Global periodicity conditions for maps and recurrences via Normal Forms

arXiv:1205.0923v1 [math.DS] 4 May 2012 Global periodicity conditions for maps and recurrences via Normal Forms Anna Cima(1), Armengol Gasull(1) and V´ıctor Ma˜nosa (2) (1) Dept. de Matem`atiques, Facultat de Ci`encies, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain [email protected]b.cat, ga[email protected]t (2) Dept. de Matem`atica Aplicada III (MA3), Control, Dynamics and Applications Group (CoDALab) Universitat Polit`ecnica de Catalunya (UPC) Colom 1, 08222 Terrassa, Spain [email protected] May 7, 2012 Abstract We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of Kk, where Kis Ror C, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values pfor which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences. 2000 Mathematics Subject Classification: 37G05, 39A11, 39A20, 37C05 Keywords: Periodic maps; Linearization; Normal Forms; Rational parametrizations; Globally periodic recurrences; Lyness recurrences. 1 1 Introduction A map Fsuch that Fp(x)≡x, for some p∈Nand for all xfor which Fpis well defined, will be called a periodic map. If pis the smallest positive integer with this property, then Fis called p-periodic. In this paper we treat the problem of characterizing the p-periodic cases in parametric families of rational maps of Kk, where Kis Ror C, having a fixed point. When Fis a p-periodic differentiable map having a fixed point, x0, it is well-known that (DF (x0))p= Id. In fact this is a simple consequence of the chain rule. As we will see in Proposition 10, m=pis the smallest positive integer number such that (DF(x0))m= Id. This simple result allows to treat in a easy way the periodicity problem when a value p such that (DF(x0))p= Id is known. For instance if Fhas a fixed point x0such that (DF (x0))2= Id then if Fis p-periodic then pmust be 2,and not p= 2m, m ∈Nas we could think in principle, and then we simply have to check whether F2= Id or not. In general, given a parametric family of maps Fa,a∈Km, the most difficult problem for finding the periodic maps is to determine which are the possible values psuch that there exists some asuch that Fais p-periodic. The tools that we will introduce in this paper will allow to find a finite set of possible values of pfor which the map can be p-periodic, converting the problem of finding these values of ainto a computational problem. Proposition 10 as well as our approach to the characterization of p-periodic maps via Normal Form Theory are based on the Montgomery-Bochner Theorem, see [23]. It will be recalled and proved in Section 2. In a few words it says that any p-periodic, C1-map with a fixed point is locally conjugated with the linear map L(x) = DF(x0)x, and so locally linearizable. Notice that the differentiability condition is necessary since it is well known that there are periodic involutions (i.e. F2= Id) given by homemorphisms with fixed points which are not linearizable, see [8]. Hence any p-periodic case in a given family with fixed points can be locally linearized. Thus, the application of a suitable Normal Form algorithm, will give necessary conditions for the existence of the linearization. As we will see, these conditions are sometimes also sufficient. We remark that this approach does not cover the problem in its full generality, because there are periodic diffeomorphisms without fixed points in Rkwith k≥7, see [18, 20]. It is well-known that the Normal Form algorithms often lead to very complicated expressions which are difficult to handle when dealing with the given parameters of the map. Sometimes, these obstructions can be significatively softened by introducing new parameters rationally depending on the old ones, and such that the coordinates of the fixed points as well as the eigenvalues of the jacobian matrix at these fixed points, depend rationally on these new parameters. This is the second main characteristic of our approach, when 2 dealing with concrete applications. The Normal Form Theory is briefly recalled in Section 3. In Section 4 we obtain some results for planar maps in the case that the linear part of Fat the fixed point is given by a matrix diag(α, β) with αβ = 1, or diag(α, 1). As first applications of the method, we get: Theorem 1. Consider a smooth complex map of the form F(x, y) =  αx +X i+j≥2 fi,jxiyj,1 αy+X i+j≥2 gi,jxiyj ,(1) where αis a primitive p-root of unity, p≥5. Then the conditions P1(F) = P2(F) = P3(F) = 0 are necessary for Fto be p-periodic, where P1(F) := (f2,1+f1,1g1,1)α4−f1,1(2f2,0−g1,1)α3+ (2g2,0f0,2−f1,1f2,0+f1,1g1,1)α2 −(f2,1+f1,1f2,0)α+f1,1f2,0, P2(F) :=g0,2g1,1α4−(g1,2+g0,2g1,1)α3+ (f1,1g1,1+ 2g2,0f0,2−g0,2g1,1)α2 +g1,1(−2g0,2+f1,1)α+f1,1g1,1+g1,2, and P3(F)is given in Appendix A. In fact, conditions P1(F) = 0 and P2(F) = 0 also work for p= 4. Theorem 2. Consider a smooth complex map of the form F(x, y) =  αx +X i+j≥2 fi,jxiyj, y +X i+j≥2 gi,jxiyj ,(2) where αis a primitive p-root of unity. Then the following are necessary conditions for F to be p-periodic: P1(F) :=f1,1= 0, P2(F) :=g0,2= 0, P3(F) :=f1,2α−2f2,0f0,2+ 2f0,2g1,1−f1,2= 0, P4(F) :=g0,3α−g0,3−f0,2g1,1= 0, P5(F) :=f1,3α2+ (−2f2,0f0,3+ 3f0,3g1,1+ 2g1,2f0,2−2f2,1f0,2−2f1,3)α+f1,3+ 2f2,0f0,3 + 2f2,1f0,2−4g2,0f2 0,2−2g1,2f0,2−3f0,3g1,1= 0, P6(F) :=g0,4α2−(f0,3g1,1+ 2g0,4+g1,2f0,2)α+g2,0f2 0,2+f0,3g1,1+g0,4+g1,2f0,2= 0. In this last case, and in contrast with the one treated in Theorem 1, it is not difficult to obtain additional periodicity conditions. Two more periodicity conditions are given in Appendix B. 3 The above results are applied in several contexts. The first application is for polynomial maps. Periodic polynomial maps are notorious examples of invertible polynomial ones, which, in turn, are the focus of many deep open problems like the Jacobian conjecture, or the linearization conjecture. This second conjecture says that if F:Cn→Cnis a pperiodic polynomial map, then there exists a polynomial automorphism ϕ(i.e. an invertible polynomial map with polynomial inverse) such that ϕ◦F◦ϕ−1is a linear map. This conjecture is true for n= 2 and as far as we know it is open for n≥3, see [15, Chaps. 8 and 9] and [21]. In Section 5 we characterize the p-periodic maps in a family of triangular maps, see Theorem 15, and we give a simple and self-contained proof of the linearization conjecture for this case. As an application of this result and Theorem 1 we prove: Proposition 3. Consider a complex polynomial map F(x, y) =  αx + 3 X i+j=2 fi,jxiyj, y/α + 3 X i+j=2 gi,jxiyj ,(3) The map is p-periodic if and only if αis a primitive p-root of the unity, and it holds one of the following conditions (i) p= 1 and F(x, y) = (x, y); (ii) p= 2,4and F(x, y) = (αx +f0,2y2, y/α)or F(x, y) = (αx, y/α +g2,0x2); (iii) p= 3, and F(x, y) = (αx +f0,3y3, y/α)or F(x, y) = (αx, y/α +g3,0x3); (iv) p≥5and F(x, y) = (αx+f0,2y2+f0,3y3, y/α)or F(x, y) = (αx, y/α+g2,0x2+g3,0x3). Similarly, as an application of Theorem 2, we prove: Proposition 4. The only p-periodic cases in the family of complex maps F(x, y) = αx +bx2+cxy +dy2 1 + m(x2+y2),y+rx2+sxy +ty2 1 + m(x2+y2), are, either F(x, y) = (x, y)when α= 1, or the ones given the polynomial maps F(x, y) = (αx +dy2, y)or F(x, y) = (αx, y +rx2)when αa primitive p-root of the unity with p > 1 and dand rarbitrary complex numbers. In all the rest of examples, given in Sections 6 and 7, the maps are the ones associated to some recurrences. Recall that given a recurrence, autonomous or not, it is said that it is globally p-periodic if for all initial conditions for which the sequence is well-defined it gives rise to a p-periodic sequence and pis the smallest positive integer number with this 4 property. We will face this question studying an associated map F. With this point of view, the recurrence will be globally periodic if and only if the map Fis periodic. The study of the global periodicity in difference equations is nowadays the subject of an active research, see for instance [1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 22, 25, 26], and references therein and several techniques have been used to approach the problem. To the best of our knowledge, this is the first time that the Normal Form Theory is used in this setting. As a second application of Theorem 1, we classify the globally periodic second order Lyness recurrences, reobtaining the results in [13] for this case: Proposition 5. The only globally periodic Lyness recurrences xn+2 =a+xn+1 xn with a∈C, are the 5-periodic case with a= 1; and the 6-periodic case with a= 0. Also as a direct consequence of Theorem 1 we get next result for some Gumovski-Miratype recurrences [17], Proposition 6. There are no globally periodic cases in the family of Gumovski-Mira recurrences xn+2 =−xn+xn+1 b+x2 n+1 , b ∈C. One of the main applications in this setting concerns the 2-periodic Lyness recurrence xn+2 =an+xn+1 xn ,where an=(afor n= 2ℓ+ 1, bfor n= 2ℓ, (4) and a, b ∈C. In Section 6.3 we solve the global periodicity problem for it by studying the family of maps Fb,a(x, y) = a+y x,a+bx +y xy , which as we will see describes the behavior of (4). Theorem 7. The only globally periodic recurrences in (4) are: (i) The cases a=b= 0 (6-periodic) and a=b= 1 (5-periodic). (ii) The cases a= (−1±i√3)/2and b=a= 1/a,10-periodic. Notice that the cases given in (i) correspond to the well-known autonomous globally periodic Lyness recurrences also appearing in Proposition 5. Finally, to show an application in K3we find the globally periodic third order Lyness recurrences, reobtaining again the result in [13]: Proposition 8. The only globally periodic third-order Lyness recurrence xn+3 =a+xn+1 +xn+2 xn , a ∈C, corresponds to a= 1 and is 8-periodic. 5 2 Some consequences of the Montgomery-Bochner Theorem The next version of Montgomery-Bochner Theorem is a simplified one, adapted to our interests. The general one applies in a much more general context, see [23]. Theorem 9 (Montgomery-Bochner). Let F:U → U be a p-periodic C1-diffeomorphism, where Uis an open set of Kk. Let x0∈ U be a fixed point of F. Then, there exists a neighbourhood of x0where Fis conjugated with the linear map L(x) = DF(x0)x. Moreover the linearization is given by the local diffeomorphism ψ(x) = 1 p p−1 X i=0 (DF (x0))−iFi(x). Proof. Since Fis p-periodic (DF(x0))p= Id. So (det(DF(x0))p= 1 and DF(x0) is invertible. Consider ψas in the statement. By the inverse function theorem it is clear that the map ψis a local diffeomophism because Dψ(x0) = Id. Moreover, using again the p-periodicity of Fwe get that ψ(F(x)) = L(ψ(x)), as we wanted to prove. As we have seen in the proof of the above theorem, if Fis a p-periodic differentiable map with a fixed point x0,then (DF(x0))p= Id. Next result relates pwith the minimum positive msuch that (DF(x0))m= Id. Proposition 10. Let Fbe a differentiable map having a fixed point x0. Assume that Fis p-periodic and let mbe the minimum positive msuch that (DF(x0))m= Id. Then p=m. Proof. By using the Montgomery-Bochner Theorem we know that Fis C1-conjugated to L(x) = DF(x0)xin a neighborhood of x0. Thus F=ψ−1◦L◦ψ, for some C1diffeomorphism ψ. Since Lm= Id if and only if Fm=ψ−1◦Lm◦ψ=ψ−1◦ψ= Id,the result follows. Corollary 11. Let Fa(x) = Lx+G(x,a),with x∈ U ⊂ Knand a∈Km,a smooth family of maps such that G(0,a)≡DxG(0,a)≡0for all a∈Km. Assume that pis the minimum positive integer number such that Lp= Id. Then if Fais periodic for some a∈Kthen it is p-periodic, i.e. Fp a= Id . In particular note that if L= Id then the only periodic case is Fa(x) = xand when L2= Id the periodicity conditions are given by Fa(Fa(x)) ≡x.For example the fact proved in [25, Ex. 2], that the only periodic map of the form F(x1, x2) = (x2+ax2 1, x1+bx1x2) corresponds to the linear case a=b= 0, follows easily using this approach. Notice that using Montgomery-Bochner Theorem a necessary condition for a map of the form Fa(x) = Lx+G(x,a),to be periodic is that Fais linearizable in a neighbourhood of 0. The linearizable cases can be detected by following the well-know Normal Form Theory, which, as far as we know, has not been used for this purpose. Some results useful for applying it will be recalled in the next section. 6 3 Periodicity conditions via Normal Form Theory We start introducing some well-known issues of Normal Form Theory, while referring the reader to [3, Sec. 2.5], for further details. Let F:= F(1) :Kk→Kk, be a family of smooth maps depending on some parameters and satisfying F(1)(0) = 0.Let F(1)(x) = F(1) 1(x) + F(1) 2(x) + ···+F(1) k(x) + O(|x|k+1) (5) be the Taylor expansion of Fat 0,where F(1) r∈ Hr,the real vector space of maps whose components are homogeneous polynomials of degree r. The aim of the Normal Form Theory is to construct a sequence of transformations Φn, starting from n= 2, such that at each step, Φnsimplifies, as much as possible, the terms of the corresponding homogeneous part of degree n. To this end, let F(1) 1(x) = DF(1)(0)x=: Lxand suppose that F(n−1)(x) = Lx+F(n−1) n(x) + O(|x|n+1), n ≥2. Consider a transformation x= Φn(y) := y+φn(y), with φn∈ Hn,such that it conjugates the map F(n−1) with a new map F(n), via the conjugation F(n−1)(Φn) = Φn(F(n)). From the above equation, it can be easily seen that F(n)(y) = Ly+L φn(y)−φn(Ly) + F(n−1) n(y) + O(|y|n+1). Clearly, if φn(y) can be chosen in such a way that ML(φn(y)) := L φn(y)−φn(Ly) = −F(n−1) n(y),(6) then F(n−1) is transformed into F(n)(y) = Ly+F(n) n+1(y) + O(|y|n+2) = Ly+O(|y|n+1). The vectorial equation (6) is the well-known homological equation associated with L= DF(1)(0), and the existence of solutions of it is the necessary and sufficient condition to be able to remove the homogeneous terms of degree n. From now, one we will assume that the linear map is diagonalizable, and so that it is L= diag(λi)k i=1. In this case, the linear operator ML:= Hn→ Hn,given in (6), has the 7 eigenvectors xmei, i = 1,2,...,k, with m= (m1, m2,...,mk)∈ Mk n:= {m∈Nksatisfying Pk i=1 mi=n};xm=xm1 1xm2 2···xmk nwhere x= (x1, x2,...,xk)∈Kk; and where eiis the i-th member of the natural basis for Kk.Hence ML(xmei) = (λi−λm)xmei,(7) where λm=λm1 1λm2 2···λmk n. Set F(n−1) n(x) = X m f(n−1) 1;mxm,X m f(n−1) 2;mxm,...,X m f(n−1) k;mxm!, and φn(x) = X m a1;mxm,X m a2;mxm,...,X m ak;mxm!, where m∈ Mk n. When λi−λm6= 0 for all the suitable values of m∈Nkand for all i= 1,2,...,k, it is said that there are no resonances. In this case the operator MLis invertible, the homological equation always has solution and so the linearization process can continue. On the contrary, if λi−λm= 0 for some m∈ Mk nand some i∈ {1,2,...,k}, then the vector λ= (λ1, λ1,...,λk) is said to be resonant of order n. In this case, by simple inspection of the homological equation, and using (7), we obtain that the the nth order obstruction equation associated to the resonance is given by (λi−λm)ai;m=−f(n−1) i;m. However, there are some maps having this resonance for which the process can continue. This happens if the right-hand side of this scalar equations vanish, namely f(n−1) i;m= 0, and these cases are the ones candidate to be linearized. Hence, we have obtained the following result Proposition 12. If L:= diag(λi)k i=1, then a necessary condition for the map (5) to be periodic is given by the nth order periodicity condition associated to the resonance condition, λi−λm= 0, given by f(n−1) i;m= 0. Remark 13. Notice that p-periodic maps with Ldiagonal are such that λp i= 1,for all i. Therefore for these maps many resonances λi−λm= 0 appear. By following the Normal Form Algorithm, it is straightforward (and well known) to see that the numerator of f(n−1) i;mis a polynomial in the coefficients of F(1). Thus, for each particular case, the above equations give periodicity conditions, which are algebraic in terms of the initial parameters of the map, once expressed in form (5). 8 To fix the ideas we give a simple example. Suppose that k= 2. Assume that L= diag(α, β). Set Φ2(y) := y+φ2(y), where φ2(x, y) := a20x2+a11xy +a02y2 b20x2+b11xy +b02y2!. Consider the map F(1)(x) = Lx+F(1) 2(x) + O(|y|3) with F(1) 2(x, y) = f20x2+f11xy +f02y2 g20x2+g11xy +g02y2!, where to simplify the notation, and from now on, if there is no possibility of confusion, we will drop the superscript (1) of the coefficients of F(1). The homological equation at order 2 is L φ2(y)−φ2(Ly) = −F(1) 2(y), and gives the following six scalar equations:        (α−α2)a20 =−f20,(β−α2)b20 =−g20, (α−αβ)a11 =−f11,(β−αβ)b11 =−g11, (α−β2)a02 =−f02,(β−β2)b02 =−g02. If no one of the six 2nd order resonance conditions:α2−α, αβ −α, β2−α, β2−β, αβ −β, and α2−β, vanish, there is no obstruction to remove the second order terms of F(1) using the conjugation Φ2. Suppose now, that the map F(1) is such that the resonance β−α2= 0 occurs. Then the scalar equation (β−α2)b20 =−g20 is an obstruction equation. But this obstruction to the linearization process disappears if g20 vanishes. In summary, if β=α2, then g20 = 0 is aperiodicity condition. 4 Proof of Theorems 1 and 2 We keep the notation introduced in the above section, i.e., F(k)is the map obtained after k−1 steps of the normal form procedure, F(k)(x) = Lx+F(k) k+1(x) + O(|x|k+2),and its coefficients are f(k) i,j and g(k) i,j . First consider the case treated in Theorem 1: F(k)(x, y) =  αx +X i+j≥k+1 f(k) i,j xiyj,1 αy+X i+j≥k+1 g(k) i,j xiyj . It is easy to check that the scalar equations associated to equation (6) are (α(1 −αn−2i−1)an−i,i =−f(n−1) n−i,i , α−1(1 −αn−2i+1)bn−i,i =−g(n−1) n−i,i , 9 We consider separately the case b= 0.In this situation x0:= (√2/2,√2/2) is a fixed point of G0. It is easy to see that (DG0(x0)p6= Id, for any positive integer p, because the matrix is not diagonalizable. So G0is not a periodic map. When b6= 0 we introduce a new parameter λ, and write b=λ/(1 + λ2) with λ2+ 1 6= 0 and λ6= 0.Notice that this parametrization covers all values of bin C\{0}. We rename the new map corresponding to Gbas gλ. The eigenvalues of its Jacobian matrix at the origin, which is always a fixed point, are λand 1/λ. The linear map Ψ(x, y) = (x−λy, x −y/λ) is a conjugation between Dgλ(0) and its diagonal form L(x, y) := (λx, y/λ). Using this conjugation we consider the map Fλ:= Ψ ◦gλ◦Ψ−1. Using Theorem 1 we impose that P2(Fλ) = 0. We get that a necessary condition for Fλto be periodic is λ2+ 12λ2+λ+ 1= 0. If λis a root of λ2+λ+1, then it is a primitive 3rd-root of the unity. Then by Corollary 11, Fλshould be globally 3-periodic. But we have already discarded this possibility. So the result follows. 6.3 Global periodicity in the 2-periodic non-autonomous Lyness recurrence In this section we study the problem of the global periodicity of the the sequence generated by the 2-periodic Lyness recurrence (4). The sequence {xn}given by this recurrence can be reobtained as (x1, x2)Ga −−→ (x2, x3)Gb −→ (x3, x4)Ga −−→ (x4, x5)Gb −→ (x5, x6)Ga −−→ ··· where Gα(x, y), with α∈ {a, b}, is the Lyness map given in (10). So the behavior of (4) is given by the dynamical system generated by the map: Gb,a(x, y) := Gb◦Ga(x, y) = a+y x,a+bx +y xy .(12) Proof of Theorem 7. As we have seen it suffices to study the perodicity problem for the map (12). It is easy to see that Gp a6= Id for p= 1,2,4. Moreover it is 3-periodic if and only if a=b= 0. Notice that this case corresponds to the globally 6-periodic recurrence. We continue searching p-periodic maps with p≥5. Following similar ideas that in the previous subsections we introduce a more suitable rational parametrization of aand b. We consider a=B3λ2+ 1+λ2B3−1 B(λ+ 1)2, b=−B+ (B2−a)2,with B(λ+ 1) 6= 0 and λ6= 0. (13) 16 Using these new parameters we cover all the values of aand bin C. Moreover the fixed point is (B, B2−a),where ais given in (13), and the eigenvalues of Gb,a at this point are λand 1/λ. After a translation (x, y)→(x−B, y −(B2−a)), which brings the fixed point to the origin, the map Gb,a conjugates, using again xand yas variables, with 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) , with linear part. LB,λ(x, y) = −x+y B,−B(λ+ 1)2x λ+λ2+λ+ 1y λ!. The linear change of variables Ψ(x, y) = x+y, (λ+ 1) Bx +1 + 1 λBygives a conjugation between LB,λ and its diagonal form L(x, y) := (λx, y/λ). Using this conjugation we consider the map FB,λ(x, y) := Ψ ◦gB,λ(x, y)◦Ψ−1(x, y), which satisfies DFB,λ(0,0) = diag (λ, 1/λ). For simplicity, we omit its explicit expression. Recall that λp−16= 0 for p= 1,2,3. By Theorem 1, when p≥5, from both conditions Pi(FB,λ) = 0, i = 1,2,we obtain the same periodicity condition C1(B, λ) = 0, where 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 Theorem 1, we obtain another polynomial restriction C2(B, λ) := P3(FB,λ) = 0. The expression of C2(B, λ) is given in Appendix C. To study the periodicity of FB,λ it suffices to deal with the two conditions C1(B, λ) = 0, C2(B, λ) = 0. Computing R(λ) := Res(C1(B, λ), C2(B, λ); B) we get R(λ) = λ36 (λ−1)24 (λ+ 1)72 λ2+ 16λ2+λ+ 124 S6(λ)T6(λ), where S(λ) = λ4+λ3+λ2+λ+ 1 and T(λ) = 3λ4+ 15λ3+ 20λ2+ 15λ+ 3. Then, a necessary condition for FB,λ to be p-periodic with p≥4 is that λis a primitive p-th of the unity and that either S(λ) = 0 or T(λ) = 0. Let us discard the former possibility. It turns out that Thas two real roots and two complex roots of modulus one. We have to prove that they are not roots of the unity. This can be seen, for instance, proving that T 17 is not divisible by any cyclotomic polynomial. This holds because if it had a cyclotomic polynomial divisor, its degree should be at most 4. The cyclotomic polynomials of degree at most 4 correspond to p∈ {1,2,3,4,5,6,8,10,12}:= D4. This is because these are the cases which correspond to cyclotomic polynomials of degree ϕ(p)≤4, being ϕthe Euler’s function, see for instance [24]. Since Res(T(λ), λp−1; λ)6= 0,for p∈ D4, the result follows. Finally, when S(λ) = 0 notice that λis a primitive 5-th root of the unity. So, by Corollary 11 if FB,λ is p-periodic it should be 5-periodic. Therefore it suffices to study whether F5 B,λ = Id or not, or equivalently whether G5 b,a = Id. Computing the numerator of the first component of G5 b,a(x, y)−(x, y) we get that it writes as a4b(1 −ab)x+O(2), where as usual O(m) denotes terms of degree at least min xand y. Hence only three possibilities for Gb,a to be 5-periodic appear: either a= 0 or b= 0 or ab = 1. The first two cases can easily discarded. It holds that G5 0,a 6= Id and G5 b,06= Id. On the other hand, when b= 1/a, a 6= 0 the numerator of the first component of G5 1/a,a(x, y)−(x, y) writes as −a(a−1)2(a2+a+ 1)2x2y+O(3). Since this last function has to vanish we get three candidates to be 5-periodic: a= 1 and a= (−1±i√3)/2 with b= 1/a =a. It is easy see that all them give rise to 5-periodic maps Gb,a. The last two correspond to the globally 10-periodic recurrence. Remark 17. The characterization of the globally periodic difference equations treated in this section can also be obtained following the approach developed in [27] that gives all the periodic QRT-maps. This result also appears in [14, p. 165] and [19]. 7 The third order Lyness recurrence We start proving a general result which will useful for solving the periodicity problem for the Lyness recurrence. Proposition 18. Consider the smooth family of maps F(x, y, z) = αx +X m fmxm, βy +X m gmxm, γz +X m hmxm!, with m∈ {(i, j, k)such that i+j+k≥2}, and where xm=xiyjzk. When α=±1,βγ = 1, and β6= 1, γ 6= 1,some necessary conditions for it to be periodic are f(2) 3,0,0=f(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 where f(2) i,j,k and g(2) i,j,k are the expressions given in the second step of the normal form procedure described in Section 3. Proof. By inspection of the 3rd order resonance conditions, we observe that when α=±1 and βγ = 1 there appear the resonances α3−α,αβγ −α,α2β−β,β2γ−β,α2γ−γand β2γ−γwhich are associated to the coefficients f(2) 3,0,0,f(2) 1,1,1,g(2) 2,1,0,g(2) 0,2,1,h(2) 2,0,1and h(2) 0,2,1 respectively. So all them must vanish to have a periodic map. Proof of Proposition 8. The dynamics of the third-order Lyness’ equation can be studied through the Lyness maps Ga(x, y, z) = y, z, a+y+z x. It is easy to see that Gp a6= Id for p= 1,2. We continue searching p-periodic maps with p≥3.It has always some fixed point (x0, x0, x0) with x2 0−2x0−a= 0 and x06= 0. Moreover the eigenvalues λof the Jacobian matrix at this points are given by the zeroes of −(λ+ 1)(λ2−(1 + 1/x0)λ+ 1) = 0. These two equations suggest us to introduce the rational parametrization of aas a=−λ2λ2−3λ+ 2 (λ2−λ+ 1)2,with λ2−λ+ 1 6= 0 and λ6= 0, which covers all values of a∈C.Then the fixed point is (x0, x0, x0) with x0=λ/(λ2−λ+1) and the eigenvalues of DGaat this point are −1, λ, 1/λ. Notice that since p≥3, we can assume λ6= 1. To apply Proposition 18 we perform the translation (x, y, z)→(x−x0, y − x0, z −x0), which brings the fixed point to the origin, obtaining gλ(x, y, z) := y, z, −λx +λ2−λ+ 1y+λ2−λ+ 1z (λ2−λ+ 1) x+λ!, with linear part Lλ(x, y) = y, z, −x+λ2−λ+ 1y λ+λ2−λ+ 1z λ!. The linear change of variables Ψ(x, y, z) = (x+y+λ2z, −x+λ y +λ z, x +λ2y+z) gives a conjugation between Lλand its diagonal form L(x, y, z) := (−x, λy, z/λ). Using the conjugation Ψ, we finally obtain a map with diagonal linear part Fλ:= Ψ ◦gλ◦Ψ−1, which is under the assumptions of Proposition 18. Applying this proposition and the Normal Form Algorithm to Fλwe can compute g(2) 2,1,0. From the equation g(2) 2,1,0= 0 we obtain that λ2−λ+ 13(λ4+ 1) = 0. 19 Thus λhas to be a primitive 8-th root of the unity. All these values of λcorrespond to the same value a= 1, which gives a globally 8-periodic recurrence. So the result follows Acknowledgements GSD-UAB and CoDALab Groups are supported by the Government of Catalonia through the SGR program. The first and second authors are also supported by MCYT through grants MTM2008-03437 and the third author by the grant DPI2011-25822. Appendix A. Expression of P3(F) when αβ = 1 Consider the map (1), applying the Normal Form Algorithm one gets that the periodicity condition associated to f(4) 3,2is given by P3(F) := f1,1g0,2g2 1,1α17 + (2g2 1,1f1,1g0,2−f3,1g0,2−f1,1g1,1g1,2−f1,1g0,2g2,1)α16 + (f3,2+ 3g2 1,1f1,1g0,2+ 3f1,1g0,2f3,0−3f1,1g0,2g2,1+ 2f1,1f2,0g0,2g1,1+ 2f2 1,1g0,2g2,0 −2f3,1g0,2+ 2f2,1f2,0g0,2+g2 1,1f1,2+ 2f2,2g1,1+f1,1g2,2+ 2f0,2f1,1g1,1g2,0−f1,1g1,1g1,2 +g2 1,1f2 1,1)α15 + (3g2 1,1f1,1g0,2+ 6f1,1g0,2f3,0−6f0,2f3,0g1,1−4f1,2f2,0g1,1−3f1,1g1,1f2,1 + 2f1,2g2,1−3f1,2f3,0+ 5g2 1,1f1,2+ 2g2 1,1f2 1,1+ 2f0,2f1,1g1,1g2,0−2f2,2f2,0−2f2 1,1g2,1 −3f3,1g0,2−2f1,1g2 0,2g2,0+ 3f1,1f2,0g0,2g1,1+ 4f2,1f2,0g0,2−3f0,2f1,1g3,0−2f1,2g0,2g2,0 + 4f2,2g1,1+f3,2+ 2g3 1,1f0,2+ 2f0,2g1,1g2,1−4f1,1f2 2,0g0,2−4f1,1g0,2g2,1+ 6f2 1,1g0,2g2,0 −3f1,1f3,1−2f1,1f2,0g1,2−2f1,1f1,2g2,0−4f4,0f0,2−2g2 1,1f0,2f2,0+ 2f1,1g2,2)α14 + (4f1,1f2,0f2,1+ 4g2 1,1f1,1g0,2+ 11f1,1g0,2f3,0−12f0,2f3,0g1,1+ 6f2 1,1f2,0g1,1 −10f1,2f2,0g1,1−8f1,1g1,1f2,1+ 2f1,2g2,1−3f1,2f3,0+ 4f1,2f2 2,0+ 3f3 1,1g2,0+ 10g2 1,1f1,2 +g2 1,1f2 1,1+ 6f2 1,1f3,0−4f0,2f1,1g1,1g2,0−2f2,2f2,0−7f2 1,1g2,1+ 12f3,0f2,0f0,2 + 4f0,2f2,0g0,2g2,0−f3,1g0,2−2f0,2g0,2g3,0−6f1,1g2 0,2g2,0+ 8f0,2f1,1f2,0g2,0+ 6f2,1f2,0g0,2 −6f0,2f1,1g3,0−4f1,2g0,2g2,0+ 6f2,2g1,1+f3,2+ 6g3 1,1f0,2+ 3f1,1g0,3g2,0+ 6f0,2g1,1g2,1 + 6f0,3g1,1g2,0−8f1,1f2 2,0g0,2−3f1,1g0,2g2,1+ 5f1,1g1,1g1,2+ 8f0,2f2 2,0g1,1+ 11f2 1,1g0,2g2,0 −5f1,1f3,1−5f1,1f2,0g1,2+ 3f1,3g2,0−8f1,1f1,2g2,0−4f4,0f0,2−12g2 1,1f0,2f2,0+ 2f0,2g3,1 −4f0,2f2,0g2,1+ 3f1,1g2,2)α13 + (6f1,1f2,0f2,1+ 9g2 1,1f1,1g0,2+ 7f1,1g0,2f3,0−24f0,2f3,0g1,1 + 19f2 1,1f2,0g1,1−16f1,2f2,0g1,1−15f1,1g1,1f2,1+ 4f1,2g2,1−5f1,2f3,0−8f0,2f3 2,0+ 4f1,2f2 2,0 + 12f3 1,1g2,0+ 15g2 1,1f1,2−5g2 1,1f2 1,1+ 12f2 1,1f3,0+ 3f0,3g3,0−33f0,2f1,1g1,1g2,0−2f2,2f2,0 −13f2 1,1g2,1+ 12f3,0f2,0f0,2+ 12f0,2f2,0g0,2g2,0+ 2f3,1g0,2−4f0,2g0,2g3,0−8f1,1g2 0,2g2,0 −6f0,2f2,1g2,0−8f1,1g2 0,2g2,0−6f0,2f2,1g2,0+ 26f0,2f1,1f2,0g2,0−14f1,1f2,0g0,2g1,1 + 2f2,1f2,0g0,2−15f0,2f1,1g3,0−2f1,2g0,2g2,0−8f1,1g2 0,2g2,0−6f0,2f2,1g2,0+ 2f2,1f2,0g0,2 + 26f0,2f1,1f2,0g2,0−14f1,1f2,0g0,2g1,1−15f0,2f1,1g3,0−2f1,2g0,2g2,0+ 2f2,2g1,1−2f3,2 + 16g3 1,1f0,2+ 6f1,1g0,3g2,0+ 14f0,2g1,1g2,1+ 12f0,3g1,1g2,0−14f1,1f2 2,0g0,2+ 3f1,1g0,2g2,1 20 + 9f1,1g1,1g1,2+ 20f0,2f2 2,0g1,1−6f2 1,1f2 2,0+ 10f2 1,1g0,2g2,0−7f1,1f3,1−9f1,1f2,0g1,2 + 3f1,3g2,0−19f1,1f1,2g2,0−6f0,3f2,0g2,0−4f4,0f0,2−30g2 1,1f0,2f2,0+ 2f0,2g3,1 −8f0,2f2,0g2,1+ 4f0,2g0,2g1,1g2,0+f1,1g2,2)α12 + (8f1,1f2,0f2,1+ 15g2 1,1f1,1g0,2 + 2f1,1g0,2f3,0−18f0,2f3,0g1,1+ 40f2 1,1f2,0g1,1−12f1,2f2,0g1,1−15f1,1g1,1f2,1−2f1,2g2,1 + 4f1,2f3,0−8f0,2f3 2,0+ 6f1,2f2 2,0+ 29f3 1,1g2,0+ 11g2 1,1f1,2−16g2 1,1f2 1,1+ 21f2 1,1f3,0 + 3f0,3g3,0−74f0,2f1,1g1,1g2,0+ 4f2,2f2,0−16f2 1,1g2,1+ 16f3,0f2,0f0,2+ 12f0,2f2,0g0,2g2,0 + 5f3,1g0,2−6f0,2g0,2g3,0−8f1,1g2 0,2g2,0−6f0,2f2,1g2,0+ 70f0,2f1,1f2,0g2,0 −35f1,1f2,0g0,2g1,1−4f2,1f2,0g0,2−19f0,2f1,1g3,0−4f2,2g1,1+ 4g1,2f0,2g2,0−3f3,2 + 24g3 1,1f0,2+ 9f1,1g0,3g2,0+ 14f0,2g1,1g2,1+ 18f0,3g1,1g2,0−8f1,1f2 2,0g0,2+ 6f1,1g0,2g2,1 + 10f1,1g1,1g1,2+ 40f0,2f2 2,0g1,1−10f2 1,1f2 2,0−3f2 1,1g0,2g2,0−8f1,1f2,0g1,2+ 3f1,3g2,0 −25f1,1f1,2g2,0−12f0,3f2,0g2,0+ 4f4,0f0,2−54g2 1,1f0,2f2,0+ 2f0,2g3,1−12f0,2f2,0g2,1 + 10f0,2g0,2g1,1g2,0−2f1,1g2,2)α11 + (−2f1,1f2,0f2,1+ 22g2 1,1f1,1g0,2−9f1,1g0,2f3,0 −10f0,2f3,0g1,1+ 54f2 1,1f2,0g1,1+ 2f1,2f2,0g1,1−8f1,1g1,1f2,1−2f1,2g2,1+ 5f1,2f3,0 −12f0,2f3 2,0−6f1,2f2 2,0+ 46f3 1,1g2,0+ 4g2 1,1f1,2−25g2 1,1f2 1,1+ 13f2 1,1f3,0+ 3f0,3g3,0 −115f0,2f1,1g1,1g2,0+ 6f2,2f2,0−9f2 1,1g2,1−8f3,0f2,0f0,2+ 4f0,2f2,0g0,2g2,0+ 5f3,1g0,2 + 2f0,2g0,2g3,0−6f0,2f2,1g2,0+ 98f0,2f1,1f2,0g2,0−52f1,1f2,0g0,2g1,1−10f2,1f2,0g0,2 −17f0,2f1,1g3,0+ 8f1,2g0,2g2,0−10f2,2g1,1+ 4g1,2f0,2g2,0−3f3,2+ 32g3 1,1f0,2 + 6f1,1g0,3g2,0+ 12f0,2g1,1g2,1−8f2 0,2g2 2,0+ 15f0,3g1,1g2,0+ 9f1,1g0,2g2,1 + 5f1,1g1,1g1,2+ 40f0,2f2 2,0g1,1−16f2 1,1f2 2,0−20f2 1,1g0,2g2,0+ 7f1,1f3,1−3f1,1f2,0g1,2 −3f1,3g2,0−19f1,1f1,2g2,0−12f0,3f2,0g2,0+ 8f4,0f0,2−62g2 1,1f0,2f2,0−2f0,2g3,1 −4f0,2f2,0g2,1+ 24f0,2g0,2g1,1g2,0−5f1,1g2,2)α10 + (−10f1,1f2,0f2,1+ 21g2 1,1f1,1g0,2 −11f1,1g0,2f3,0+ 14f0,2f3,0g1,1+ 53f2 1,1f2,0g1,1+ 16f1,2f2,0g1,1+ 5f1,1g1,1f2,1 −8f1,2g2,1+ 11f1,2f3,0+ 4f0,2f3 2,0−8f1,2f2 2,0+ 53f3 1,1g2,0−9g2 1,1f1,2−29g2 1,1f2 1,1 + 2f2 1,1f3,0−6f0,3g3,0−130f0,2f1,1g1,1g2,0+ 6f2,2f2,0+f2 1,1g2,1−16f3,0f2,0f0,2 −24f0,2f2,0g0,2g2,0+ 2f3,1g0,2+ 6f0,2g0,2g3,0+ 4f1,1g2 0,2g2,0+ 98f0,2f1,1f2,0g2,0 −56f1,1f2,0g0,2g1,1−10f2,1f2,0g0,2−3f0,2f1,1g3,0+ 4f1,2g0,2g2,0−10f2,2g1,1 + 4g1,2f0,2g2,0+ 28g3 1,1f0,2−4f0,2g1,1g2,1−8f2 0,2g2 2,0+ 3f0,3g1,1g2,0+ 14f1,1f2 2,0g0,2 + 3f1,1g0,2g2,1−3f1,1g1,1g1,2+ 28f0,2f2 2,0g1,1−4f2 1,1f2 2,0−38f2 1,1g0,2g2,0+ 13f1,1f3,1 + 5f1,1f2,0g1,2−6f1,3g2,0−f1,1f1,2g2,0+ 8f4,0f0,2−56g2 1,1f0,2f2,0−4f0,2g3,1 + 4f0,2f2,0g2,1+ 26f0,2g0,2g1,1g2,0−5f1,1g2,2)α9+ (−16f1,1f2,0f2,1+ 17g2 1,1f1,1g0,2 −10f1,1g0,2f3,0+ 22f0,2f3,0g1,1+ 36f2 1,1f2,0g1,1+ 24f1,2f2,0g1,1+ 15f1,1g1,1f2,1 −2f1,2g2,1+ 2f1,2f3,0+ 8f0,2f3 2,0−14f1,2f2 2,0+ 45f3 1,1g2,0−13g2 1,1f1,2−22g2 1,1f2 1,1 −17f2 1,1f3,0−6f0,3g3,0−103f0,2f1,1g1,1g2,0+ 14f2 1,1g2,1−24f3,0f2,0f0,2−f3,1g0,2 −28f0,2f2,0g0,2g2,0+ 10f0,2g0,2g3,0+ 10f1,1g2 0,2g2,0+ 6f0,2f2,1g2,0+ 58f0,2f1,1f2,0g2,0 −41f1,1f2,0g0,2g1,1−4f2,1f2,0g0,2+ 16f0,2f1,1g3,0+ 6f1,2g0,2g2,0−4f2,2g1,1+ 3f3,2 + 22g3 1,1f0,2−6f1,1g0,3g2,0−10f0,2g1,1g2,1−8f2 0,2g2 2,0−9f0,3g1,1g2,0+ 16f1,1f2 2,0g0,2 21 −8f1,1g1,1g1,2−4f0,2f2 2,0g1,1+ 6f2 1,1f2 2,0−36f2 1,1g0,2g2,0+ 10f1,1f3,1+ 10f1,1f2,0g1,2 −6f1,3g2,0+ 20f1,1f1,2g2,0+ 18f0,3f2,0g2,0+ 4f4,0f0,2−26g2 1,1f0,2f2,0−4f0,2g3,1 + 16f0,2f2,0g2,1+ 36f0,2g0,2g1,1g2,0−2f1,1g2,2)α8+ (−10f1,1f2,0f2,1+ 8g2 1,1f1,1g0,2 −3f1,1g0,2f3,0+ 24f0,2f3,0g1,1+ 12f2 1,1f2,0g1,1+ 14f1,2f2,0g1,1+ 16f1,1g1,1f2,1 −2f1,2g2,1−f1,2f3,0+ 16f0,2f3 2,0−2f1,2f2 2,0+ 25f3 1,1g2,0−14g2 1,1f1,2−13g2 1,1f2 1,1 −21f2 1,1f3,0−6f0,3g3,0−62f0,2f1,1g1,1g2,0−6f2,2f2,0+ 15f2 1,1g2,1−12f3,0f2,0f0,2 −28f0,2f2,0g0,2g2,0−3f3,1g0,2+ 6f1,1g2 0,2g2,0+ 6f0,2f2,1g2,0−22f1,1f2,0g0,2g1,1 + 2f2,1f2,0g0,2+ 21f0,2f1,1g3,0−4f1,2g0,2g2,0+ 2f2,2g1,1−4g1,2f0,2g2,0+ 3f3,2 + 10g3 1,1f0,2−9f1,1g0,3g2,0−18f0,2g1,1g2,1−8f2 0,2g2 2,0−21f0,3g1,1g2,0+ 12f1,1f2 2,0g0,2 −5f1,1g0,2g2,1−9f1,1g1,1g1,2−24f0,2f2 2,0g1,1+ 20f1,12f2 2,0−29f2 1,1g0,2g2,0+f1,1f3,1 + 9f1,1f2,0g1,2−3f1,3g2,0+ 26f1,1f1,2g2,0+ 18f0,3f2,0g2,0−4f4,0f0,2−4g2 1,1f0,2f2,0 −2f0,2g3,1+ 12f0,2f2,0g2,1+ 22f0,2g0,2g1,1g2,0+f1,1g2,2)α7+ (2f1,1f2,0f2,1 + 3g2 1,1f1,1g0,2+f1,1g0,2f3,0+ 12f0,2f3,0g1,1−f2 1,1f2,0g1,1+ 4f1,2f2,0g1,1 + 11f1,1g1,1f2,1+ 4f1,2g2,1−7f1,2f3,0+ 8f0,2f3 2,0+ 4f1,2f2 2,0+ 8f3 1,1g2,0−7g2 1,1f1,2 −3g2 1,1f2 1,1−16f2 1,1f3,0+ 3f0,3g3,0−14f0,2f1,1g1,1g2,0−6f2,2f2,0+ 13f2 1,1g2,1 + 4f3,0f2,0f0,2−2f3,1g0,2−2f0,2g0,2g3,0+ 4f1,1g2 0,2g2,0+ 6f0,2f2,1g2,0 −22f0,2f1,1f2,0g2,0+ 6f2,1f2,0g0,2+ 20f0,2f1,1g3,0−2f1,2g0,2g2,0+ 6f2,2g1,1 + 2f3,2+ 4g3 1,1f0,2−6f1,1g0,3g2,0−10f0,2g1,1g2,1−15f0,3g1,1g2,0+ 2f1,1f2 2,0g0,2 −4f1,1f2,0g0,2g1,1−4g1,2f0,2g2,0−3f1,1g0,2g2,1−5f1,1g1,1g1,2−28f0,2f2 2,0g1,1 + 16f2 1,1f2 2,0−10f2 1,1g0,2g2,0−5f1,1f3,1+ 5f1,1f2,0g1,2+ 3f1,3g2,0+ 23f1,1f1,2g2,0 + 12f0,3f2,0g2,0−4f4,0f0,2+ 14g2 1,1f0,2f2,0+ 2f0,2g3,1+ 8f0,2f2,0g2,1+ 3f1,1g2,2 + 16f0,2g0,2g1,1g2,0)α6+ (8f1,1f2,0f2,1+ 2f1,1g0,2f3,0+ 2f0,2f3,0g1,1−6f2 1,1f2,0g1,1 −8f1,2f2,0g1,1+ 3f1,1g1,1f2,1+ 2f1,2g2,1−4f1,2f3,0+ 10f1,2f2 2,0−3f3 1,1g2,0 −3g2 1,1f1,2−5f2 1,1f3,0+ 3f0,3g3,0−4f2,2f2,0+ 4f2 1,1g2,1+ 8f3,0f2,0f0,2−f3,1g0,2 + 4f0,2f2,0g0,2g2,0−4f0,2g0,2g3,0−30f0,2f1,1f2,0g2,0+f1,1f2,0g0,2g1,1+ 4f2,1f2,0g0,2 + 6f0,2f1,1g3,0−4f1,2g0,2g2,0+ 4f2,2g1,1−4g1,2f0,2g2,0−f3,2−3f1,1g0,3g2,0 −6f0,2g1,1g2,1−9f0,3g1,1g2,0−4f1,1f2 2,0g0,2−2f1,1g0,2g2,1−2f1,1g1,1g1,2+ 6f2 1,1f2 2,0 −16f0,2f2 2,0g1,1−3f2 1,1g0,2g2,0−8f1,1f3,1+ 3f1,3g2,0+ 7f1,1f1,2g2,0−6f0,3f2,0g2,0 −4f4,0f0,2+ 10g2 1,1f0,2f2,0+ 2f0,2g3,1−4f0,2f2,0g2,1+ 2f1,1g2,2)α5+ (10f1,1f2,0f2,1 +f1,1g0,2f3,0−2f0,2f3,0g1,1−6f1,2f2,0g1,1+ 2f1,2g2,1−f1,2f3,0−4f0,2f3 2,0 + 6f1,2f2 2,0−2f3 1,1g2,0+g2 1,1f2 1,1+ 3f2 1,1f3,0+ 3f0,3g3,0+ 6f0,2f1,1g1,1g2,0+ 2f2,2f2,0 +f2 1,1g2,1+ 8f3,0f2,0f0,2+ 8f0,2f2,0g0,2g2,0−4f0,2f1,1f2,0g2,0+ 2f1,1f2,0g0,2g1,1 + 2f2,1f2,0g0,2+ 2f0,2f1,1g3,0+ 2f2,2g1,1−f3,2−4f1,1f2 2,0g0,2−4f2 1,1f2 2,0+ 2f2 1,1g0,2g2,0 −3f1,1f3,1−f1,1f2,0g1,2+ 3f1,3g2,0+f1,1f1,2g2,0−6f0,3f2,0g2,0+ 6g2 1,1f0,2f2,0 + 2f0,2g3,1−4f0,2f2,0g2,1+f1,1g2,2)α4+ (−f3 1,1g2,0+ 4f0,2f2 2,0g1,1−f1,1f2,0g1,2 −8f2 1,1f2 2,0−2f1,1f2 2,0g0,2−2f0,2f1,1g3,0+f1,2f3,0−4f1,2f2,0g1,1+ 2f2,2f2,0 22 −f1,1g1,1f2,1+ 2f2 1,1f3,0+f2 1,1f2,0g1,1−f3,2−f2 1,1g2,1−6f0,3f2,0g2,0−4f0,2f2,0g2,1 −3f1,1f1,2g2,0−2f0,2f3,0g1,1−4f0,2f3 2,0−f1,1f3,1+ 2f1,1f2,0f2,1)α3+ (−2f2 1,1f2 2,0 + 4f0,2f2 2,0g1,1+f1,2f3,0+f3 1,1g2,0+f2 1,1f3,0+ 4f0,2f1,1f2,0g2,0+ 2f2,2f2,0+f1,1f3,1 −2f1,2f2 2,0+ 2f2 1,1f2,0g1,1)α2+ (−2f1,2f2 2,0−2f1,1f2,0f2,1−f2 1,1f3,0)α+ 2f2 1,1f2 2,0. Appendix B. Expression of P7(F) and P8(F) when β= 1 Applying the Normal Form Algorithm to the map (2) we get that, when α36= 1, then f(4) 1,4=P7(F)/(α3−1) and g(4) 0,5=P8(F)/(α3−1) where P7(F) := f1,4α3+ (3f0,3g1,2−3f1,4−2f0,3f2,1−2f2,0f0,4+ 2g1,3f0,2−2f2,2f0,2 + 4f0,4g1,1)α2+ (−4g1,3f0,2+ 4f0,3f2,1−8f0,4g1,1−6f0,3g1,2−4g2,1f2 0,2 −10f0,3f0,2g2,0+ 3f1,4+ 3f3,0f2 0,2+ 4f2,0f0,4+ 4f2,2f0,2)α−f1,4−2f2,2f0,2 + 2g1,3f0,2+ 4g2,1f2 0,2−2f2,0f0,4+ 2f2,02f0,22−3f3,0f2 0,2−8g1,1f2,0f2 0,2 + 8f2 0,2g2 1,1+ 4f0,4g1,1+ 3f0,3g1,2+ 10f0,3f0,2g2,0−2f0,3f2,1, P8(F) := g0,5α3+ (−3g0,5−f0,4g1,1−g1,3f0,2−f0,3g1,2)α2+ (2f0,3g1,2+ 2f0,4g1,1 +g2,1f2 0,2+ 2f0,3f0,2g2,0+ 2g1,3f0,2+ 3g0,5)α−f0,3g1,2−g0,5−2f0,3f0,2g2,0 −f0,4g1,1+g1,1f2,0f2 0,2−g1,3f0,2−g2,1f2 0,2−2f2 0,2g2 1,1. Appendix C. Expression of C2(B, λ) in the proof of Theorem 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 References [1] R.M. Abu-Saris. A self-invertibility condition for global periodicity of difference equations. Appl. Math. Lett. 19 (2006), 1078–1082 [2] R.M. Abu-Saris, Q.M. Al-Hassan. On global periodicity of difference equations. J. Math. Anal. Appl. 283 (2003), 468–477. [3] D.K. Arrowsmith, C.M. Place. An introduction to dynamical systems. Cambridge University Press, Cambridge 1990. [4] F. Balibrea, A. Linero. Some new results and open problems on periodicity of difference equations. Grazer Math. Ber. 350 (2006), 15-38. [5] F. Balibrea, A. Linero. On the global periodicity of some difference equations of third order. J. Difference Equ. Appl. 13 (2007), 1011-1027. [6] I. Bajo, E. Liz. Periodicity on discrete dynamical systems generated by a class of rational mappings. J. Difference Equ. Appl. 12 (2006), no. 12, 1201-1212. [7] L. Berg, S. Stevi´c. Periodicity of some classes of holomorphic difference equations. J. Difference Equations and Appl. 12 (2006), 827–835. [8] R.H. Bing. A homeomorphism between the 3-sphere and the sum of two solid horned spheres. Annals of Mathematics. 56 (1952). 354-362. [9] A. Caro, A. Linero. Existence and uniqueness of p-cycles of second and third order. J. Difference Equ. Appl. 15 (2009), 489-500. [10] A. Caro, A. Linero. General cycles of potential form. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2735-2749. [11] A. Cima, A. Gasull, F. Ma˜nosas. On periodic rational difference equations of order k. J. Difference Equ. Appl. 10 (2004), 549–559. [12] A. Cima, A. Gasull, F. Ma˜nosas. Global linearization of periodic difference equations. Discrete Contin. Dynam. Systems A 32 (2012), 1575–1595. [13] M. Cs¨ornyei, M. Laczkovich. Some periodic and non-periodic recursions. Monatsh. Math. 132 (2001), 215-236. [14] J. J. Duistermaat. “Discrete Integrable Systems: QRT Maps and Elliptic Surfaces”. Springer Monographs in Mathematics. Springer, New York, 2010. 24 [15] A. van den Essen. “Polynomial Automorphisms and the Jacobian Conjecture”. Progress in Mathematics 190, Birkh¨auser Verlag, Basel, 2000. [16] E.A. Grove, G. Ladas. “Periodicities in Nonlinear Difference Equations Equations”. Advances in discrete Math. and Appl, vol. 4. Chapman & Hall/CRC Press, Boca Raton FL, 2005. [17] I. Gumovski, Ch. Mira. “Recurrences and discrete dynamic systems”. Lecture Notes in Mathematics 809. Springer Verlag, Berlin, 1980. [18] R. Haynes, S. Kwasik, J. Mast, R. Schultz. Periodic maps R7without fixed points. Math. Proc. Cambridge Philos. Soc. 132 (2002), 131-136. [19] D. Jogia, J.A.G. Roberts, F. Vivaldi. An algebraic geometric approach to integrable maps of the plane. J. Phys. A 39 (2006), 1133-1149. [20] J.M. Kister. Differentiable periodic actions on E8without fixed points. Amer. J. Math. 85 (1963), 316-319. [21] S. Maubach. “Polynomial Endomorphisms and Kernels of Derivations”. Ph. D. Theses University of Nijemegen, Nijmegen, 2003. [22] B.D. Mestel. On globally periodic solutions of the difference equation xn+1 = f(xn)/xn−1. J. Difference Equ. Appl. 9 (2003), 201–209. [23] D. Montgomery, L. Zippin. “Topological Transformation Groups”. Interscience, New York, 1955. [24] I. Niven, H.S. Zukerman, H.L. Montgomery. “An Introduction to the Theory of Numbers”. Fifth edition, John Wiley & Sons, Inc., New York, 1991. [25] J. Rubi´o-Masseg´u. On the global periodicity of discrete dynamical systems and application to rational difference equations. J. Math. Anal. Appl. 343 (2008), 182-189. [26] J. Rubi´o-Masseg´u, V. Ma˜nosa. Normal forms for rational difference equations with applications to the global periodicity problem. J. Math. Anal. Appl. 332 (2007), 896918. [27] T. Tsuda. Integrable mappings via rational elliptic surfaces. J. Phys. A: Math. Gen. 37 (2004), 2721–2730. 25