This is a preprint of: “Continua of periodic points for planar integrable rational maps”, Armengol Gasull, Mireia Llorens, V´ıctor Ma˜ nosa, Int. J. Difference Equ., vol. 11(1), 37–63, 2016. Continua of periodic points for planar integrable rational maps Armengol Gasull and Mireia Llorens Universitat Aut`onoma de Barcelona Department de Matem`atiques Facultat de Ci`encies, 08193 Bellaterra, Spain gasull@mat.uab.cat and mllorens@mat.uab.cat ˜ V´ıctor Manosa Universitat Polit`ecnica de Catalunya Department de Matem`atiques Colom 1, 08222 Terrassa, Spain victor.manosa@upc.edu Abstract We present three alternative methodologies to find continua of periodic points with a prescribed period for rational maps having rational first integrals. The first two have been already used for other authors and apply when the maps are bi- rational and the generic level sets of the corresponding first integrals have either genus 0 or 1. As far as we know, the third one is new and it works for rational maps without imposing topological properties to the invariant level sets. It is based on a computational point of view, and relies on the use of resultants in a suitable setting. We apply them to several examples, including the 2-periodic Lyness composition maps and some of the celebrated McMillan-Gumowski-Mira maps. AMS Subject Classifications: 37C25, 39A20, 39A23. Keywords: Integrable rational maps, birational maps, periodic orbits. 1 Introduction A planar rational map F : U → K2 , where U ⊆ K2 is an open set and K ∈ {R, C}, is called birational if it has a rational inverse F −1 . Such a map is integrable if there exists 2 A. Gasull, M. Llorens, V. Ma˜nosa a non-constant function V : U → K such that V (F (x, y)) = V (x, y), which is called a first integral of F . If a map F possesses a first integral V , then the level sets of V are invariant under F . We consider integrable rational (or birational) maps that have a rational first inte- gral V (x, y) = V1 (x, y)/V2 (x, y), thus preserving the fibration given by the pencil of algebraic curves Ch := {V1 (x, y) − h V2 (x, y) = 0}, for h ∈ Im(V ). (1.1) When each curve Ch is invariant under the iterates of F it is said that the map preserves the fibration given by the curves {Ch }. Integrable planar birational maps typically exhibit continua of periodic orbits lying in some of the level curves of their first integrals. In fact, a part of the very few globally periodic cases( [12]), a typical phase portrait includes curves densely filling an open set of the phase space and each one of them full of periodic points with the same period. Moreover, in this case the set of periods of the map is infinite and it contains, among others, all the periods greater than a particular one. Nevertheless, it is usually very dif- ficult to find any of these continua of periodic points by using either the direct approach of solving the system F p (x, y) = (x, y), for a given p ∈ N, or any of their equivalent expressions F m (x, y) = F m−p (x, y), for some 1 ≤ m ≤ p, where F 0 = Id, F 1 = F and F k = F ◦ F k−1 , k ≥ 2. Similarly, F −1 is the inverse of F and F −k = (F −1 )k . In this paper we present three alternative effective methodologies for finding con- tinua of periodic points with a prescribed period for rational maps having rational first integrals. The first two have been already used for other authors and only apply when the maps are birational and the generic level sets of the corresponding first integrals have either genus 0 or 1. To the best of our knowledge, the third one is new and it works for rational maps, non-necessarily birational, without imposing any restriction to the invariant level sets. It is based on a computational point of view, and relies on the use of resultants in a suitable setting. Concerning the first approach, in case that the curves Ch in the fibration (1.1) have generically genus 0, it is well-known that each map F|Ch is conjugate to a M¨obius trans- formation ( [30]). This fact permits, when the maps F|{|V |=|h|} are conjugate to rotations, to give an explicit expression of rotation number ρ(h), and to determine the curves in (1.1) filled with p-periodic points by solving the equations q ρ(h) = for all 1 ≤ q ≤ p − 1, coprime with p. (1.2) p See the proof of Theorem 2.1 in Section 2 for more details. The second approach deals with the most studied case which corresponds to the sit- uation where the invariant curves in the preserved fibration (1.1) have generic genus 1 Continua of periodic points for planar integrable rational maps 3 (also called elliptic fibrations). This case is important because birational maps preserv- ing elliptic fibrations appear in many physical applications( [28]). Some classical maps like the Lyness and the integrable McMillan-Gumowski-Mira ones, which are included in a large family known as QRT maps preserve elliptic fibrations, see [27] and the ref- erences therein. Although in the elliptic case there is not an explicit expression of the rotation number function it is possible identify the energy levels where the periodic orbits are located by using the group estructure of the elliptic curves in the fibration (1.1), see for instance [3, 15, 33]. In Section 3 we give an example of this approach. As we have already said, the last method presented is purely computational, it works for rational maps that are not necessarily birational (a trivial example of rational map, not birational, and with a rational first integral V (x, y) = y, is F (x, y) = (x2 , y)), and it is only based on the assumption that the continua of periodic orbits lie on the level sets of the rational first integral. This new method, together with some examples of application, is developed in Section 4. Another advantage of our approach is that it can be extended to integrable rational maps in higher dimensions. In this paper we are not concerned with this question but we believe that, using our point of view, we could also reproduce the results of [29] on this subject, which were obtained using more algebraic tools. Observe, however, that removing the hypotheses on F to be rational, one should not expect that the continua of periodic orbits of integrable maps in the plane coincide with the level sets of their first integrals, as discussed in Section 5. Throughout the paper we will assume that F is a rational map with a rational first integral V = V1 /V2 , where both F and V are defined in an open and dense set of the domain of definition of the dynamical system, the good set G(F ) ⊂ K2 . In the real case we also will assume that there exists a set H ⊂ Im(V ) ⊂ R such that the set U := {(x, y) ∈ R2 : V (x, y) ∈ H} ∩ G(F ) is a non empty open set of R2 . Sections 2 and 3 deal with the genus 0 and genus 1 situations, respectively. Both have the same schemes, first we recall the methods and then we apply them to some interesting maps. Finally, in Section 4, we introduce and apply to several examples our computational method to find continua of periodic points. As an illustration of the applicability of our results we dedicate Section 4.4.2 to describe the continua of real 5-periodic points for a McMillan-Gumovski-Mira map. 2 The genus 0 case 2.1 Explicit expression of the rotation number Suppose that each generic curve Ch := {V1 (x, y) − hV2 (x, y) = 0} for h in an open set H ⊂ K is irreducible in C and has genus 0. Then the Cayley-Riemann Theorem ensures that it can be rationally parameterized, that is, that there exist rational func- 4 A. Gasull, M. Llorens, V. Ma˜nosa tions P1,h (t), P2,h (t) ∈ C(t) such that for almost all of the values t ∈ K we have (P1,h (t), P2,h (t)) ∈ Ch ; and reciprocally, for almost all point (x, y) ∈ Ch , there exists t ∈ K such that (x, y) = Ph (t) := (P1,h (t), P2,h (t)). A rational parametrization of Ch is not unique but it is always possible to obtain a proper one, that is, a parametrization P such that P −1 is also rational, which is unique modulus M¨obius transformations. The reader is referred to [30] for a further insight about these facts. Given a birational map F preserving each curve Ch of the fibration (1.1) defined in K2 , a simple computation shows that if we take a proper parametrization of each curve Ph (t), the action of F|Ch is conjugate to the M¨obius transformation defined in b := K ∪ {∞}, given by Mh = P −1 ◦ Fh ◦ Ph , see for instance [22, Proposition 1]. K h The dynamics of M¨obius maps is well-known. With respect the searching of pe- riodic orbits, when Mh is conjugate to a rotation, the rotation number ρ(h) of Mh , which is the one of F|Ch , can be explicitly obtained. Indeed, consider the map M (t) = (at + b)/(ct + d) where a, b, c, d ∈ K with c 6= 0, and defined for t ∈ K. b If √ a+d+ ∆ ∆ = (d − a) + 4bc 6= 0 and ξ = 2 √ is such that |ξ| = 1, a+d− ∆ then M is conjugated to a rotation in Kb with rotation number ρ := arg(ξ) (mod 2π). In particular, M is periodic with minimal period p if and only if ξ is a p-primitive root of the unity. See for instance [7, Section 2.2]. Once we have the expression of the rotation number function θ(h) associated to each map F|Ch , we can characterize the continua of periodic points by solving Equation (1.2). 2.2 The Bastien and Rogalski maps As an example of this specific methodology we consider the one-parameter family of maps   a − y + y2 Fa (x, y) = y, , (2.1) x defined in (R+ )2 with a > 1/4. This family has been introduced and studied by G. Bastien and M. Rogalski in [4]. Each map Fa has the first integral x2 + y 2 − x − y + a Va (x, y) = , (2.2) xy so it preserves the fibration of (R+ )2 given by the conics:  Ch = x2 + y 2 − x − y + a − hxy = 0, for h > 2 − 1/a . (2.3) A similar study could be done extending this map to C2 and taking a ∈ C. Continua of periodic points for planar integrable rational maps 5 By using the parametrization by lines method ( [30]) it is not difficult to find the following proper rational parametrization of the curves Ch in the fibration (2.3), see [22] for more details: 2δt − ah2 − (1 + δ)h − 2 + 4a P1,h (t) = + a, 2(−t2 + ht − 1) (−ah + δ − 1) t2 + (4 a − 2) t − ah − δ − 1 P2,h (t) = , 2(−t2 + ht − 1) whose inverse is given by −2 δ x + (ah2 + (δ + 1)h − 4 a + 2) y − ah + 2 a + δ − 1 Ph −1 (x, y) = . (ah2 + (1 − δ)h − 4 a + 2) x + 2 δ y + ah − 2 a − δ + 1 For a fixed value of a > 1/4 and for 2 − 1/a < h < 2, the level curves Ch are ellipses. A computation gives (h + 1)t − 1 Mh (t) = Ph−1 ◦ Fa ◦ Ph (t) = , (2.4) t+1 so if 2 − 1/a < h < 2 then Fa|Ch is conjugate to Mh|Rb , which is, in turn, conjugate to a rotation with rotation number √ ! h − i 4 − h2 ρ(h) = arg mod 2π, (2.5) 2 see [4] and also [22]. By solving the Equation (1.2) for p ≤ 25, we will obtain: Theorem 2.1. The curves Ch of the fibration (2.3) are filled of periodic points with minimal period p for p ≤ 25, if and only if 2 − 1/a < h < 2, and it is satisfied cp (h) = 0 for: c3 (h) = h + 1, c4 (h) = h, c5 (h) = h2 + h − 1, c6 (h) = h − 1, c7 (h) = h3 + h2 − 2h − 1, c8 (h) = h2 − 2, c9 (h) = h3 − 3h + 1, c10 (h) = h2 − h − 1, c11 (h) = h5 + h4 − 4h3 − 3h2 + 3h + 1, c12 (h) = h2 − 3, c13 (h) = h6 + h5 − 5h4 − 4h3 + 6h2 + 3h − 1, c14 (h) = h3 − h2 − 2h + 1, c15 (h) = h4 − h3 − 4h2 + 4h + 1, c16 (h) = h4 − 4h2 + 2, 6 A. Gasull, M. Llorens, V. Ma˜nosa c17 (h) = h8 + h7 − 7h6 − 6h5 + 15h4 + 10h3 − 10h2 − 4h + 1, c18 (h) = h3 − 3h − 1, c19 (h) = h9 + h8 − 8h7 − 7h6 + 21h5 + 15h4 − 20h3 − 10h2 + 5h + 1, c20 (h) = h4 − 5h2 + 5, c21 (h) = h6 − h5 − 6h4 + 6h3 + 8h2 − 8h + 1, c22 (h) = h5 − h4 − 4h3 + 3h2 + 3h − 1, c23 (h) = h11 + h10 − 10h9 − 9h8 + 36h7 + 28h6 − 56h5 − 35h4 + 35h3 + 15h2 − 6h − 1, c24 (h) = h4 − 4h2 + 1, c25 (h) = h10 − 10h8 + 35h6 + h5 − 50h4 − 5h3 + 25h2 + 5h − 1. Observe that the above conditions characterizing the existence of continua of pe- riodic points in the curves Ch do not depend on the parameter a. This is due to the appearance of a simply as an additive constant in the expression of the curves. Of course, this fact is also reflected on the rotation number function (2.5) which is indepen- dent of a. For the sake of brevity we only display the values for p ≤ 25 but the method allows to go much further. Proof of Theorem 2.1. Observe that since the rotation number function is given by (2.5), then Equation (1.2) is satisfied, and the level h is filled by points of minimal period p, if and only if √ h − i 4 − h2 ξp := , (2.6) 2 is a primitive p-root of the unity. These conditions are contained in the set of solutions of the system  2 ξ − hξ + 1 = 0, ξ p − 1 = 0. Therefore, we can obtain the values of h for which (2.6) holds by looking for those factors of φp (h) := Res(ξ 2 − hξ + 1, ξ p − 1; ξ) that do not appear in φk (h) for any k < p that is a divisor of p. Here, as usual, Res(·, ·; ξ) denotes the resultant of both polynomials with respect to ξ. Recall that this resultant only vanishes when these two polynomials have a common (real or complex) zero, see for instance [20]. We only detail the computations when p = 25: φ25 (h) = Res(ξ 2 − hξ + 1, ξ 25 − 1; ξ) 2 = (2 − h) h2 + h − 1 h10 − 10h8 + 35h6 + h5 − 50h4 − 5h3 2 +25h2 + 5h − 1 . We can exclude the factor 2−h because when h = 2, Fa|C2 is not conjugate to a rotation. The vanishing of the factor h2 +h−1 corresponds to periodic points of minimal period 5. Thus the condition characterizing periodic points of minimal period 25 is c25 (h) = 0. Continua of periodic points for planar integrable rational maps 7 3 The elliptic case 3.1 Periodic orbits seen as finite order points of a group If a curve Ch := {V1 (x, y) − hV2 (x, y) = 0} for h ∈ H has genus 1 (i.e. is an elliptic curve) then it has an associated group structure( [31]). It is known that this fact allows to characterize the dynamics of any birational map preserving it. As a consequence, we will locate the level sets where the periodic orbits lay. We start recalling the chord-tangent group law associated with an elliptic curve C ∈ CP2 , where as usual, we consider the extension of the algebraic curve in K2 to CP2 given by the homogenization process, [31, Section I.2]. Given two points P and Q in C we define the addition P + Q as in Figure 3.1: (i) Select a point O ∈ C to be the neutral element of the inner addition; (ii) Take the chord passing through P and Q (the tangent line if P = Q). It will always intersect C at a unique third point denoted by P ∗ Q; (iii) The point P + Q is then defined as O ∗ (P ∗ Q). The curve endowed with this inner addition (C, +, O) is an abelian group. Figure 3.1: Group law with an affine neutral element O. The relation between the dynamics of a birational map preserving an elliptic curve and the group law of the curve is given by a result of Jogia, Roberts and Vivaldi [18, Theorem 3], JRV Theorem from now on, which states that if F is a birational map over a field K not of characteristic 2 or 3, that preserves an elliptic curve C defined over K. Then there exist points O, Q ∈ C such that the map can be expressed in terms of the group law + on C as either (a) F|C : P 7→ P + Q, or (b) F|C : P 7→ i(P ) + Q, where i is a periodic automorphism whose period is either 2, 3, 4 or 6, and in this case F has the same period as i. When F is a birational map preserving an elliptic curve (C, +, O) whose dynamics corresponds to case (a) of the JRV Theorem, then F|C n (P ) = P +n Q. In this case notice that P is a p-periodic point if and only if p z }| { p Q = Q + · · · + Q = O, (3.1) 8 A. Gasull, M. Llorens, V. Ma˜nosa that is, if and only if Q is a finite order point of the group (C, +, O). Equation (3.1) can be used to determine the invariant curves where are located the periodic points. See for instance [3, 15, 33] to see how to use this point of view in the case of the paradigmatic example given by the Lyness maps. In the next section we give an example of how to take advantage of equation (3.1) to characterize the invariant curves Ch where lie the periodic orbits for a particular family of maps, the 2-periodic Lyness composition maps. 3.2 The 2-periodic Lyness composition maps We consider the 2-parametric family of maps given by  a + y a + bx + y  Fb,a (x, y) := , , (3.2) x xy with a, b ∈ C, with ab 6= 0, that we will call 2-periodic Lyness composition maps because they appear as composition maps, to describe the sequences generated by the 2-periodic difference equation ( an + un+1 a for n = 2` + 1, un+2 = , where an = (3.3) un b for n = 2`. More concretely, Fb,a (x, y) := (Fb ◦ Fa )(x, y) where Fa and Fb are the Lyness maps Fα (x, y) = (y, (α + y)/x) , associated to the autonomous Lyness recurrence un+2 = (α + un+1 )/un . These maps have been introduced and studied in [6, 10, 19]. Each map Fb,a has the first integral (bx + a) (ay + b) (ab + ax + by) Vb,a (x, y) = , (3.4) xy so it preserves each curve of the fibration given by the curves Ch = {(bx + a)(ay + b)(ax + by + ab) − hxy = 0} , h ∈ C. (3.5) By taking homogeneous coordinates [x : y : z] ∈ CP2 these curves extend to fh := {(bx + az)(ay + bz)(ax + by + abz) − hxyz = 0, x, y, z ∈ C}, C and the map Fb,a also extends to a polynomial one in CP2 Feb,a ([x : y : z]) = [ayz + y 2 : az 2 + bxz + yz : xy], (3.6) except at the points [x : 0 : 0], [−a : 0 : b] and [0 : −a : 1]. Continua of periodic points for planar integrable rational maps 9 Some computations (see the Appendix for a brief comment about them) show that, fh are elliptic except when h = 0 or h is a solution of when ab 6= 0, the curves C h4 + p3 (a, b)h3 + p2 (a, b)h2 + p1 (a, b)h + p0 (a, b) = 0, (3.7) for some polynomials pj (a, b) given in this appendix. In particular we get that equation (3.5) gives an invariant fibration with generic fibers of genus 1. Observe that for all the values of h, the curves C fh contain the three points at infinity [1 : 0 : 0], [0 : 1 : 0], [b : −a : 0]. It is not difficult to prove that taking O := [0 : 1 : 0], for any elliptic level h, the map (3.6) can be re-written as Feb,a|Cfh ([x : y : z]) = [x : y : z] + [1 : 0 : 0], (3.8) see [6]. Now, we can use Equation (3.1), which in our case writes as: p · [1 : 0 : 0] = [0 : 1 : 0], (3.9) in order to obtain the curves where are located the continua of periodic orbits with a prescribed period p. The main result is the following: Theorem 3.1. Consider the family of complex maps Fa,b given in (3.2), with ab 6= 0. The following statements hold: (i) The map F1,1 is globally 5-periodic. (ii) There are no elliptic curves Ch filled with 2 or 3 periodic points of these maps. (iii) The elliptic curves Ch filled with p-periodic points of these maps with period p ∈ {4, . . . , 10}, are given by those values a, b, and h 6= 0 such that h is not a solution of Equation (3.7) and cp (a, b, hp ) = 0, where: c4 (a, b, h) = h − BA, c5 (a, b, h) = (ab − 1) h + BA,   c6 (a, b, h) = h2 + a2 b2 − 2a3 − 2b3 + 3ab h + BA −a2 b2 + a3 + b3 − 2ab , c7 (a, b, h) = −h3 + a3 b3 − 4a2 b2 + 3a3 + 3b3 −3ab h2 −BA −4a2 b2 + 3a3 + 3b3 − 3ab h + B 3 A3 , c8 (a, b, h) = h + −2a3 − 2b3 + 5ab − 1 h2  3 +BA −a2 b2 + a3 + b3 − 4ab + 2 h − B 2 A2 , c9 (a, b, h) = −h5 + −4a2 b2 + 5a3 + 5b3 − 6ab h4 + a5 b5 − 10a4 b4 + 17a5 b2 + 17a2 b5 − 10a6 − 38a3 b3 −10b6 + 24a4 b + 24ab4 − 15a2 b2 h3 + BA 10a4 b4 − 17a5 b2 −17a2 b5 + 10a6 + 37a3 b3 + 10b6 − 26a4 b − 26ab4 + 19a2 b2 h2 − 4a4 b4 − 9a5 b2 − 9a2 b5 + 5a6 + 20a3 b3 + 5b6 − 14a4 b −14ab4 + 12a2 b2 B 2 A2 h + a4 b4 − 2a5 b2 − 2a2 b5 + a6 +5a3 b3 + b6 − 3a4 b − 3ab4 + 3a2 b2 B 3 A3 , 10 A. Gasull, M. Llorens, V. Ma˜nosa c10 (a, b, h) = (−ab − 1) h5 + −a3 b3 + 4a4 b + 4ab4 − 12a2 b2 + 5a3 + 5b3 −5ab) h4 + −a5 b5 + 6a6 b3 + 6a3 b6 − 6a7 b − 35a4 b4 − 6ab7 +41a5 b2 + 41a2 b5 − 10a6 − 56a3 b3 − 10b6 + 20a4 b + 20ab4 −10a2 b2 h3 + BA a5 b5 − 5a6 b3 − 5a3 b6 + 4a7 b + 30a4 b4 + 4ab7 −37a5 b2− 37a2 b5 + 10a6 + 55a3 b3 + 10b6 − 20a4 b − 20ab4 +10a2 b2 h2 − a5 b5 − 2a6 b3 − 2a3 b6 + a7 b + 12a4 b4 + ab7 −15a5 b2 − 15a2 b5 + 5a6 + 25a3 b3 + 5b6 − 10a4 b − 10ab4 +5a2 b2 B 2 A2 h + B 5 A5 , with A = a2 − b and B = a − b2 . Remark 3.2. (a) With respect statement (ii), it is easy to verify that indeed the maps Fb,a have not 2-periodic orbits. Moreover, when ab 6= 0, they have not 3-periodic orbits [6, Proposition 20]. In fact, F0,0 is globally 3-periodic. (b) The computations of the expressions cp (a, b, h) are formal, so they only charac- terize the curves filled with p-periodic points, provided that they are elliptic ones. For instance, when AB = 0, the condition c4 (a, b, h) = 0 is h = 0, and one ob- tains a value that does not correspond neither to an elliptic curve nor to a continua of 4-periodic points. That is the reason why we need to include the characteriza- tion of the elliptic curves as a hypothesis of statement (iii). (c) If we restrict our attention to a, b ∈ R and to continua of real p-periodic points it may happen some of the curves Chp have no real points. (d) We have also obtained the conditions for p ∈ {11, 12, 13, 14}. They are poly- nomials in h, with polynomials coefficients in a and b, and respective degrees 8, 7, 11 and 10. We omit their explicit expressions for the sake of shortness. They will appear in [21]. To prove the statements (ii) and (iii) of the above result we need to compute the elements of the form P + nQ, so we briefly recall the computation procedure. Consider the points Q = [1 : 0 : 0] and O = [0 : 1 : 0] which belong to the elliptic curves (Ch , +, O). For any point P ∈ Ch , the point P + Q is given by P + Q = (P ∗ Q) ∗ O. So, if we consider a point [x0 : y0 : 1], to compute [x1 : y1 : 1] = [x0 : y0 : 1] + Q, we first calculate [x0 : y0 : 1] ∗ Q by taking the horizontal line y = y0 passing through [x0 : y0 : 1] and Q, which cuts Ch in a second point [x1 : y0 : 1] = [x0 : y0 : 1] ∗ Q. Finally, to get [x1 : y0 : 1] ∗ O, we consider the vertical line x = x1 passing through [x1 : y0 : 1] which cuts Ch at the new point [x1 : y1 : 1] = [x0 : y0 : 1] + Q. Observe that since Q is tangent to the asymptote y = −b/a of Ch , the point Q ∗ Q, which is necessary to get 2Q = (Q∗Q)∗O, is obtained by computing the intersection of the line y = −b/a with Ch which is the affine point [x1 : −b/a : 1]. The expression of 2Q is finally obtained by computing the intersection of the line x = x1 , passing through [x1 : −b/a : 1], with Ch . Continua of periodic points for planar integrable rational maps 11 Proof of Theorem 3.1. (i) It is a simple computation. Moreover it is a consequence of the well-known fact that the Lyness map F1 is globally 5-periodic. (ii-iii) Applying the formal rules explained in the preamble to the proof we obtain, with the assistance of a computer algebra system, the points ±nQ, for n ≤ 5. We get Q = [1 : 0 : 0], [0 : −a : 1], 2Q = " # h − aA −h + AB 3Q = : :1 , bA abA " # h − A (a2 b − b2 + a) b ((ab − 1) Ah + BA2 ) 4Q = :− 2 :1 , a (−h + aA) h − (−b2 + 2a) Ah + aBA2 " −a (ah2 − A (2a2 − b) h + BA3 ) 5Q = : (−h + A (a2 b − b2 + a)) (−h + BA) # (−h + aA) p(a, b, h) :1 , b (−h + A (a2 b − b2 + a)) ((ab − 1) h + BA)   with p(a, b, h) = h2 + a2 b2 − 2a3 − 2b3 + 3ab h + BA −a2 b2 + a3 + b3 − 2ab . To compute the points −nQ, we use that −P = P ∗ (O ∗ O) (see [32, page 18]). In our case O ∗ O = [−a/b : 0 : 1]. We arrive to: −Q =[−b : 0 : 1], " # h − BA h + bB −2Q = :− :1 , abB aB " # aB ((ab − 1) h + BA) −h + B (−ab2 + a2 − b) −3Q = − : :1 , (h + bB) (−h + BA) b (h + bB) " − (h + bB) q0 (a, b, h) −4Q = : a (−h + B (−ab + a2 − b)) ((ab − 1) h + (−b2 + a) A) 2 # b (bh2 − B (−2b2 + a) h + AB 3 ) :1 , (−h + BA) (−h + B (−ab2 + a2 − b)) " (h − B (−ab2 + a2 − b)) q1 (a, b, h) −5Q = : b (bh2 − B (−2b2 + a) h + AB 3 ) q2 (a, b, h) # (−h + BA) q3 (a, b, h) : 1 , where ab ((ab − 1) h + BA) q4 (a, b, h) 12 A. Gasull, M. Llorens, V. Ma˜nosa   q0 (a, b, h) =h2 + a2 b2 − 2a3 − 2b3 + 3ab h + BA −a2 b2 + a3 + b3 − 2ab ,  q1 (a, b, h) = − h3 + a3 b3 − 4a2 b2 + 3a3 + 3b3 − 3ab h2  − BA −4a2 b2 + 3a3 + 3b3 − 3ab h + B 3 A3 ,   q2 (a, b, h) =h2 + a2 b2 − 2a3 − 2b3 + 3ab h + BA −a2 b2 + a3 + b3 − 2ab ,   q3 (a, b, h) =h3 − B 2a2 − 3b h2 + −a3 b2 + a4 − 3a2 b + 3b2 B 2 h − b2 AB 3 ,  q4 (a, b, h) =bh2 − B −2b2 + a h + AB 3 . The expressions of the points ±nQ are formal. They give affine points when all the denominators do not vanish, and they give infinite points otherwise. For instance, observe that 3Q is an infinite point if and only A = 0. Let us prove our results. Observe first that always 2Q = [0 : −a : 1] 6= O. The condition 3Q = O reads as 3Q = [a(h − aA) : −h + AB : abA] = [0 : 1 : 0]. Since ab 6= 0, we get that A = 0, but then 3Q = [ah : −h : 0] 6= O. Hence item (ii) follows. The p-periodicity conditions are obtained imposing that p + 1 p E Q = −E Q, p = 4, 5, . . . , 10, 2 2 where E(x) denotes the integer part of x. For instance, when p = 4, the condition 2Q = −2Q is [0 : −a : 1] = [h − BA : −(h + bB)b : abB], which holds if and only if h − BA = 0, giving rise to condition c4 (a, b, h) = 0. Similarly, to characterize the elliptic curves containing 6-periodic points we get that 3Q = −3Q is equivalent to:    h − aA aB ((ab − 1) h + AB)  =− , bA (h + bB) (−h + AB) 2 2  −h + AB = −h + B (−ab + a − b) .   abA b (h + bB) From the above equations we arrive to c6 (a, b, h) = 0. The other conditions can be deduced similarly. 4 A computational approach 4.1 Resultant periodicity conditions (RPC) Let Λ ∈ Kn be a set of parameters and let FΛ : U ⊂ K2 7→ K2 be a family of planar integrable rational maps with rational first integrals VΛ . We present a method, alternative Continua of periodic points for planar integrable rational maps 13 to the one of solving any of the system FΛ p (x, y) = (x, y), for finding continua of p- periodic points lying on the level sets of VΛ . The method consists to find the values of h for which the system  (FΛp )1 (x, y) = x, (4.1) VΛ (x, y) = h, has continua of solutions, where the subscript 1 (resp. 2) indicates the first (resp. sec- ond) component of a point. In this way the new unknowns of the problem are simply the values h of the level sets of VΛ (x, y) = h. These values of h have to be such that   φ(y, h) := Res numer (FΛp )1 (x, y) − x , numer (VΛ (x, y) − h) ; x (4.2) vanishes identically. In other words we need to collect the factors of the above resultant that only depend on h, if any. Denote by Dp (Λ, h) the product of these factors. We introduce the functions dp (Λ, h) as those factors of Dp (Λ, h) that remain after removing from this polynomial all the factors that already appear in some Dk (Λ, h) where k is either 1 or a proper divisor of p. We call the conditions dp (Λ, h) = 0 the resultant d-periodicity conditions associated to the first integral VΛ (RPC from now on). We remark that sometimes can be computationally more efficient to take the equa- tion (FΛp )2 (x, y) = y, instead of the first one in (4.1), or to do in (4.2) the resultant with respect to y instead to the resultant with respect to x. The main fact is that the energy levels filled with periodic points must satisfy the RPC. Nevertheless, a priori there is no guarantee that all the solutions of them are formed by periodic points because they satisfy only a necessary condition and, more- over, the above resultants could contain some spurious factors. Anyway, in all the ex- amples given in this section we check that the RPC obtained give actual continua of p-periodic points. In the birational case, the direct method for finding p-periodic points tries to solve some the systems FΛ m (x, y) = FΛm−p (x, y), for some natural number m < p, instead of system when m = p. Recall also that in the genus 1 case, it is much more difficult to study the periodicity condition pQ = O that the equivalent one mQ = −(p − m)Q, again for some natural number m, near p/2. Similarly, for birational maps, when com- puting the resultant (4.2) we reach to some computational obstructions that many times can be overcome by using the alternative equivalent formulation  φm (y, h) := Res numer (FΛm )1 (x, y) − (FΛm−p )1 (x, y) , (4.3) numer (VΛ (x, y) − h) ; x) , for some suitable m near p/2. All the RPC computed in these section have been obtained analyzing the factors of all the resultants (4.3), which have been calculated by using the computer algebra software MAPLE v.17 with an Intel Core 2Duo P8700 processor, and by exhausting the limits that this technology allows. The main computational obstruction comes from the fact that the size of the expressions of the resultants requires large resources to guarantee the allocation of enough memory for the computation. 14 A. Gasull, M. Llorens, V. Ma˜nosa 4.2 The feasibility region If we set K = C the RPC are always well defined, but when K = R, then some RPC dp (Λ, h) = 0 can have no real points. So we are only interesed on the cases where there exists a pair (Λ, h) ∈ Rn × R satisfying some RPC, dp (Λ, h) = 0, at some real points. We will call the feasibility region, to the subset R = {(Λ, h) ∈ Rn × R : {VΛ (x, y) = h} ∩ R2 6= ∅}. 4.3 Auxiliary results. Of course, we can apply the method blindly, but we insist that it will only work in the case that the continua of periodic points coincide with the level sets of the first integral. In the case of maps defined in R2 , we can guarantee this fact if for any fixed Λ there exists a Lie symmetry XΛ of the map FΛ with the same first integral VΛ (a Lie Symmetry is a vector field which is related with the dynamics of the map since FΛ maps any orbit of the differential system determined by the vector field, to another orbit of this system, see [9, 17]). From the results in [9] the existence of such a vector field guarantees that the action of the map on any connected components of the level sets of the first integral is conjugated to a linear transformation. Hence if on a closed invariant connected component of an energy level there is no singular point of the associated Lie symmetry XΛ and there is a p-periodic point of FΛ then the whole level set is filled with p-periodic points. It can be seen that if a map F has a first integral V ∈ C m+1 (U) on U ⊂ R2 and it preserves a measure absolutely continuous with respect the Lebesgue measure with non- vanishing density ν ∈ C m in U, it holds that the vector field X = ν −1 (−Vy , Vx ) is always a Lie Symmetry of F in U. As a consequence of this fact, and of the results of [5, 8], we have the following proposition that characterizes the existence of a continua of periodic points over the level sets of the first integral and proves some regularity properties of the rotation number function. Proposition 4.1. Let F be a C 2 -measure preserving map with an invariant measure with non-vanishing density ν ∈ C 1 (U) and with a first integral V ∈ C 2 (U). If V has a connected component γh of a level set {V (x) = h}, which is invariant by F and diffeomorphic to S1 , then F |γh is conjugate to a rotation. Moreover, if ρh denotes the rotation number of this map, then the function ρ(h) : h → ρh is continuous and tends to θ/(2π) when γh approaches to an non-degenerated fixed point with eigenvalues e±iθ , where 0 6= θ ∈ R. Finally, if F |γh has rotation number ρh = q/p ∈ Q, with gcd(p, q) = 1, then F has a continuum of p-periodic points in γh ⊂ U. Continua of periodic points for planar integrable rational maps 15 4.4 The McMillan-Gumovski-Mira maps In this section we consider a one-parameter family of maps defined in R2 , considered by E. M. McMillan in [24] and by I. Gumovski and C. Mira in [16],  y  Fa (x, y) = y, −x + (4.4) a + y2 for a > 0, that we will call McMillan-Gumovski-Mira maps. For any a > 0 this is a birational map with rational inverse Fa −1 (x, y) = ((x − ya − yx2 )/(a + x2 ), x). Both maps Fa and Fa−1 share the polynomial first integral Va (x, y) := a(x2 + y 2 ) + x2 y 2 − xy. (4.5) A computation shows that det(DFa (x, y)) ≡ 1, so Fa is an integrable map preserv- ing the Lebesgue measure and it has the Lie Symmetry Xa (x, y) = (x − 2ay − 2x2 y, 2ax − y + 2xy 2 ), (4.6) see [9, Example 2]. By Proposition 4.1, on any simple closed invariant connected com- ponent of the level curves, the map is conjugated to a rotation (such connected compo- nents exist, for instance, as a consequence of Lemma 4.3 below). Therefore, those levels for which the associated rotation number is rational are filled with periodic points. 4.4.1 Computation of the RPC for the McMillan-Gumovski-Mira maps The approach developed in the previous section allow us to prove the following result. Theorem 4.2. The RPC of the McMillan-Gumovski-Mira maps associated to the first integral (4.5) for p ≤ 10 are given by dp (a, h) = 0, where: d3 (a, h) =h + a(a + 1), d4 (a, h) =h + a2 ,   d5 (a, h) =h3 + a (3a − 1) h2 + a 3a3 − 2a2 − a − 1 h + a4 a2 − a − 1 ,  d6 (a, h) =h2 + 2a2 + 1 h + a4 ,  d7 (a, h) =h6 + 2a (3a + 1) h5 + a 15a3 + 10a2 − a + 3 h4 + a 20a5 + 20a4   −4a3 + 8a2 + a + 1 h3 + a4 15a4 + 20a3 − 6a2 + 6a + 2 h2   + a4 6a6 + 10a5 − 4a4 + a2 − a − 1 h + a9 a3 + 2a2 − a − 1 ,   d8 (a, h) =(2h2 + 4a2 + 1 h + a2 2a2 − 1 )(h4 + 4h3 a2 + 6h2 a4   + a2 2a2 − 2a + 1 2a2 + 2a + 1 h + a8 ), 16 A. Gasull, M. Llorens, V. Ma˜nosa  d9 (a, h) =h9 + 3a (3a − 1) h8 + a 36a3 − 24a2 − 7 h7 + a 84a5 − 84a4 − 41a2  +6a − 5) h6 + a 126a7 − 168a6 − 99a4 + 30a3 − 19a2 + 4a − 1 h5  + a2 126a8 − 210a7 − 125a5 + 60a4 − 26a3 + 8a2 − 3a + 1 h4  + a3 84a9 − 168a8 − 85a6 + 60a5 − 14a4 − 3a2 − 1 h3  + a7 36a7 − 84a6 − 27a4 + 30a3 − a2 − 8a − 1 h2   + a8 9a8 − 24a7 − a5 + 6a4 + a3 − 4a2 + 1 h + a15 a3 − 3a2 + 1 ,    d10 (a, h) =h6 + 6a2 + 3 h5 + 15a4 + 12a2 + 1 h4 + 2a4 10a2 + 9 h3     + 3a4 5a2 − 1 a2 + 1 h2 + a4 a2 + 1 6a4 − 3a2 + 1 h + a12 . Proof. For p = 2 we have that numer((Fa 2 )1 (x, y) − x) = −xa − y (xy − 1) . Then Res(numer((Fa 2 )1 (x, y) − x), Va (x, y) − h, x)   = a + y 2 4 ay 4 + 4 a2 y 2 −4 hy 2 − 4 ah − y 2 . Notice that in this case there is no factor without dependence on the variable y, so we can ensure that there are not energy levels formed by continua of 2-periodic points. We continue with p = 3. In this case  numer((Fa 3 )1 (x, y) − x) = − (x + y) a3 + −x3 − x2 y − 2 xy 2 − 2 y 3 − x a2   − y 2 yx3 + 2 y 2 x2 + xy 3 + y 4 − 2 x2 − 1 a − xy 2 xy + y 2 − 1 (xy − 1) , and Res(numer((Fa 3 )1 (x, y) − x), Va (x, y) − h, x) 4  = (a(a + 1) + h)2 y 2 + a y4 + 2 y2a − y2 − h . The only factor which is independent of y is a(a + 1) + h. Therefore the expression of d3 (a, h) follows. To compute the RPC corresponding to p = 5 we consider the equation Fa 3 (x, y) = Fa −2 (x, y). Then  numer((Fa 3 )1 (x, y) − (Fa −2 )1 (x, y)) = (x − y) a6 + (x − y) 3x2 + 3y 2 − 1 a5 + (x − y) 3x4 + 9x2 y 2 + 3y 4 − 2x2 − 3xy − 2y 2 − 1 a4 + (x − y)  × x6 + 9x4 y 2 + 9x2 y 4 + y 6 − x4 − 6x3 y − 4x2 y 2 − 6xy 3 − y 4 − x2 + 2xy − y 2 a3 + (x − y) 3y 2 x6 + 9x4 y 4 + 3x2 y 6 − 3x5 y − 2x4 y 2 − 12x3 y 3 − 2x2 y 4 − 3xy 5 +2x3 y + 2x2 y 2 + 2xy 3 − x2 + xy  − y 2 a2 + xy (x − y) (xy − 1) 3x4 y 2 + 3x2 y 4 −3x3 y − x2 y 2 − 3xy 3 + xy − 1 a + x3 y 3 (x − y) (xy − 1)3 and Res(numer((Fa 3 )1 (x, y) − (Fa −2 )1 (x, y)), Va (x, y) − h, x) = 6  d25 (a, h) y 2 + a y 4 + 2 ay 2 − y 2 − h . Continua of periodic points for planar integrable rational maps 17 Hence d5 (a, h) = 0 is the searched condition. Proceeding similarly for all the other values of p ≤ 10 we prove the theorem. 4.4.2 The feasibility region for the McMillan-Gumovski-Mira map Now we determine the feasibility region of a map Fa , that is, those pairs (a, h) ∈ R2 such that {Va (x, y) = h} ∩ R2 = {a(x2 + y 2 ) + x2 y 2 − xy = h} ∩ R2 6= ∅. Observe that the fixed points of a McMillan-Gumovski-Mira p map Fa are (0, 0) for any a > 0; and (±x0 (a), ±x0 (a)), where x0 (a) = 1/2 − a, when 0 < a < 1/2. These fixed points are critical points of the first integral (4.5), and also the unique sin- gular points of the Lie Symmetry (4.6). A straightforward analysis gives: Lemma 4.3. The fixed points of the map (4.4) are critical points of the first integral (4.5). Furthermore (i) If 0 < a < 1/2, the origin is a saddle of Va (x, y) and the points (±x0 (a), ±x0 (a)) are absolute minima. (ii) If a ≥ 1/2 the origin is an absolute minimum of Va (x, y). The value of the first integral Va at the absolute minima is given by ( −a2 + a − 1/4 if 0 < a < 1/2, hc (a) := 0 if a ≥ 1/2. (iii) When 0 < a < 1/2 and −a2 + a − 1/4 < h < 0 the level sets of Va (x, y) = h are formed by two disjoint closed curves, each one of them diffeomorphic to S1 . (iv) When h > 0 the level sets of Va (x, y) = h are closed curves, diffeomorphic to S1 . (v) When 0 < a < 1/2 the level set Va (x, y) = 0 is formed by the point (0, 0) and two closed loops starting and ending at this point. When a ≥ 1/2 the level set is the point (0, 0). As a consequence of the above result the solutions of the RPC, dp (a, h) = 0, cor- respond with nonempty curves {(x, y) ∈ R2 , Va (x, y) = h} for all h ≥ hc (a) and therefore, the region R = {(a, h), a > 0 and h ≥ hc (a)} ⊂ R+ × R, is the feasibility region for the map Fa . It is displayed in the blue-shaded region of Figure 4.1. Hence, Corollary 4.4. If there exist level curves {Va (x, y) = h} ⊂ R2 filled with p-periodic points, then these curves correspond with some pairs (a, h) ∈ {dp (a, h) = 0} ∩ R. Moreover, as a consequence of the existence of the Lie symmetry (4.6), whose sin- gular points are the fixed points of Fa , by applying Corollary 4.5 we have: Corollary 4.5. If (a, h) ∈ {dp (a, h) = 0} ∩ R, and there exists a p-periodic point of Fa on the curve {Va (x, y) = h}, then all the curve is filled with p-periodic points. 18 A. Gasull, M. Llorens, V. Ma˜nosa Figure 4.1: Intersection of the curve d5 (a, h) = 0 with the feasibility region of the McMillan-Gumovski-Mira map Fa (shaded in blue). 4.4.3 Analysis of the 5-periodic RPC. In this section we show how to determine the number and shape of the level curves associated to a particular RPC. Our approach, together with some well-known tricks (see Lemma 4.7 below), reduce this problem to a routine task that can be done for each period. In this section, as an illustration, we study with detail the case p = 5. The results are summarized as follows: Theorem 4.6. Consider the McMillan-Gumovski-Mira map (4.4) with a > 0 and define √ √ √ 3− 5 3+ 5 1+ 5 a1 = ≈ 0.048 < a2 = ≈ 0.328 < a3 = ≈ 1.618. 16 16 2 Then the set of real 5-periodic points is given by the smooth non-empty level sets Va (x, y) = h for the values of h satisfying d5 (a, h) = 0, with d5 given in Theorem 4.2. It is formed by: (i) Five disjoint closed curves diffeomorphic to S1 when a ∈ (0, a1 ). They correspond to two negative and one positive values of h. (ii) Three disjoint closed curves diffeomorphic to S1 when a ∈ [a1 , a2 ). They corre- spond to one negative and one positive values of h. (iii) One closed curve diffeomorphic to S1 when a ∈ [a2 , a3 ). It corresponds one posi- tive value of h. (iv) The empty set when a ≥ a3 . Continua of periodic points for planar integrable rational maps 19 a ∈ (0, a1 ) a ∈ [a1 , a2 ) a ∈ [a2 , a3 ) Figure 4.2: Level sets full of 5-periodic points according different values of a. In the left hand-side figure there are 5 disjoint simple closed curves, but two of them seem to collide at the origin because they pass very near it. Moreover, the clockwise rotation number on the level sets corresponding to the smallest value of h given in item (i) is 2/5 and on all the other level sets is 1/5. The overall situation is displayed and illustrated in Figures 4.1, 4.2 and 4.3. To prove Theorem 4.6 we will use the following result which is a combination of the results obtained in [13] and [14]. Denote ∆h (P ) the discriminant of the polynomial P (h) = an hn + · · · + a1 h + a0 , that is n(n−1) 1 ∆h (P ) = (−1) 2 Res(P (h), P 0 (h); h). an Lemma 4.7. Let Ga (h) = gn (a)hn + gn−1 (a)hn−1 + · · · + g1 (a)h + g0 (a), be a a family of real polynomials depending also polynomially on a real parameter a. Set Ia = (φ(a), +∞) where φ(a) is a continuous function. Suppose that there exists an open interval Λ ⊂ R such that: (i) There exists a0 ∈ Λ, such that Ga0 (h) has exactly r ≥ 0 simple roots in Ia0 . (ii) For all a ∈ Λ, Ga (φ(a)) · gn (a) 6= 0. (iii) For all a ∈ Λ, ∆h (Ga ) 6= 0. Then, for all a ∈ Λ, Ga (h) has exactly r simple roots in Ia . Proof. Observe that from the hypothesis (ii) the roots (real or complex) of Ga (h) depend continuously on a since gn (a) 6= 0. The same hypothesis ensures that there are no real roots trespassing the border of Ia when varying the parameter a. Then, the variation 20 A. Gasull, M. Llorens, V. Ma˜nosa of the number of real roots must be caused by the appearance of some multiple real root, due either to the collision of some real roots or to the collision of some couple of conjugated complex roots. In any case, since by (iii), ∆h (Ga ) 6= 0, multiple roots never appear. Hence the number of real roots of Ga (h) remains constant for all a ∈ Λ. Figure 4.3: Clockwise rotation number 2/5 of a 5-periodic orbit {(xn , yn )}n on a con- nected component of a level set with h < 0. The dots are the points of the orbit and the numbers near them indicate the subscript n mod 5. Proof of Theorem 4.6. Observe that d5 (a, h) can be rewritten as the one-parametric fam- ily of polynomials in h: Ga (h) := d5 (a, h) = g3 (a)h3 + g2 (a)h2 + g1 (a)h + g0 (a), where g3 (a) = 1, g2 (a) = a(3a−1), g1 (a) = a(3a3 −2a2 −a−1), g0 (a) = a4 (a2 −a−1). Remember that (a, h) ∈ R if and only if h ∈ [hc (a), +∞). We have to study the number of real zeros of Ga (h) = 0 in the feasibility region. We will use Lemma 4.7. First, observe that for all a > 0, ∆h (Ga (h)) = a3 (32a2 + 13a + 4) > 0. Therefore hypothesis (iii) is always satisfied. We split the proof in two cases: a ∈ (0, 1/2) and a ≥ 1/2. Case a ∈ (0, 1/2). In this situation Ia = (−a2 + a − 1/4, +∞). Since g3 (a) 6= 0, by the lemma, the number of real simple roots in Ga (h) is constant on any open interval I where Ga (φ(a)) 6= 0. A computation shows that  1 3 1 Ga (φ(a)) = Ga − a2 + a − = a − a2 − 4 8 64 Continua of periodic points for planar integrable rational maps 21 vanishes at a1 and a2 . Hence we know that Ga has a constant number of real roots in Ia for a in each of the intervals (0, a1 ), (a1 , a2 ) and (a2 , 1/2). Moreover, observe that since g0 (a) 6= 0 for all a ∈ (0, 1/2), the number of positive and negative solutions of Ga (h) = 0 neither changes for a in any of the three open intervals. Hence to know the number of level sets candidate to be full of 5-periodic points, and the sign of the corresponding values of h, it suffices to study one concrete value of a in each of the intervals (0, a1 ), (a1 , a2 ) and (a2 , 1/2) together with the values a = a1 and a = a2 . We only detail the computations taking a = 1/50 ∈ (0, a1 ). For this value, 47 2 127597 2549 G1/50 (h) = h3 − h − h− . 2500 6250000 15625000000 It is easy to prove that this polynomial has three simple roots, two negative and one pos- itive, all located in I1/50 . Therefore the same holds for all a ∈ (0, a1 ). By items (iii) and (iv) of Lemma 4.3, these values give rise to five disjoint closed curves, diffeomorphic to S1 , candidates to be full of 5-periodic points. To ensure that the non-empty level curves corresponding to d5 (a, h) = 0, are full of 5-periodic points (recall that conditions dk (a, h) = 0 are only necessary conditions of periodicity) we need to continue our analysis. The rotation numbers of Fa on each of the found level sets can be obtained either following the orbit of a given 5-periodic point, see Figure 4.3, or using the properties of the rotation number function introduced in Proposition 4.1. In particular we need to control the limits of the rotation numbers when we level sets tend to infinity, or to the level sets containing the fixed points. By using the same tools that in [11, Proposition 4] we can prove that the rotation number function ρ(h) tends to 1/4 when h tends to infinity, tends to 0 when h tends to 0 (this level set corresponds to two homoclinic orbits tending to the origin) and tends to arccos(4a − 1)/(2π) when the the ovals of the level sets degenerate to the fixed points (±x0 (a), ±x0 (a)). From these limiting behavior of ρ(h) and because it is continuous, we obtain that for a ∈ (0, a1 ) there are at least two values of h < 0 such that ρh is 1/5 and 2/5, since these two values belong to the interval (0, arccos(4a−1)/(2π)), and there is one positive value of h such that ρh = 1/5, because this value belongs to the interval (0, 1/4). Similarly, when a ∈ [a1 , a2 ) there are at least one positive and one negative value of h both corresponding to the rotation number 1/5 and when a ∈ [a2 , 1/2) there is at least one positive value of h also corresponding to the rotation number 1/5. In short, when a ∈ (0, 1/2), we have proved that the found real level sets given by d5 (a, h) = 0 are actually full of 5-periodic points and have the rotation numbers described in the statement of the theorem. Case a ≥ 1/2. Here Ia = (0, +∞). Note that Ga (0) = g0 (a) = a4 (a2 − a − 1), has only one positive root, a = a3 . So we have to determine the number of roots of Ga in (0, ∞) when either a ∈ (1/2, a3 ) or when a ∈ (a3 , ∞). Hence, we can reduce the 22 A. Gasull, M. Llorens, V. Ma˜nosa problem to study the values a = 1/2, a = a3 and two (rational) numbers, one in each of the intervals (1/2, a3 ) and (a3 , ∞). The rest of the study follows as in the previous case and we omit the details. 4.5 The Bastien-Rogalski and the Lyness 2-periodic maps revisited We consider again the one-parameter family of maps (2.1) and the family of maps (3.2) studied in Sections 2.2 and 3.2, respectively. Applying the method described in this section we obtain: Theorem 4.8. (i) The RPC associated to the first integral (2.2) for the family of Bastien and Rogalski maps (2.1), dp (a, h) = 0, coincide for each p ≤ 15 with the corresponding expression cp (a, h) = 0 given in Theorem 2.1. (ii) The RPC associated to the first integral (3.4) for the family of Lyness 2-periodic maps (3.2) coincide, for each p ≤ 10, with the corresponding expression cp (a, h) = 0 given in Theorem 3.1. On one hand the methods used to prove Theorems 2.1 and 3.1 allow to obtain condi- tions for the existence of continua of periodic points for bigger values of p that the ones obtained with the method introduced in this section. Nevertheless, on the other hand, this new approach is merely computational and it does not require the knowledge nei- ther of the proper rational parametrization in the first case, nor of the expression of the map in terms of the group operation on the elliptic curve (3.8) in the second situation. 5 A final comment All the methods presented in this paper strongly rely on the fact that the continua of periodic points lie on the level sets of the first integral. This is a feature that integrable birational maps, among others, accomplish. But this is not the scenario that one should expect when integrable diffeomorphisms are considered. Indeed, let us consider a real planar map F that preserves a fibration of curves given by {V = h}{h∈H} where H ⊂ R is an open set and such that for each h ∈ H each curve {V = h} ∼ = S1 . We can think that Fh := F|V =h defines an one-parametric family of circle maps. If in addition the maps Fh are orientable diffeomorphisms, then generically (i.e. in an open and dense set in the space of orientable diffeomorphisms with an appropriate topology, [26]) the persistence of the rotation number is known to hold, [2, Theorem A] and [25] (see also [1]), so that generically the function ρ(h) giving the rotation number associated to each map Fh is either constant or a cantor function (very far, thus, from scenario that appear when birational maps with rational first integrals are considered). In the latter case, if there exists h0 ∈ H such that ρ(h0 ) = q/p, then there exists an open set h0 ∈ U such that for all h ∈ U ⊂ H the function ρ(h) ≡ q/p. So at each Continua of periodic points for planar integrable rational maps 23 level set there are p-periodic points, [26]. By continuity of the solutions of the system Fhp (x, y) = Id with respect the parameter h, we get that the periodic points at each level set form a continuum. The fact that generically these periodic points are hyperbolic implies that this continuum do not coincide with the level sets {V = h}. Acknowledgements. The authors are supported by Ministry of Economy and Competitiveness of the Span- ish Government through grants MTM2013-40998-P (first author) and DPI2011-25822 (third author). The first author is also supported by the grant 2014-SGR-568 from AGAUR, Generalitat de Catalunya and BREUDS project FP7-PEOPLE-2012-IRSES- 318999. The third author is supported by the grant 2014-SGR-859 from AGAUR, Gen- eralitat de Catalunya. Appendix The non-elliptic levels of the curves (3.5) can be found either by looking for the singular points (see [6]), or computing the roots of the denominator of the j-invariant associated to them [31]. Here we present a Magma code that allows to do this computation. K:=FunctionField(Rationals(),3); A:=AffineSpace(K,2); C:=Curve(A,(b*x+a)*(a*y+b)*(a*x+b*y+a*b)-h*x*y; CP:=ProjectiveClosure(C); P:=CP![0,1,0]; E:=EllipticCurve(CP,P); Factorization(Denominator(jInvariant(E))); Processing this code in the Magma’s online calculator page [23], one obtains that the non-elliptic cases occur either when ab = 0 or when ab 6= 0 and h = 0 or h is a root of the polynomial (3.7), h4 + p3 (a, b)h3 + p2 (a, b)h2 + p1 (a, b)h + p0 (a, b) = 0, where p3 (a, b) , p2 (a, b) , p1 (a, b) and p0 (a, b) are given by: p3 (a, b) = −4 a2 b2 − 4 a3 − 4 b3 + ab, p2 (a, b) = 6 a4 b4 + 4 a5 b2 + 4 a2 b5 + 6 a6 − 21 a3 b3 + 6 b6 − 3 a4 b − 3 ab4 , p1 (a, b) = −4 a6 b6 + 4 a7 b4 + 4 a4 b7 + 4 a8 b2 − 25 a5 b5 + 4 a2 b8 − 4 a9 +18 a6 b3 + 18 a3 b6 − 4 b9 + 3 a7 b − 21 a4 b4 + 3 ab7 ,   3 p0 (a, b) = −b2 + a a2 − b −a2 b2 + a3 + b3 . 24 A. Gasull, M. Llorens, V. Ma˜nosa References [1] Ll. Alsed`a, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one. 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