Global eriodicity and complete integrability of discrete dynamical systems

GLOBAL ERIODICITY AND COMPLETE INTEGRABILITY OF DISCRETE DYNAMICAL SYSTEMS ANNA CIMA(1), ARMENGOL GASULL(1), AND V´ ICTOR MA˜ NOSA (2) Abstract. Consider the discrete dynamical system generated a map F. It is said that it is globally periodic if there exists a natural number psuch that Fp(x)≡x. On the other hand it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing in the ones coming from difference equations. 1. Introduction and Main Result Let F:U ⊆ Kk→ U,be a map where Uis an open set of Kkand Kdenotes either Ror C.This paper concerns with some features of the dynamical system generated by F. We start recalling some definitions. We say that a map Fis globally periodic if it exists some p∈Nsuch that Fp(x) = xfor all x∈ U,where Fp=Fp) ◦···◦ F. In particular notice that globally periodic maps have to be bijective and F−1=Fp−1.This name comes from the fact that if Fis globally periodic then the discrete dynamical system (DDS for short) generated by Fhas all its orbits as periodic orbits. The study of globally periodic maps is a current subject of interest of the dynamical systems community, see for instance [1, 2, 3, 4, 5, 8, 18, 22]. Indeed, the functional equation Fp= Id is also known in the literature as Babbage equation, see [17] and it is one of the oldest iterative functional equations ever discussed. 2000 Mathematics Subject Classification. 39A11, 39A20. Key words and phrases. Globally periodic discrete dynamical system, first integrals, completely integrable systems, difference equations, invariants for difference equations. The authors are partially supported by the Government of Catalonia’s grant 2001SGR-00173, and CICYT through grants BFM2002-04236-C02-2 (first and second authors) and DPI2002-04018-C02-01 (third author). This work was partially done while the third author was visiting Barcelona’s CRM (Centre de Recerca Matem`atica), having the support of the UPC’s “Programa d’ajuts a la mobilitat”; V. Ma˜nosa acknowledges both institutions. 1 2 ANNA CIMA, ARMENGOL GASULL, AND V´ ICTOR MA˜ NOSA A discrete dynamical system generated by a map F:U ⊆ Kk→ U is called completely integrable if there exists kfunctionally independent first integrals of the DDS. Recall that a first integral of the DDS generated by Fis a non constant K–valued function Hwhich is constant on the orbits of the DDS. That is, H(F(x)) = H(x),for all x∈ U. Integrable dynamical systems have a distinguished role as models of natural phenomena and are one of the main subjects of contemporaneous nonlinear science, from the theoretic and the applied view point. Nowadays, discrete– time integrable systems is also an area of a deep research activity, see for instance [7, 14, 15]. Note that there is a minor, but important, difference between the definition of complete integrable discrete and complete integrable continuous dynamical systems. For the later case the number of functionally independent first integrals has to be just k−1. In this paper we relate the globally periodicity of a map with the complete integrability of its associated DDS, see the Theorem 1 below. Somehow, our result extents to DDS a common knowledge result in continuous dynamics, namely that the existence of several continua of periodic solutions is related with the existence of several first integrals. Along the paper we only consider the positive iterates of F. Thus, indeed we deal with discrete semi– dynamical systems. Nevertheless, for the sake of simplicity, we keep the name of discrete dynamical system. The main result of this paper is the following Theorem: Theorem 1. Let F:U ⊆ Kk→ U an injective differentiable map defined in an open set U. The following statements hold: (a) If the DDS generated by Fis completely integrable and there exists a set of kof its first integrals H1, H2,···, Hksuch that Card Ãk \ i=1{Hi=ci}∩U!≤K, for all ci∈K,(1) being Ka given positive integer, then Fis globally periodic. (b) If Fis globally periodic, then the DDS generated by Fis completely integrable. Statement (a) of Theorem 1 is not difficult to be proved. Its proof is given in Section 2. We also present a complete integrable DDS (see Example 2.1) showing that the uniform finiteness condition (1) is necessary in order to get global periodicity. Statement (b) is the most difficult part of the Theorem and it is proved in Section 3. Our proof is constructive, i.e. it provides a method for obtaining a complete set of first integrals for DDS generated GLOBAL PERIODICITY AND COMPLETE INTEGRABILITY 3 by global periodic maps. We end Section 3 with several examples which illustrate the procedure. The key idea to prove part (b) of Theorem 1 is the following: Assume Fp=Id and let F:U → Kkp be defined by F(x) = (x, F(x), . . . , Fp−1(x)). Since F(F(x)) = (F(x), F2(x), . . . , Fp−1(x), x),each S: (Kk)p→(Kk)p satisfying S(v1, v2, . . . , vp) = S(v2, v3, . . . , vp, v1) gives a first integral of the DDS generated by F, S ◦ F,whenever it is not a constant function. In order to prove the existence of many of such first integrals (functionally independent if some minimum regularity conditions are fulfilled) we can consider as first integrals the ones obtained when the functions Sare chosen from a basis of the symmetric polynomials in kp =Nvariables. The difficult part is to see that among all these Nfunctions, kof them are functionally independent. Observe that, essentially, Theorem 1 establishes the equivalence between complete integrability (with the uniform finiteness condition (1)) and global periodicity. There is only one small point missing: when Fis globally periodic we know that its associated DDS is completely integrable but condition (1) is not guaranteed for a given set of first integrals. Indeed, in Example 3.3, we show that there are completely integrable maps being globally periodic but having sets of functionally independent first integrals which do not fulfill condition (1). In the particular case of the example we also show that there is another set of functionally independent first integrals which satisfies condition (1). We think that this will be the situation when we deal with Fbeing polynomial or rational maps, but for general Fit seems much more difficult to be proved. We do not consider this problem in the paper. In Section 4 we adapt and improve the general method to DDS which came from difference equations. A full collection of periodic recurrences, especially rational, are presented. Section 5 is devoted to consider the case where Fis non–smooth or even discontinuous. From the proof of Theorem 1 it is easy to see that for a F globally periodic, the condition on the regularity of Fgiven in its statement is only needed to ensure the existence of exactly kfunctionally independent first integrals. Without this condition in general we can only prove the existence of some first integrals of the DDS given by F. We end this section with two examples to illustrate how the method works in the non–smooth or the discontinuos cases. Acknowledgement. We acknowledge F. Ma˜nosas for stimulating and helpful indications. 4 ANNA CIMA, ARMENGOL GASULL, AND V´ ICTOR MA˜ NOSA 2. Complete integrability implies global periodicity Proof of Theorem 1 (a). Consider ci∈K,i= 1, . . . , k such that L= Tk i=1{Hi=ci}∩U 6=∅.Thus Lis an invariant non–empty finite set of points for the DDS generated by Fand card(L) = KL≤K. Hence the dynamics over it decomposes into subsystems which are either periodic or pre–periodic (the ones which, with a finite number of iterates, reach a periodic orbit as a ω–limit set). Since Fis injective this last option can be discarded. Furthermore the period of the periodic subsystems in Lhas to be a divisor of KL!.Thus a period p(not necessarily minimal) which satisfies the statement is p=K!. ¤ Notice that the above proof does not use the regularity of F. As we will see this hypothesis is included in the statement of the Theorem to enunciate part (b); see also Section 5. Next example shows that the uniform finiteness condition (1) is necessary in statement (a) of Theorem 1. Example 2.1: Consider the map F:R2→R2, F(x, y) = (x+ 2π, y). This map Fis bijective in the whole R2and it is clearly not p–periodic for any p∈N.Moreover, the DDS generated by it is completely integrable. In fact H1(x, y) = y−sin(x)and H2(x, y) = yare two functionally independent first integrals. Finally, it is clear that for some k1and k2the finiteness condition is no satisfied, i.e. the set {H1(x, y) = k1}∩{H2(x, y) = k2}has infinite many points. Recall that if a set of rsmooth functions, H1, H2, . . . , Hrfrom W ⊂ Kkinto Kis such that the rank of the matrix (DH)xis rfor almost all x∈ W (here H= (H1, H2, . . . , Hr)) then these functions are functionally independent, see [13, 26]. This property together with the assumption that the first integral are rational functions allow to obtain alternative hypotheses to (1) in Theorem 1 (a). The results are stated in the following corollary: Corollary 2. If F:U ⊆ Kk→ U is an injective map on Uand there exists krational first integrals H1, H2,··· , Hkfor the DDS generated by Fsuch that det (∇H1,∇H2, . . . , ∇Hk)6= 0 in U,(2) then Fis globally periodic. Proof. Set Hi=Pi/Qiand di= max c∈K{deg(Pi−cQi)}. Using B´ezout Theorem we know that Tm i=1{Hi=ci}contains either a number less or equal than Qn i=1 dipoints or an infinite number of them. Condition (2) prevents this second possibility. Thus, the result is obtained applying Theorem 1 (a), with K=Qn i=1 di.¤ GLOBAL PERIODICITY AND COMPLETE INTEGRABILITY 5 Remark 3. The above results obtained by taking rational first integrals can also be extended to Khovansky–type first integrals. The reason is that the maximum number of solutions of systems of equations given by such type of functions can also be uniformly bounded in terms of some generalized degrees, see [16, 24]. Recall that a map f:Rk→Rkis of Khovansky–type (and write f∈Kk m,n where m= (m1, . . . , mk)) if f= (f1, . . . , fk)where each fi∈R[x, y]is a polynomial with degree mi,x∈Rk,y= (y1, . . . , yn)∈Rnand yi=ehai,xi with ai∈Rk. Note that when n= 0,Kk m,0is the set of polynomial maps. For instance, for f∈K2 m,n (planar Khovansky maps) the maximum number of solutions of a system defined by f=p(for any p∈R2), whenever it is finite, is bounded by K=m1m2(1 + m1+m2)n2n(n−1)/2. 3. Global periodicity implies complete integrability Proposition 4. Let F:U ⊂ Kk→ U be a globally p–periodic map on U. Let Φ : Up= p z }| { U ×U ×···×U −→ K be invariant by the permutation of the indices S(y1, y2, . . . , yp) = (y2, y3, . . . , yp, y1), where each yi∈ U, i.e. Φ(S(y)) = Φ(y).Then, whenever it is not a constant function, Hφ(x) = Φ(x, F(x), . . . , Fp−1(x)) is a first integral of the DDS generated by F. Proof. Since Fis p–periodic, and Φ is invariant under the index permutation Swe get: HΦ(F(x)) = Φ(F(x), F2(x), . . . , Fp(x)) = = Φ(F(x), F2(x), . . . , Fp−1(x), x) = = Φ(S(x, F(x), . . . , Fp−1(x))) = = Φ(x, F(x), . . . , Fp−1(x)) = HΦ(x). ¤ Remark 5. The case where the above procedure gives rise to a constant function HΦcan not be discarded, as the following example shows: F(x, y) = (y, c/(xy)), F2(x, y)=(c/(xy), x), F 3(x, y) = (x, y),Φ(y1, y2, y3) = y1y2y3 and Hφ(F(x, y)) = c. From the above result, each function Φ invariant by the permutation S gives rise to a candidate to be a first integral of the DDS generated by F. 6 ANNA CIMA, ARMENGOL GASULL, AND V´ ICTOR MA˜ NOSA As we will see, to our purposes it will suffice to consider Φ being symmetric polynomials in N=kp variables. The set of all symmetric polynomials in Nvariables forms an algebra of dimension N. There are several homogeneous basis for the symmetric polynomials. We shall use two of them: the so called elementary symmetric polynomials in Nvariables: σ1(y1, . . . , yN) := y1+···+yN, σ2(y1, . . . , yN) := y1y2+y1y3+···+yN−1yN, . . .. . .. . . σN−1(y1, . . . , yN) := y1y2. . . yN−1+y1y2. . . yN−2yN+···+y2y3. . . yN, σN(y1, . . . , yN) := y1y2. . . yN. and the power sums Si(y1, y2, . . . , yN) = PN j=1 yi j,i.e., S1(y1, . . . , yN) := y1+···+yN, S2(y1, . . . , yN) := y2 1+y2 2+···+y2 N, . . .. . .. . . SN(y1, . . . , yN) := yN 1+yN 2+···+yN N. The functions σiare the coefficients (modulus a signum) of the polynomial p(w)=(w−y1)·····(w−yN), in fact p(w) = wN+···+(−1)N−iσi(y1, . . . , yN)wN−i+···+(−1)NσN(y1,. . . , yN). Set x= (x1, . . . , xk)∈Kk, and introduce the notation (Fi 1(x), Fi 2(x), . . . , Fi k(x)) := Fi(x).We also have the following Corollary of Proposition 4. Corollary 6. Let F:U → U be a globally p–periodic map on a set U ⊂ Kk and let Ψ : U → Kbe an arbitrary function. Then, whenever they are not constant functions, Hσi,Ψ(x) = σi(Ψ(x),Ψ(F(x)), . . . , Ψ(Fp−1(x))) and HSi,Ψ(x) = Si(Ψ(x),Ψ(F(x)),...,Ψ(Fp−1(x))) (3) are first integrals of the DDS generated by F. Proof. It follows directly from Proposition 4, taking Φ(y1, . . . , yp) = σi(Ψ(y1),...,Ψ(yn)),or Φ(y1, . . . , yp) = Si(Ψ(y1), . . . , Ψ(yn)).¤ The following result proves part (b) of Theorem 1. Theorem 7. Let F:U ⊆ Kk→ U be a globally p–periodic differentiable map. Then DDS generated by Fis completely integrable. Furthermore the set of kfunctionally independent first integrals of Fis constructed from the GLOBAL PERIODICITY AND COMPLETE INTEGRABILITY 7 set of functions (3) given in Corollary 6 with Ψ = Id.As a consequence they are analytic (respectively, rational) if Fis analytic (respectively, rational). Proof. Set N=kp and consider F(x) := (x, F(x), F2(x), . . . , Fp−1(x)).For each i= 1,2, . . . , N let yibe defined through (y1, y2, . . . , yN) := (x, F(x), F2(x), . . . , Fp−1(x)). It can occur that for some indices r, s, yr≡ys.Assume that among the N components y1, y2, . . . , yNthere are `which are different. Let Y1, Y2, . . . , Y` denote these components. Clearly `≥k. For each i= 1,2, . . . , ` let nibe the number of variables in (y1, . . . , yN) which coincides with Yi. Now consider the homogeneous symmetric polynomials S1, S2, . . . , SN introduced before Si(y1, y2, . . . , yN) = yi 1+yi 2+. . . +yi N.Among these Npolynomials choose the `first ones and consider S1, S2, . . . , S`.Also for i= 1,2, . . . , `, let Hi(x) be defined by Hi(x) = Si(F(x)). Then, S= (S1, S2, . . . , S`) and H= (H1, H2, . . . , H`) define some maps: S:Kpk →K`, H :U ⊂ Kk→K`. Recall that from Proposition 4 the functions Hi,whenever they are not constant, are first integrals of the DDS generated by F. In order to prove that they are functionally independent it suffices to show that the matrix (DH)xhas rank kfor almost all x∈ U.To this end we describe the map H as ¯ S◦Y◦F where F:Uk→Kkp , Y :Kkp →K`,¯ S:K`→K` and Yand ¯ Sare defined through: Y(y1, y2, . . . , yN) = (Y1, Y2, . . . , Y`) and ¯ S= ( ¯ S1,¯ S2, . . . , ¯ S`) with ¯ Si(w1, w2, . . . , w`) = n1wi 1+n2wi 2+···+n`wi `. Then by the chain rule we have that for each x∈ U : (DH)x= (D¯ S)(Y◦F(x)) (DY )F(x)(DF)x. The matrix (DF)xhas the maximum rank equals to kfor all x∈ U,because the first kcomponents of Fgives us the identity map. The matrix of (DY ) has maximum rank `at each point of Kk. We claim that (D¯ S)(Y◦F(x)) has also the maximum rank equal `for almost all x∈ U.If the claim is true, since `≥kthe result follows. In order to prove the claim notice that for i= 1,2, . . . , `, ¯ Si(w1, w2, . . . , wn) = n1wi 1+n2wi 2+···+n`wi `, 8 ANNA CIMA, ARMENGOL GASULL, AND V´ ICTOR MA˜ NOSA and so (DS)(w1,w2,...,wn)=       n1n2··· n` 2n1w12n2w2··· 2n`w` 3n1w2 13n2w2 2··· 3n`w3 ` . . .. . .. . .. . . `n1wl−1 1`n2w`−1 2··· `n`w`−1 `        . To compute the determinant of this matrix we observe that det(D¯ S)(w1,w2,...,wn)=`!n1n2···n`det(V) where det(V) = Y i<j, j=1,...,` (wi−wj) is the known Vandermonde’s determinant. Taking (w1, w2, . . . , wn) = (Y◦ F)(x),since among the components of Y◦F there are no repetitions, this determinant does not vanish identically and the result follows. ¤ Remark 8. The Montgomery-Bochner Theorem, see [23] asserts that if x∗∈Kkis a fixed point of a Ck, k ≥1or analytic map Fsuch that Fp=Id then there is a neigbourhood of x∗where the DDS generated by F is Ckor analytic conjugated to the DDS generated by its linear part L(x) := DF(x∗)xnear 0.Thus the existence of a complete set of integrals for the DDS generated by F(only in a neigbourhood of x∗) is also a consequence of the study of the same problem for the linear DDS generated by L(x)near 0. In order to illustrate the method developed in Theorem 7 we begin with a trivial but pedagogical example. Example 3.1: Consider the 3-periodic linear map: F(x, y, z)=(y, z, x). Here k= 3, p = 3 and N= 9.Since F(x, y, z) = (x, y, z, y, z, x, z, x, y), `is just 3.Following the proof of the above theorem we have to consider S1, S2and S3of 9coordinates and do the appropriate substitutions. We obtain the following set of functionally independent first integrals in U= K3: H1(x, y, z) = 3(x+y+z), H2(x, y, z) = 3 ¡x2+y2+z2¢, H3(x, y, z) = 3 ¡x3+y3+z3¢. GLOBAL PERIODICITY AND COMPLETE INTEGRABILITY 9 Example 3.2: Let Fbe given by: F(x, y) = µy 1 + x+y,−x 1 + x+y¶. It is easy to check that F4=Id in U=K2\{(1+x+y)(1+2y)(1−x+y) = 0} and that F(x, y) = =µx, y, y 1 + x+y,−x 1 + x+y,−x 1+2y,−y 1+2y,−y 1−x+y,x 1−x+y¶. Thus a system of two independent first integrals is: H1(x, y)=(x+y)2y 1+2y+ (x−y)2x (1 −x+y)(1 + x+y), Hσ8(x, y) = µx2y2 (1 −x+y) (1 + 2y) (1 + x+y)¶2 , where Hσ8(x) = σ8(F(x)). The above example is given by the flow at time 2π/4 of the planar isochronous center ˙x=−y+x2,˙y=x(1 + y),see [11]. Recall that given an ordinary differential equation defined in an invariant open set U ⊂ Rkit is called isochronous, or T−isochronous in Uif there exists T > 0 such that ϕ(T, x)≡xfor all x∈ U,where ϕis the flow associated to the ordinary differential equation. Thus isochronous flows give rise to a full collection of p−periodic maps in U; it suffices to consider, for any p∈N, Fp(x) := ϕ(T/p, x) and by the flow property Fp p(x)≡xfor all x∈ U.For instance another 4-periodic map is given by F(x, y) = Ã−y 3 p1+4y3−4x3,x 3 p1+4y3−4x3!. It cames from the flow at time (2π)/4 of the 2π−isochronous center ˙x= −y+ 4x2y2,˙y=x+ 4xy3.Similar computations to the ones used in the above example give a complete set of integrals for the DDS generated by F. From the above considerations and Theorem 7 we get the following corollary: Corollary 9. Let Fp(x) := ϕ(T/p, x)be a map defined from Uinto itself, where ϕ(t, x)is the flow of a T−isochronous ordinary differential equation defined in Uand p∈N.Then the DDS generated by Fpis completely integrable. Remark 10. The method given by Theorem 7 gives a constructive way for obtaining kfunctionally independent first integrals H1, . . . , Hkfor the 16 ANNA CIMA, ARMENGOL GASULL, AND V´ ICTOR MA˜ NOSA 4.1. Invariants for non-minimal recurrences. Given a difference equation having an invariant it is very easy to extend this invariant to any equivalent difference equation to the given one: it suffices to perform the change of variables which transforms one equation into the other in the given invariant. This section is devoted to explain how to extend the invariant of a difference equation (6) to the difference equations generated by a given one (7). Proposition 14. Consider the difference equation (7): xn+kr =f(xn, xn+r, xn+2r, . . . , xn+(k−1)r), and let I:Kk→Kbe an invariant of the generating difference equation xn+k=f(xn, xn+1, xn+2, . . . , xn+k−1), then J(x) = I(φ(x))I(φ◦F(x)) ···I(φ◦Fr−1(x)) is an invariant of (7), where x= (x1, . . . , xkr)∈Kkr,φ:Kkr →Kkis the projection map given by φ(y1, y2, . . . , ykr) = (y1, y1+r, y1+2r, . . . , y1+(k−1)r), and F:Kkr →Kkr is the map which generates the DDS associated to (7) F(y1, . . . , ykr) = (y2, y3, . . . , ykr−1, ykr, f(y1, y1+r, y1+2r, . . . , y1+(k−1)r)). Proof. Observe that I(φ◦Fr(x)) = I(φ(x)), since I(φ◦Fr(x)) = I(φ(x1+r, x2+r, x3+r, . . . , xkr+r)) = =I(x1+r, x1+2r, . . . , x1+kr) = =I(x1, x1+r, x1+2r, . . . , x1+(k−1)r) = I(φ(x)). Therefore J(F(x)) = I(φ◦F(x)) ···I(φ◦Fr−1(x))I(φ◦Fr(x)) = =I(φ◦F(x)) ···H(φ◦Fr−1(x))I(φ(x)) = J(x). ¤ We end this subsection with some applications of the above result. Example 4.2: (i) Setting r= 2,the 6–periodic recurrence (i) of Proposition 13, generates a 12–periodic recurrence xn+4 =xn+2/xn,which has the invariants Ji(xn, xn+1, xn+2, xn+3) = ISi(xn, xn+2)ISi(xn+1, xn+3), i = 1,2, where the functions ISiare given in (i) of Proposition 13. GLOBAL PERIODICITY AND COMPLETE INTEGRABILITY 17 (ii) Taking r= 3 the 5–periodic Lyness recurrence, generates a 15– periodic recurrence given by xn+6 = (1 + xn+3)/xn.One of its invariants is J(xn, xn+1, xn+2, xn+3, xn+4, xn+5) = =Iσ5(xn, xn+3)Iσ5(xn+1, xn+4)Iσ5(xn+2, xn+5), where Iσ5(x, y) = (1 + x)(1 + y)(1 + x+y)/(xy). (iii) The M¨obius recurrence zn+1 =(−1 + i)zn+ 2i zn−1 + i=: f(zn) is a 4–periodic recurrence with invariant given by I(zn) = µzn−1−i zn+1+i¶4 . Taking r= 3, we obtain that the 12–periodic recurrence zn+3 =f(zn) has the invariant J(zn, zn+1, zn+2) = I(zn)I(zn+1)I(zn+2). 5. The non–differentiable case In the case that the map Fis not differentiable or even discontinuous, part (b) of Theorem 1 does not apply. However the following result shows that if Fis globally periodic then the DDS generated by Fhas at least k (some of them maybe functionally dependent) non constant first integrals of the form Hσi. Theorem 15. Let F:U ⊆ Kk→ U be a globally p–periodic map. Define the set of functions Σ := {Hσi}{i=1,...,kp}, where Hσi(x) = σi(x, F (x),..., Fp−1(x)) and σiis the basis of the symmetric polynomials in kp variables given in Section 3. Then, among all the elements of Σthere are at least k non constant first integrals for the DDS generated by F. Proof. By Theorem 7 we already know that all the non-constant functions in Σ are first integrals of the DDS generated by F. Let us prove that at least kof them are effectively non constant. Set x= (x1,··· , xk). Notice that, modulus a sign, the functions in Σ are the coefficients of the polynomial P(w, x) := (w−x1)...(w−xk)(w−F1(x)) . . . (w−Fk(x))(w−F2 1(x))· . . . (w−F2 k(x)) . . . (w−Fp−1 1(x)) . . . (w−Fp−1 k(x)) =wkp +···+ (−1)iHσi(x)wkp−i+···+ (−1)kpHσnk (x). First we prove that that at least there is one non constant function in Σ. Assume, by contradiction, that all the functions in Σ are constant functions. Then P(w, x) = P(w) is a polynomial of degree kp with constant coefficients having infinitely many zeros (for instance w=x1is a zero for any value of 18 ANNA CIMA, ARMENGOL GASULL, AND V´ ICTOR MA˜ NOSA x1.). Thus Phas to be identically zero, in contradiction with the fact that its coefficient of the (kp)th–order term is 1. We continue again by contradiction. Assume that the statement is not true and let Hσi1, Hσi2, . . . , Hσik−1 (with i`6=isif `6=s) such that can depend on x,while the others coefficients of P(w, x) are constants. For the sake of simplicity, write ˜ Hi(x) = (−1)iHσi(x). Since P(w, x) vanishes for w=xifor all i= 1,2, . . . , k we can consider the system L1of equations P(x`,x) = 0 for `= 1,2, . . . , k −1,i.e., L1:(˜ Hi1(x)xkp−i1 `+˜ Hi2(x)xkp−i2 `+···+˜ Hik−1(x)xkp−ik−1 `= =−Pj /∈{i1,i2,...,ik−1}cjxkp−j ` , looking at ˜ Hir(x), r = 1, . . . , k −1,as the unknowns. The determinant of the matrix of the coefficients of this linear system is: ¯¯¯¯¯¯¯¯¯¯ xkp−i1 1xkp−i2 1. . . xkp−ik−1 1 xkp−i1 2xkp−i2 2. . . xkp−ik−1 2 . . .. . .. . .. . . xkp−i1 k−1xkp−i2 k−1. . . xkp−ik−1 k−1 ¯¯¯¯¯¯¯¯¯¯ := D(x1, x2, . . . , xk−1). The monomials of the polynomial D(x1, x2, . . . , xk−1) have 1 or -1 as a coefficient and two of these monomials can not coincide (it is because not two of the exponents i1, i2, . . . , ik−1coincide). In particular D(x1, x2, . . . , xk−1) is not identically zero and so, for almost all the values of x1, x2, . . . , xk−1,it is possible to isolate ˜ Hir(x) for r= 1,2, . . . , k−1 in terms of x1, x2, . . . , xk−1. It implies that in fact each ˜ Hir(x), r = 1,2, . . . , k −1 does not depend on xk. Now we can repeat the above process taking the systems of equations : Lr:{P(x`,x) = 0 for `∈ {1,2, . . . , k}but `6=k−r+ 1} for each r∈ {1,2, . . . , k}. System Lrimplies that each ˜ Hij(x),for j= 1,2, . . . , k −1 does not depend on xk+1−r.Hence, for each j= 1,2, . . . , k −1,˜ Hij(x) does not depend on x1, x2, . . . , xk,i.e. it is constant. Therefore all the functions in Σ are constant, which is contradiction with our first step, where we have shown that at least one of the elements in Σ has to be a non constant function. Thus we can ensure that among all the elements of Σ at least k of them are non constant functions, as desired. ¤ Following the guidelines of Theorem 15 and the spirit of Section 4, we give several invariants of two non–smooth globally periodic difference equations. GLOBAL PERIODICITY AND COMPLETE INTEGRABILITY 19 Note that in the non-differentiable case it is a difficult task to elucidate whether several functions are functionally independent or not. We do not consider this question in the study of the following examples. In [2], Abu–Saris and Al–Hassan, introduced the following second order globally 4–periodic difference equation: xn+2 =g(xn+1)/xn,where g(x) = (aif x∈Q, bif x∈R\Q,(8) and a, b ∈Q\{0}. We obtain the following result. Proposition 16. The following functions are invariants for the difference equation (8): Iσ1(xn, xn+1) = xn+xn+1 +g(xn+1) xn +g(xn) xn+1 . Iσ2(xn, xn+1) = xnxn+1 +g(xn) + g(xn+1) + xn+1g(xn+1) xn +xng(xn) xn+1 + g(xn)g(xn+1) xnxn+1 . Iσ3(xn, xn+1) = xng(xn) + xn+1g(xn+1) + g(xn)g(xn+1) xn +g(xn)g(xn+1) xn+1 . Iσ4(xn, xn+1) = g(xn)g(xn+1). Proof. The proof is an immediate check. It suffices to note that g(g(xn+1)/xn) = g(xn) by using a case by case study. Thus, by using Theorem 15 we obtain that Iσk(xn, xn+1) = σk(xn, xn+1, xn+2, xn+3) =σkµxn, xn+1,g(xn+1) xn ,g(xn) xn+1 ¶, for k= 1, . . . , 4 are invariants of the difference equation (8). ¤ In [8], Cs¨ornyei and Laczkovich studied second order globally 5–periodic recurrence: xn+2 = max(0, xn+1)−xn,(9) which has an associated non–smooth dynamical system F(x, y) = ((y, y −x) if y≥0, (y, −x) if y < 0. Observe that, according to [25], equation (9) is the ultradiscrete version of the Lyness recurrence. 20 ANNA CIMA, ARMENGOL GASULL, AND V´ ICTOR MA˜ NOSA Define the following subsets of R2: D1={(x, y) : x≥0, y ≥0, y ≤x}, D2={(x, y) : x≥0, y ≤0}, D3={(x, y) : x≤0, y ≤0}, D4={(x, y) : x≤0, y ≥0}, D5={(x, y) : x≥0, y ≥0, y ≥x}. Note that the DDS generated by Fbehaves as D1 F −→ D2 F −→ D3 F −→ D4 F −→ D5 F −→ D1, and each time that Factues is indeed a linear map. By using again Theorem 11 with the symmetric polynomials Siwith 5 indeterminates, we obtain a set of continuous invariants of the difference equation (9), which naturally have different expressions on each different subset Di. Proposition 17. The functions Iσk(x, y)listed below, are invariants for the difference equation (9). Iσ1(x, y) =                xif (x, y)∈D1 x−yif (x, y)∈D2, −x−yif (x, y)∈D3, y−xif (x, y)∈D4, yif (x, y)∈D5. Iσ2(x, y) =      −x2+ 2xy −2y2if (x, y)∈D1, −2x2+ 2xy −y2if (x, y)∈D5, −x2−y2otherwise. Iσ3(x, y) =                −x(x2−2xy + 2y2)if (x, y)∈D1, −x3+x2y−xy2+y3if (x, y)∈D2, x3+x2y+xy2+y3if (x, y)∈D3, x3−x2y+y2x−y3if (x, y)∈D4, −y(y2−2xy + 2x2)if (x, y)∈D5. Iσ4(x, y) =      −x2+ 2xy −2y2if (x, y)∈D1, −2x2+ 2xy −y2if (x, y)∈D5, x2y2otherwise. Iσ5(x, y) =                xy2(x−y)2if (x, y)∈D1, x2y2(x−y)if (x, y)∈D2, −x2y2(x+y)if (x, y)∈D3, x2y2(y−x)if (x, y)∈D4, x2y(x−y)2if (x, y)∈D5. GLOBAL PERIODICITY AND COMPLETE INTEGRABILITY 21 Proof. Once again the proof is only a (tedious) check. We give the details for obtaining the first invariant Iσ1.Setting xn:= xand xn+1 := y, we obtain xn+2 = max(0, y)−x,xn+3 = max(0,max(0, y)−x)−yand xn+4 = max(0,max(0,max(0, y)−x)−y)−max(0, y) + x. Thus, by using Theorem 11 we have the following invariant: Iσ1(xn, xn+1) = σ1(xn, xn+1, xn+2, xn+3, xn+4) = =xn+xn+1 +xn+2 +xn+3 +xn+4, which can be written as Iσ1(x, y) = x+max (0,max (0, y)−x)+max (0,max (0,max (0, y)−x)−y). By studying the above function in each sector Di, i = 1, . . . , 5 we obtain the expression given in the statement. The other four invariants can be obtained by following the same procedure but using σi, i = 2,...,5 instead of σ1.¤ References [1] R.M. Abu–Saris. 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(1)Dept. de Matem` atiques, Facultat de Ci` encies, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain E-mail address:[email protected], [email protected] (2) Dept. de Matem` atica Aplicada III, Control, Dynamics and Applications Group (CoDALab), Universitat Polit` ecnica de Catalunya, Colom 1, 08222 Terrassa, Spain E-mail address:[email protected]