by Fa(Fa(x)) ≡x.Fo example he ac p o ed
in [25, Ex. 2], ha he only pe iodic map o he o m F(x1, x2) = (x2+ax2
1, x1+bx1x2)
co esponds o he linea case a=b= 0, ollows easily using his app oach.
No ice ha using Mon gome y-Bochne Theo em a necessa y condi ion o a map o he
o m Fa(x) = Lx+G(x,a), o be pe iodic is ha Fais linea izable in a neighbou hood
o 0. The linea izable cases can be de ec ed by ollowing he well-know No mal Fo m
Theo y, which, as a as we know, has no been used o his pu pose. Some esul s use ul
o applying i will be ecalled in he nex sec ion.
6
3 Pe iodici y condi ions ia No mal Fo m Theo y
We s a in oducing some well-known issues o No mal Fo m Theo y, while e e ing he
eade o [3, Sec. 2.5], o u he de ails.
Le F:= F(1) :Kk→Kk, be a amily o smoo h maps depending on some pa ame e s
and sa is ying F(1)(0) = 0.Le
F(1)(x) = F(1)
1(x) + F(1)
2(x) + ···+F(1)
k(x) + O(|x|k+1) (5)
be he Taylo expansion o Fa 0,whe e F(1)
∈ H , he eal ec o space o maps whose
componen s a e homogeneous polynomials o deg ee .
The aim o he No mal Fo m Theo y is o cons uc a sequence o ans o ma ions Φn,
s a ing om n= 2, such ha a each s ep, Φnsimpli ies, as much as possible, he e ms o
he co esponding homogeneous pa o deg ee n. To his end, le F(1)
1(x) = DF(1)(0)x=:
Lxand suppose ha
F(n−1)(x) = Lx+F(n−1)
n(x) + O(|x|n+1), n ≥2.
Conside a ans o ma ion
x= Φn(y) := y+φn(y),
wi h φn∈ Hn,such ha i conjuga es he map F(n−1) wi h a new map F(n), ia he
conjuga ion
F(n−1)(Φn) = Φn(F(n)).
F om he abo e equa ion, i can be easily seen ha
F(n)(y) = Ly+L φn(y)−φn(Ly) + F(n−1)
n(y) + O(|y|n+1).
Clea ly, i φn(y) can be chosen in such a way ha
ML(φn(y)) := L φn(y)−φn(Ly) = −F(n−1)
n(y),(6)
hen F(n−1) is ans o med in o
F(n)(y) = Ly+F(n)
n+1(y) + O(|y|n+2) = Ly+O(|y|n+1).
The ec o ial equa ion (6) is he well-known homological equa ion associa ed wi h L=
DF(1)(0), and he exis ence o solu ions o i is he necessa y and su icien condi ion o be
able o emo e he homogeneous e ms o deg ee n.
F om now, one we will assume ha he linea map is diagonalizable, and so ha i is
L= diag(λi)k
i=1. In his case, he linea ope a o ML:= Hn→ Hn,gi en in (6), has he
7
eigen ec o s xmei, i = 1,2,...,k, wi h m= (m1, m2,...,mk)∈ Mk
n:= {m∈Nksa is ying
Pk
i=1 mi=n};xm=xm1
1xm2
2···xmk
nwhe e x= (x1, x2,...,xk)∈Kk; and whe e eiis he
i- h membe o he na u al basis o Kk.Hence
ML(xmei) = (λi−λm)xmei,(7)
whe e λm=λm1
1λm2
2···λmk
n.
Se
F(n−1)
n(x) = X
m
(n−1)
1;mxm,X
m
(n−1)
2;mxm,...,X
m
(n−1)
k;mxm!,
and
φn(x) = X
m
a1;mxm,X
m
a2;mxm,...,X
m
ak;mxm!,
whe e m∈ Mk
n.
When λi−λm6= 0 o all he sui able alues o m∈Nkand o all i= 1,2,...,k,
i is said ha he e a e no esonances. In his case he ope a o MLis in e ible, he
homological equa ion always has solu ion and so he linea iza ion p ocess can con inue. On
he con a y, i λi−λm= 0 o some m∈ Mk
nand some i∈ {1,2,...,k}, hen he ec o
λ= (λ1, λ1,...,λk) is said o be esonan o o de n. In his case, by simple inspec ion
o he homological equa ion, and using (7), we ob ain ha he he n h o de obs uc ion
equa ion associa ed o he esonance is gi en by
(λi−λm)ai;m=− (n−1)
i;m.
Howe e , he e a e some maps ha ing his esonance o which he p ocess can con inue.
This happens i he igh -hand side o his scala equa ions anish, namely (n−1)
i;m= 0, and
hese cases a e he ones candida e o be linea ized. Hence, we ha e ob ained he ollowing
esul
P oposi ion 12. I L:= diag(λi)k
i=1, hen a necessa y condi ion o he map (5) o be
pe iodic is gi en by he n h o de pe iodici y condi ion associa ed o he esonance condi ion,
λi−λm= 0, gi en by (n−1)
i;m= 0.
Rema k 13. No ice ha p-pe iodic maps wi h Ldiagonal a e such ha λp
i= 1, o all i.
The e o e o hese maps many esonances λi−λm= 0 appea .
By ollowing he No mal Fo m Algo i hm, i is s aigh o wa d (and well known) o
see ha he nume a o o (n−1)
i;mis a polynomial in he coe icien s o F(1). Thus, o each
pa icula case, he abo e equa ions gi e pe iodici y condi ions, which a e algeb aic in e ms
o he ini ial pa ame e s o he map, once exp essed in o m (5).
8
To ix he ideas we gi e a simple example. Suppose ha k= 2. Assume ha L=
diag(α, β). Se Φ2(y) := y+φ2(y), whe e
φ2(x, y) := a20x2+a11xy +a02y2
b20x2+b11xy +b02y2!.
Conside he map F(1)(x) = Lx+F(1)
2(x) + O(|y|3) wi h
F(1)
2(x, y) = 20x2+ 11xy + 02y2
g20x2+g11xy +g02y2!,
whe e o simpli y he no a ion, and om now on, i he e is no possibili y o con usion, we
will d op he supe sc ip (1) o he coe icien s o F(1).
The homological equa ion a o de 2 is L φ2(y)−φ2(Ly) = −F(1)
2(y), and gi es he
ollowing six scala equa ions:







(α−α2)a20 =− 20,(β−α2)b20 =−g20,
(α−αβ)a11 =− 11,(β−αβ)b11 =−g11,
(α−β2)a02 =− 02,(β−β2)b02 =−g02.
I no one o he six 2nd o de esonance condi ions:α2−α, αβ −α, β2−α, β2−β, αβ −β,
and α2−β, anish, he e is no obs uc ion o emo e he second o de e ms o F(1) using
he conjuga ion Φ2.
Suppose now, ha he map F(1) is such ha he esonance β−α2= 0 occu s. Then
he scala equa ion (β−α2)b20 =−g20 is an obs uc ion equa ion. Bu his obs uc ion o
he linea iza ion p ocess disappea s i g20 anishes. In summa y, i β=α2, hen g20 = 0 is
ape iodici y condi ion.
4 P oo o Theo ems 1 and 2
We keep he no a ion in oduced in he abo e sec ion, i.e., F(k)is he map ob ained a e
k−1 s eps o he no mal o m p ocedu e, F(k)(x) = Lx+F(k)
k+1(x) + O(|x|k+2),and i s
coe icien s a e (k)
i,j and g(k)
i,j . Fi s conside he case ea ed in Theo em 1:
F(k)(x, y) = 
αx +X
i+j≥k+1
(k)
i,j xiyj,1
αy+X
i+j≥k+1
g(k)
i,j xiyj
.
I is easy o check ha he scala equa ions associa ed o equa ion (6) a e
(α(1 −αn−2i−1)an−i,i =− (n−1)
n−i,i ,
α−1(1 −αn−2i+1)bn−i,i =−g(n−1)
n−i,i ,
9
We conside sepa a ely he case b= 0.In his si ua ion x0:= (√2/2,√2/2) is a ixed
poin o G0. I is easy o see ha (DG0(x0)p6= Id, o any posi i e in ege p, because he
ma ix is no diagonalizable. So G0is no a pe iodic map.
When b6= 0 we in oduce a new pa ame e λ, and w i e b=λ/(1 + λ2) wi h λ2+ 1 6= 0
and λ6= 0.No ice ha his pa ame iza ion co e s all alues o bin C {0}. We ename he
new map co esponding o Gbas gλ. The eigen alues o i s Jacobian ma ix a he o igin,
which is always a ixed poin , a e λand 1/λ. The linea map Ψ(x, y) = (x−λy, x −y/λ)
is a conjuga ion be ween Dgλ(0) and i s diagonal o m L(x, y) := (λx, y/λ). Using his
conjuga ion we conside he map Fλ:= Ψ ◦gλ◦Ψ−1. Using Theo em 1 we impose ha
P2(Fλ) = 0. We ge ha a necessa y condi ion o Fλ o be pe iodic is
λ2+ 12λ2+λ+ 1= 0.
I λis a oo o λ2+λ+1, hen i is a p imi i e 3 d- oo o he uni y. Then by Co olla y 11,
Fλshould be globally 3-pe iodic. Bu we ha e al eady disca ded his possibili y. So he
esul ollows.
6.3 Global pe iodici y in he 2-pe iodic non-au onomous Lyness ecu -
ence
In his sec ion we s udy he p oblem o he global pe iodici y o he he sequence gene a ed
by he 2-pe iodic Lyness ecu ence (4). The sequence {xn}gi en by his ecu ence can
be eob ained as
(x1, x2)Ga
−−→ (x2, x3)Gb
−→ (x3, x4)Ga
−−→ (x4, x5)Gb
−→ (x5, x6)Ga
−−→ ···
whe e Gα(x, y), wi h α∈ {a, b}, is he Lyness map gi en in (10). So he beha io o (4) is
gi en by he dynamical sys em gene a ed by he map:
Gb,a(x, y) := Gb◦Ga(x, y) = a+y
x,a+bx +y
xy .(12)
P oo o Theo em 7. As we ha e seen i su ices o s udy he pe odici y p oblem o he
map (12). I is easy o see ha Gp
a6= Id o p= 1,2,4. Mo eo e i is 3-pe iodic i and only
i a=b= 0. No ice ha his case co esponds o he globally 6-pe iodic ecu ence. We
con inue sea ching p-pe iodic maps wi h p≥5.
Following simila ideas ha in he p e ious subsec ions we in oduce a mo e sui able
a ional pa ame iza ion o aand b. We conside
a=B3λ2+ 1+λ2B3−1
B(λ+ 1)2,
b=−B+ (B2−a)2,wi h B(λ+ 1) 6= 0 and λ6= 0.
(13)
16

Using hese new pa ame e s we co e all he alues o aand bin C. Mo eo e he ixed
poin is (B, B2−a),whe e ais gi en in (13), and he eigen alues o Gb,a a his poin a e
λand 1/λ. A e a ansla ion (x, y)→(x−B, y −(B2−a)), which b ings he ixed poin
o he o igin, he map Gb,a conjuga es, using again xand yas a iables, wi h
gB,λ(x, y) = 

y−Bx
x+B,−B2(λ+ 1)2x−Bλ2+λ+ 1y+λ xy
B(λ+ 1)2y+λ(x+B)
,
wi h linea pa .
LB,λ(x, y) = −x+y
B,−B(λ+ 1)2x
λ+λ2+λ+ 1y
λ!.
The linea change o a iables Ψ(x, y) = x+y, (λ+ 1) Bx +1 + 1
λBygi es a conjuga-
ion be ween LB,λ and i s diagonal o m L(x, y) := (λx, y/λ). Using his conjuga ion we
conside he map
FB,λ(x, y) := Ψ ◦gB,λ(x, y)◦Ψ−1(x, y),
which sa is ies DFB,λ(0,0) = diag (λ, 1/λ). Fo simplici y, we omi i s explici exp ession.
Recall ha λp−16= 0 o p= 1,2,3.
By Theo em 1, when p≥5, om bo h condi ions Pi(FB,λ) = 0, i = 1,2,we ob ain he
same pe iodici y condi ion C1(B, λ) = 0, whe e
C1(B, λ) := B6λ10 + 9B6λ9+ 35B6λ8+ 80B6λ7+ 124B6λ6+ 2B3λ9+ 142B6λ5+ 8B3λ8
+124B6λ4+ 18B3λ7+ 80B6λ3+ 32B3λ6+ 35B6λ2+ 40B3λ5+ 9B6λ
+32B3λ4+λ7+B6+ 18B3λ3+ 3λ6+ 8B3λ2+ 2λ5+ 2B3λ+ 3λ4+λ3.
Using again Theo em 1, we ob ain ano he polynomial es ic ion C2(B, λ) := P3(FB,λ) =
0. The exp ession o C2(B, λ) is gi en in Appendix C. To s udy he pe iodici y o FB,λ i
su ices o deal wi h he wo condi ions
C1(B, λ) = 0, C2(B, λ) = 0.
Compu ing R(λ) := Res(C1(B, λ), C2(B, λ); B) we ge
R(λ) = λ36 (λ−1)24 (λ+ 1)72 λ2+ 16λ2+λ+ 124 S6(λ)T6(λ),
whe e S(λ) = λ4+λ3+λ2+λ+ 1 and T(λ) = 3λ4+ 15λ3+ 20λ2+ 15λ+ 3. Then, a
necessa y condi ion o FB,λ o be p-pe iodic wi h p≥4 is ha λis a p imi i e p- h o he
uni y and ha ei he S(λ) = 0 o T(λ) = 0. Le us disca d he o me possibili y.
I u ns ou ha Thas wo eal oo s and wo complex oo s o modulus one. We ha e
o p o e ha hey a e no oo s o he uni y. This can be seen, o ins ance, p o ing ha T
17
is no di isible by any cyclo omic polynomial. This holds because i i had a cyclo omic
polynomial di iso , i s deg ee should be a mos 4. The cyclo omic polynomials o deg ee
a mos 4 co espond o p∈ {1,2,3,4,5,6,8,10,12}:= D4. This is because hese a e he
cases which co espond o cyclo omic polynomials o deg ee ϕ(p)≤4, being ϕ he Eule ’s
unc ion, see o ins ance [24]. Since
Res(T(λ), λp−1; λ)6= 0, o p∈ D4,
he esul ollows.
Finally, when S(λ) = 0 no ice ha λis a p imi i e 5- h oo o he uni y. So, by
Co olla y 11 i FB,λ is p-pe iodic i should be 5-pe iodic. The e o e i su ices o s udy
whe he F5
B,λ = Id o no , o equi alen ly whe he G5
b,a = Id. Compu ing he nume a o o
he i s componen o G5
b,a(x, y)−(x, y) we ge ha i w i es as a4b(1 −ab)x+O(2), whe e
as usual O(m) deno es e ms o deg ee a leas min xand y. Hence only h ee possibili ies
o Gb,a o be 5-pe iodic appea : ei he a= 0 o b= 0 o ab = 1.
The i s wo cases can easily disca ded. I holds ha G5
0,a 6= Id and G5
b,06= Id. On he
o he hand, when b= 1/a, a 6= 0 he nume a o o he i s componen o G5
1/a,a(x, y)−(x, y)
w i es as −a(a−1)2(a2+a+ 1)2x2y+O(3). Since his las unc ion has o anish we ge
h ee candida es o be 5-pe iodic: a= 1 and a= (−1±i√3)/2 wi h b= 1/a =a. I is easy
see ha all hem gi e ise o 5-pe iodic maps Gb,a. The las wo co espond o he globally
10-pe iodic ecu ence.
Rema k 17. The cha ac e iza ion o he globally pe iodic di e ence equa ions ea ed in
his sec ion can also be ob ained ollowing he app oach de eloped in [27] ha gi es all he
pe iodic QRT-maps. This esul also appea s in [14, p. 165] and [19].
7 The hi d o de Lyness ecu ence
We s a p o ing a gene al esul which will use ul o sol ing he pe iodici y p oblem o
he Lyness ecu ence.
P oposi ion 18. Conside he smoo h amily o maps
F(x, y, z) = αx +X
m
mxm, βy +X
m
gmxm, γz +X
m
hmxm!,
wi h m∈ {(i, j, k)such ha i+j+k≥2}, and whe e xm=xiyjzk. When α=±1,βγ = 1,
and β6= 1, γ 6= 1,some necessa y condi ions o i o be pe iodic a e
(2)
3,0,0= (2)
1,1,1=g(2)
2,1,0=g(2)
0,2,1=h(2)
2,0,1=h(2)
0,1,2= 0,
18
whe e (2)
i,j,k and g(2)
i,j,k a e he exp essions gi en in he second s ep o he no mal o m p oce-
du e desc ibed in Sec ion 3.
P oo . By inspec ion o he 3 d o de esonance condi ions, we obse e ha when α=±1
and βγ = 1 he e appea he esonances α3−α,αβγ −α,α2β−β,β2γ−β,α2γ−γand
β2γ−γwhich a e associa ed o he coe icien s (2)
3,0,0, (2)
1,1,1,g(2)
2,1,0,g(2)
0,2,1,h(2)
2,0,1and h(2)
0,2,1
espec i ely. So all hem mus anish o ha e a pe iodic map.
P oo o P oposi ion 8. The dynamics o he hi d-o de Lyness’ equa ion can be s udied
h ough he Lyness maps
Ga(x, y, z) = y, z, a+y+z
x.
I is easy o see ha Gp
a6= Id o p= 1,2. We con inue sea ching p-pe iodic maps wi h
p≥3.I has always some ixed poin (x0, x0, x0) wi h x2
0−2x0−a= 0 and x06= 0.
Mo eo e he eigen alues λo he Jacobian ma ix a his poin s a e gi en by he ze oes
o −(λ+ 1)(λ2−(1 + 1/x0)λ+ 1) = 0. These wo equa ions sugges us o in oduce he
a ional pa ame iza ion o aas
a=−λ2λ2−3λ+ 2
(λ2−λ+ 1)2,wi h λ2−λ+ 1 6= 0 and λ6= 0,
which co e s all alues o a∈C.Then he ixed poin is (x0, x0, x0) wi h x0=λ/(λ2−λ+1)
and he eigen alues o DGaa his poin a e −1, λ, 1/λ. No ice ha since p≥3, we can
assume λ6= 1. To apply P oposi ion 18 we pe o m he ansla ion (x, y, z)→(x−x0, y −
x0, z −x0), which b ings he ixed poin o he o igin, ob aining
gλ(x, y, z) := y, z, −λx +λ2−λ+ 1y+λ2−λ+ 1z
(λ2−λ+ 1) x+λ!,
wi h linea pa
Lλ(x, y) = y, z, −x+λ2−λ+ 1y
λ+λ2−λ+ 1z
λ!.
The linea change o a iables Ψ(x, y, z) = (x+y+λ2z, −x+λ y +λ z, x +λ2y+z)
gi es a conjuga ion be ween Lλand i s diagonal o m L(x, y, z) := (−x, λy, z/λ). Using he
conjuga ion Ψ, we inally ob ain a map wi h diagonal linea pa Fλ:= Ψ ◦gλ◦Ψ−1, which
is unde he assump ions o P oposi ion 18.
Applying his p oposi ion and he No mal Fo m Algo i hm o Fλwe can compu e g(2)
2,1,0.
F om he equa ion g(2)
2,1,0= 0 we ob ain ha
λ2−λ+ 13(λ4+ 1) = 0.
19
Thus λhas o be a p imi i e 8- h oo o he uni y. All hese alues o λco espond o he
same alue a= 1, which gi es a globally 8-pe iodic ecu ence. So he esul ollows
Acknowledgemen s
GSD-UAB and CoDALab G oups a e suppo ed by he Go e nmen o Ca alonia h ough
he SGR p og am. The i s and second au ho s a e also suppo ed by MCYT h ough
g an s MTM2008-03437 and he hi d au ho by he g an DPI2011-25822.
Appendix A. Exp ession o P3(F) when αβ = 1
Conside he map (1), applying he No mal Fo m Algo i hm one ge s ha he pe iodici y
condi ion associa ed o (4)
3,2is gi en by
P3(F) := 1,1g0,2g2
1,1α17 + (2g2
1,1 1,1g0,2− 3,1g0,2− 1,1g1,1g1,2− 1,1g0,2g2,1)α16
+ ( 3,2+ 3g2
1,1 1,1g0,2+ 3 1,1g0,2 3,0−3 1,1g0,2g2,1+ 2 1,1 2,0g0,2g1,1+ 2 2
1,1g0,2g2,0
−2 3,1g0,2+ 2 2,1 2,0g0,2+g2
1,1 1,2+ 2 2,2g1,1+ 1,1g2,2+ 2 0,2 1,1g1,1g2,0− 1,1g1,1g1,2
+g2
1,1 2
1,1)α15 + (3g2
1,1 1,1g0,2+ 6 1,1g0,2 3,0−6 0,2 3,0g1,1−4 1,2 2,0g1,1−3 1,1g1,1 2,1
+ 2 1,2g2,1−3 1,2 3,0+ 5g2
1,1 1,2+ 2g2
1,1 2
1,1+ 2 0,2 1,1g1,1g2,0−2 2,2 2,0−2 2
1,1g2,1
−3 3,1g0,2−2 1,1g2
0,2g2,0+ 3 1,1 2,0g0,2g1,1+ 4 2,1 2,0g0,2−3 0,2 1,1g3,0−2 1,2g0,2g2,0
+ 4 2,2g1,1+ 3,2+ 2g3
1,1 0,2+ 2 0,2g1,1g2,1−4 1,1 2
2,0g0,2−4 1,1g0,2g2,1+ 6 2
1,1g0,2g2,0
−3 1,1 3,1−2 1,1 2,0g1,2−2 1,1 1,2g2,0−4 4,0 0,2−2g2
1,1 0,2 2,0+ 2 1,1g2,2)α14
+ (4 1,1 2,0 2,1+ 4g2
1,1 1,1g0,2+ 11 1,1g0,2 3,0−12 0,2 3,0g1,1+ 6 2
1,1 2,0g1,1
−10 1,2 2,0g1,1−8 1,1g1,1 2,1+ 2 1,2g2,1−3 1,2 3,0+ 4 1,2 2
2,0+ 3 3
1,1g2,0+ 10g2
1,1 1,2
+g2
1,1 2
1,1+ 6 2
1,1 3,0−4 0,2 1,1g1,1g2,0−2 2,2 2,0−7 2
1,1g2,1+ 12 3,0 2,0 0,2
+ 4 0,2 2,0g0,2g2,0− 3,1g0,2−2 0,2g0,2g3,0−6 1,1g2
0,2g2,0+ 8 0,2 1,1 2,0g2,0+ 6 2,1 2,0g0,2
−6 0,2 1,1g3,0−4 1,2g0,2g2,0+ 6 2,2g1,1+ 3,2+ 6g3
1,1 0,2+ 3 1,1g0,3g2,0+ 6 0,2g1,1g2,1
+ 6 0,3g1,1g2,0−8 1,1 2
2,0g0,2−3 1,1g0,2g2,1+ 5 1,1g1,1g1,2+ 8 0,2 2
2,0g1,1+ 11 2
1,1g0,2g2,0
−5 1,1 3,1−5 1,1 2,0g1,2+ 3 1,3g2,0−8 1,1 1,2g2,0−4 4,0 0,2−12g2
1,1 0,2 2,0+ 2 0,2g3,1
−4 0,2 2,0g2,1+ 3 1,1g2,2)α13 + (6 1,1 2,0 2,1+ 9g2
1,1 1,1g0,2+ 7 1,1g0,2 3,0−24 0,2 3,0g1,1
+ 19 2
1,1 2,0g1,1−16 1,2 2,0g1,1−15 1,1g1,1 2,1+ 4 1,2g2,1−5 1,2 3,0−8 0,2 3
2,0+ 4 1,2 2
2,0
+ 12 3
1,1g2,0+ 15g2
1,1 1,2−5g2
1,1 2
1,1+ 12 2
1,1 3,0+ 3 0,3g3,0−33 0,2 1,1g1,1g2,0−2 2,2 2,0
−13 2
1,1g2,1+ 12 3,0 2,0 0,2+ 12 0,2 2,0g0,2g2,0+ 2 3,1g0,2−4 0,2g0,2g3,0−8 1,1g2
0,2g2,0
−6 0,2 2,1g2,0−8 1,1g2
0,2g2,0−6 0,2 2,1g2,0+ 26 0,2 1,1 2,0g2,0−14 1,1 2,0g0,2g1,1
+ 2 2,1 2,0g0,2−15 0,2 1,1g3,0−2 1,2g0,2g2,0−8 1,1g2
0,2g2,0−6 0,2 2,1g2,0+ 2 2,1 2,0g0,2
+ 26 0,2 1,1 2,0g2,0−14 1,1 2,0g0,2g1,1−15 0,2 1,1g3,0−2 1,2g0,2g2,0+ 2 2,2g1,1−2 3,2
+ 16g3
1,1 0,2+ 6 1,1g0,3g2,0+ 14 0,2g1,1g2,1+ 12 0,3g1,1g2,0−14 1,1 2
2,0g0,2+ 3 1,1g0,2g2,1
20
+ 9 1,1g1,1g1,2+ 20 0,2 2
2,0g1,1−6 2
1,1 2
2,0+ 10 2
1,1g0,2g2,0−7 1,1 3,1−9 1,1 2,0g1,2
+ 3 1,3g2,0−19 1,1 1,2g2,0−6 0,3 2,0g2,0−4 4,0 0,2−30g2
1,1 0,2 2,0+ 2 0,2g3,1
−8 0,2 2,0g2,1+ 4 0,2g0,2g1,1g2,0+ 1,1g2,2)α12 + (8 1,1 2,0 2,1+ 15g2
1,1 1,1g0,2
+ 2 1,1g0,2 3,0−18 0,2 3,0g1,1+ 40 2
1,1 2,0g1,1−12 1,2 2,0g1,1−15 1,1g1,1 2,1−2 1,2g2,1
+ 4 1,2 3,0−8 0,2 3
2,0+ 6 1,2 2
2,0+ 29 3
1,1g2,0+ 11g2
1,1 1,2−16g2
1,1 2
1,1+ 21 2
1,1 3,0
+ 3 0,3g3,0−74 0,2 1,1g1,1g2,0+ 4 2,2 2,0−16 2
1,1g2,1+ 16 3,0 2,0 0,2+ 12 0,2 2,0g0,2g2,0
+ 5 3,1g0,2−6 0,2g0,2g3,0−8 1,1g2
0,2g2,0−6 0,2 2,1g2,0+ 70 0,2 1,1 2,0g2,0
−35 1,1 2,0g0,2g1,1−4 2,1 2,0g0,2−19 0,2 1,1g3,0−4 2,2g1,1+ 4g1,2 0,2g2,0−3 3,2
+ 24g3
1,1 0,2+ 9 1,1g0,3g2,0+ 14 0,2g1,1g2,1+ 18 0,3g1,1g2,0−8 1,1 2
2,0g0,2+ 6 1,1g0,2g2,1
+ 10 1,1g1,1g1,2+ 40 0,2 2
2,0g1,1−10 2
1,1 2
2,0−3 2
1,1g0,2g2,0−8 1,1 2,0g1,2+ 3 1,3g2,0
−25 1,1 1,2g2,0−12 0,3 2,0g2,0+ 4 4,0 0,2−54g2
1,1 0,2 2,0+ 2 0,2g3,1−12 0,2 2,0g2,1
+ 10 0,2g0,2g1,1g2,0−2 1,1g2,2)α11 + (−2 1,1 2,0 2,1+ 22g2
1,1 1,1g0,2−9 1,1g0,2 3,0
−10 0,2 3,0g1,1+ 54 2
1,1 2,0g1,1+ 2 1,2 2,0g1,1−8 1,1g1,1 2,1−2 1,2g2,1+ 5 1,2 3,0
−12 0,2 3
2,0−6 1,2 2
2,0+ 46 3
1,1g2,0+ 4g2
1,1 1,2−25g2
1,1 2
1,1+ 13 2
1,1 3,0+ 3 0,3g3,0
−115 0,2 1,1g1,1g2,0+ 6 2,2 2,0−9 2
1,1g2,1−8 3,0 2,0 0,2+ 4 0,2 2,0g0,2g2,0+ 5 3,1g0,2
+ 2 0,2g0,2g3,0−6 0,2 2,1g2,0+ 98 0,2 1,1 2,0g2,0−52 1,1 2,0g0,2g1,1−10 2,1 2,0g0,2
−17 0,2 1,1g3,0+ 8 1,2g0,2g2,0−10 2,2g1,1+ 4g1,2 0,2g2,0−3 3,2+ 32g3
1,1 0,2
+ 6 1,1g0,3g2,0+ 12 0,2g1,1g2,1−8 2
0,2g2
2,0+ 15 0,3g1,1g2,0+ 9 1,1g0,2g2,1
+ 5 1,1g1,1g1,2+ 40 0,2 2
2,0g1,1−16 2
1,1 2
2,0−20 2
1,1g0,2g2,0+ 7 1,1 3,1−3 1,1 2,0g1,2
−3 1,3g2,0−19 1,1 1,2g2,0−12 0,3 2,0g2,0+ 8 4,0 0,2−62g2
1,1 0,2 2,0−2 0,2g3,1
−4 0,2 2,0g2,1+ 24 0,2g0,2g1,1g2,0−5 1,1g2,2)α10 + (−10 1,1 2,0 2,1+ 21g2
1,1 1,1g0,2
−11 1,1g0,2 3,0+ 14 0,2 3,0g1,1+ 53 2
1,1 2,0g1,1+ 16 1,2 2,0g1,1+ 5 1,1g1,1 2,1
−8 1,2g2,1+ 11 1,2 3,0+ 4 0,2 3
2,0−8 1,2 2
2,0+ 53 3
1,1g2,0−9g2
1,1 1,2−29g2
1,1 2
1,1
+ 2 2
1,1 3,0−6 0,3g3,0−130 0,2 1,1g1,1g2,0+ 6 2,2 2,0+ 2
1,1g2,1−16 3,0 2,0 0,2
−24 0,2 2,0g0,2g2,0+ 2 3,1g0,2+ 6 0,2g0,2g3,0+ 4 1,1g2
0,2g2,0+ 98 0,2 1,1 2,0g2,0
−56 1,1 2,0g0,2g1,1−10 2,1 2,0g0,2−3 0,2 1,1g3,0+ 4 1,2g0,2g2,0−10 2,2g1,1
+ 4g1,2 0,2g2,0+ 28g3
1,1 0,2−4 0,2g1,1g2,1−8 2
0,2g2
2,0+ 3 0,3g1,1g2,0+ 14 1,1 2
2,0g0,2
+ 3 1,1g0,2g2,1−3 1,1g1,1g1,2+ 28 0,2 2
2,0g1,1−4 2
1,1 2
2,0−38 2
1,1g0,2g2,0+ 13 1,1 3,1
+ 5 1,1 2,0g1,2−6 1,3g2,0− 1,1 1,2g2,0+ 8 4,0 0,2−56g2
1,1 0,2 2,0−4 0,2g3,1
+ 4 0,2 2,0g2,1+ 26 0,2g0,2g1,1g2,0−5 1,1g2,2)α9+ (−16 1,1 2,0 2,1+ 17g2
1,1 1,1g0,2
−10 1,1g0,2 3,0+ 22 0,2 3,0g1,1+ 36 2
1,1 2,0g1,1+ 24 1,2 2,0g1,1+ 15 1,1g1,1 2,1
−2 1,2g2,1+ 2 1,2 3,0+ 8 0,2 3
2,0−14 1,2 2
2,0+ 45 3
1,1g2,0−13g2
1,1 1,2−22g2
1,1 2
1,1
−17 2
1,1 3,0−6 0,3g3,0−103 0,2 1,1g1,1g2,0+ 14 2
1,1g2,1−24 3,0 2,0 0,2− 3,1g0,2
−28 0,2 2,0g0,2g2,0+ 10 0,2g0,2g3,0+ 10 1,1g2
0,2g2,0+ 6 0,2 2,1g2,0+ 58 0,2 1,1 2,0g2,0
−41 1,1 2,0g0,2g1,1−4 2,1 2,0g0,2+ 16 0,2 1,1g3,0+ 6 1,2g0,2g2,0−4 2,2g1,1+ 3 3,2
+ 22g3
1,1 0,2−6 1,1g0,3g2,0−10 0,2g1,1g2,1−8 2
0,2g2
2,0−9 0,3g1,1g2,0+ 16 1,1 2
2,0g0,2
21

−8 1,1g1,1g1,2−4 0,2 2
2,0g1,1+ 6 2
1,1 2
2,0−36 2
1,1g0,2g2,0+ 10 1,1 3,1+ 10 1,1 2,0g1,2
−6 1,3g2,0+ 20 1,1 1,2g2,0+ 18 0,3 2,0g2,0+ 4 4,0 0,2−26g2
1,1 0,2 2,0−4 0,2g3,1
+ 16 0,2 2,0g2,1+ 36 0,2g0,2g1,1g2,0−2 1,1g2,2)α8+ (−10 1,1 2,0 2,1+ 8g2
1,1 1,1g0,2
−3 1,1g0,2 3,0+ 24 0,2 3,0g1,1+ 12 2
1,1 2,0g1,1+ 14 1,2 2,0g1,1+ 16 1,1g1,1 2,1
−2 1,2g2,1− 1,2 3,0+ 16 0,2 3
2,0−2 1,2 2
2,0+ 25 3
1,1g2,0−14g2
1,1 1,2−13g2
1,1 2
1,1
−21 2
1,1 3,0−6 0,3g3,0−62 0,2 1,1g1,1g2,0−6 2,2 2,0+ 15 2
1,1g2,1−12 3,0 2,0 0,2
−28 0,2 2,0g0,2g2,0−3 3,1g0,2+ 6 1,1g2
0,2g2,0+ 6 0,2 2,1g2,0−22 1,1 2,0g0,2g1,1
+ 2 2,1 2,0g0,2+ 21 0,2 1,1g3,0−4 1,2g0,2g2,0+ 2 2,2g1,1−4g1,2 0,2g2,0+ 3 3,2
+ 10g3
1,1 0,2−9 1,1g0,3g2,0−18 0,2g1,1g2,1−8 2
0,2g2
2,0−21 0,3g1,1g2,0+ 12 1,1 2
2,0g0,2
−5 1,1g0,2g2,1−9 1,1g1,1g1,2−24 0,2 2
2,0g1,1+ 20 1,12 2
2,0−29 2
1,1g0,2g2,0+ 1,1 3,1
+ 9 1,1 2,0g1,2−3 1,3g2,0+ 26 1,1 1,2g2,0+ 18 0,3 2,0g2,0−4 4,0 0,2−4g2
1,1 0,2 2,0
−2 0,2g3,1+ 12 0,2 2,0g2,1+ 22 0,2g0,2g1,1g2,0+ 1,1g2,2)α7+ (2 1,1 2,0 2,1
+ 3g2
1,1 1,1g0,2+ 1,1g0,2 3,0+ 12 0,2 3,0g1,1− 2
1,1 2,0g1,1+ 4 1,2 2,0g1,1
+ 11 1,1g1,1 2,1+ 4 1,2g2,1−7 1,2 3,0+ 8 0,2 3
2,0+ 4 1,2 2
2,0+ 8 3
1,1g2,0−7g2
1,1 1,2
−3g2
1,1 2
1,1−16 2
1,1 3,0+ 3 0,3g3,0−14 0,2 1,1g1,1g2,0−6 2,2 2,0+ 13 2
1,1g2,1
+ 4 3,0 2,0 0,2−2 3,1g0,2−2 0,2g0,2g3,0+ 4 1,1g2
0,2g2,0+ 6 0,2 2,1g2,0
−22 0,2 1,1 2,0g2,0+ 6 2,1 2,0g0,2+ 20 0,2 1,1g3,0−2 1,2g0,2g2,0+ 6 2,2g1,1
+ 2 3,2+ 4g3
1,1 0,2−6 1,1g0,3g2,0−10 0,2g1,1g2,1−15 0,3g1,1g2,0+ 2 1,1 2
2,0g0,2
−4 1,1 2,0g0,2g1,1−4g1,2 0,2g2,0−3 1,1g0,2g2,1−5 1,1g1,1g1,2−28 0,2 2
2,0g1,1
+ 16 2
1,1 2
2,0−10 2
1,1g0,2g2,0−5 1,1 3,1+ 5 1,1 2,0g1,2+ 3 1,3g2,0+ 23 1,1 1,2g2,0
+ 12 0,3 2,0g2,0−4 4,0 0,2+ 14g2
1,1 0,2 2,0+ 2 0,2g3,1+ 8 0,2 2,0g2,1+ 3 1,1g2,2
+ 16 0,2g0,2g1,1g2,0)α6+ (8 1,1 2,0 2,1+ 2 1,1g0,2 3,0+ 2 0,2 3,0g1,1−6 2
1,1 2,0g1,1
−8 1,2 2,0g1,1+ 3 1,1g1,1 2,1+ 2 1,2g2,1−4 1,2 3,0+ 10 1,2 2
2,0−3 3
1,1g2,0
−3g2
1,1 1,2−5 2
1,1 3,0+ 3 0,3g3,0−4 2,2 2,0+ 4 2
1,1g2,1+ 8 3,0 2,0 0,2− 3,1g0,2
+ 4 0,2 2,0g0,2g2,0−4 0,2g0,2g3,0−30 0,2 1,1 2,0g2,0+ 1,1 2,0g0,2g1,1+ 4 2,1 2,0g0,2
+ 6 0,2 1,1g3,0−4 1,2g0,2g2,0+ 4 2,2g1,1−4g1,2 0,2g2,0− 3,2−3 1,1g0,3g2,0
−6 0,2g1,1g2,1−9 0,3g1,1g2,0−4 1,1 2
2,0g0,2−2 1,1g0,2g2,1−2 1,1g1,1g1,2+ 6 2
1,1 2
2,0
−16 0,2 2
2,0g1,1−3 2
1,1g0,2g2,0−8 1,1 3,1+ 3 1,3g2,0+ 7 1,1 1,2g2,0−6 0,3 2,0g2,0
−4 4,0 0,2+ 10g2
1,1 0,2 2,0+ 2 0,2g3,1−4 0,2 2,0g2,1+ 2 1,1g2,2)α5+ (10 1,1 2,0 2,1
+ 1,1g0,2 3,0−2 0,2 3,0g1,1−6 1,2 2,0g1,1+ 2 1,2g2,1− 1,2 3,0−4 0,2 3
2,0
+ 6 1,2 2
2,0−2 3
1,1g2,0+g2
1,1 2
1,1+ 3 2
1,1 3,0+ 3 0,3g3,0+ 6 0,2 1,1g1,1g2,0+ 2 2,2 2,0
+ 2
1,1g2,1+ 8 3,0 2,0 0,2+ 8 0,2 2,0g0,2g2,0−4 0,2 1,1 2,0g2,0+ 2 1,1 2,0g0,2g1,1
+ 2 2,1 2,0g0,2+ 2 0,2 1,1g3,0+ 2 2,2g1,1− 3,2−4 1,1 2
2,0g0,2−4 2
1,1 2
2,0+ 2 2
1,1g0,2g2,0
−3 1,1 3,1− 1,1 2,0g1,2+ 3 1,3g2,0+ 1,1 1,2g2,0−6 0,3 2,0g2,0+ 6g2
1,1 0,2 2,0
+ 2 0,2g3,1−4 0,2 2,0g2,1+ 1,1g2,2)α4+ (− 3
1,1g2,0+ 4 0,2 2
2,0g1,1− 1,1 2,0g1,2
−8 2
1,1 2
2,0−2 1,1 2
2,0g0,2−2 0,2 1,1g3,0+ 1,2 3,0−4 1,2 2,0g1,1+ 2 2,2 2,0
22
− 1,1g1,1 2,1+ 2 2
1,1 3,0+ 2
1,1 2,0g1,1− 3,2− 2
1,1g2,1−6 0,3 2,0g2,0−4 0,2 2,0g2,1
−3 1,1 1,2g2,0−2 0,2 3,0g1,1−4 0,2 3
2,0− 1,1 3,1+ 2 1,1 2,0 2,1)α3+ (−2 2
1,1 2
2,0
+ 4 0,2 2
2,0g1,1+ 1,2 3,0+ 3
1,1g2,0+ 2
1,1 3,0+ 4 0,2 1,1 2,0g2,0+ 2 2,2 2,0+ 1,1 3,1
−2 1,2 2
2,0+ 2 2
1,1 2,0g1,1)α2+ (−2 1,2 2
2,0−2 1,1 2,0 2,1− 2
1,1 3,0)α+ 2 2
1,1 2
2,0.
Appendix B. Exp ession o P7(F) and P8(F) when β= 1
Applying he No mal Fo m Algo i hm o he map (2) we ge ha , when α36= 1, hen
(4)
1,4=P7(F)/(α3−1) and g(4)
0,5=P8(F)/(α3−1) whe e
P7(F) := 1,4α3+ (3 0,3g1,2−3 1,4−2 0,3 2,1−2 2,0 0,4+ 2g1,3 0,2−2 2,2 0,2
+ 4 0,4g1,1)α2+ (−4g1,3 0,2+ 4 0,3 2,1−8 0,4g1,1−6 0,3g1,2−4g2,1 2
0,2
−10 0,3 0,2g2,0+ 3 1,4+ 3 3,0 2
0,2+ 4 2,0 0,4+ 4 2,2 0,2)α− 1,4−2 2,2 0,2
+ 2g1,3 0,2+ 4g2,1 2
0,2−2 2,0 0,4+ 2 2,02 0,22−3 3,0 2
0,2−8g1,1 2,0 2
0,2
+ 8 2
0,2g2
1,1+ 4 0,4g1,1+ 3 0,3g1,2+ 10 0,3 0,2g2,0−2 0,3 2,1,
P8(F) := g0,5α3+ (−3g0,5− 0,4g1,1−g1,3 0,2− 0,3g1,2)α2+ (2 0,3g1,2+ 2 0,4g1,1
+g2,1 2
0,2+ 2 0,3 0,2g2,0+ 2g1,3 0,2+ 3g0,5)α− 0,3g1,2−g0,5−2 0,3 0,2g2,0
− 0,4g1,1+g1,1 2,0 2
0,2−g1,3 0,2−g2,1 2
0,2−2 2
0,2g2
1,1.
Appendix C. Exp ession o C2(B, λ) in he p oo o Theo em 7
C2(B, λ) := 8B12λ25 + 125B12λ24 + 912B12λ23 + 4140B12λ22 + 13091B12λ21 + 23B9λ24
+30388B12λ20 + 264B9λ23 + 52493B12λ19 + 1457B9λ22 + 64792B12λ18
+5130B9λ21 + 44963B12λ17 + 12792B9λ20 + 17B6λ23 −22114B12λ16
+23399B9λ19 + 152B6λ22 −126694B12λ15 + 30518B9λ18 + 685B6λ21
−230443B12λ14 + 23012B9λ17 + 2027B6λ20 −285544B12λ13 −7945B9λ16
+4241B6λ19 −265465B12λ12 −59005B9λ15 + 6222B6λ18 + 23B3λ21
−182980B12λ11 −111409B9λ14 + 5530B6λ17 + 126B3λ20 −80299B12λ10
−140407B9λ13 −138B6λ16 + 356B3λ19 −280B12λ9−131599B9λ12
−10552B6λ15 + 644B3λ18 + 37544B12λ8−90967B9λ11 −21809B6λ14
+723B3λ17 + 40086B12λ7−40111B9λ10 −28180B6λ13 + 253B3λ16 + 8λ19
+26571B12λ6−883B9λ9−26229B6λ12 −844B3λ15 + 29λ18 + 12701B12λ5
+17318B9λ8−17474B6λ11 −2101B3λ14 + 36λ17 + 4481B12λ4+ 18449B9λ7
−6870B6λ10 −2804B3λ13 + 34λ16 + 1143B12λ3+ 12036B9λ6+ 902B6λ9
−2563B3λ12 −33λ15 + 199B12λ2+ 5561B9λ5+ 4012B6λ8−1532B3λ11 −71λ14
+21B12λ+ 1827B9λ4+ 3635B6λ7−364B3λ10 −137λ13 +B12 + 404B9λ3
+2079B6λ6+ 335B3λ9−92λ12 + 53B9λ2+ 839B6λ5+ 493B3λ8−56λ11
+3B9λ+ 232B6λ4+ 348B3λ7+ 8λ10 + 37B6λ3+ 149B3λ6+ 29λ9+ 2B6λ2
+35B3λ5+ 25λ8+ 3B3λ4+ 9λ7+λ6.
23
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