This is a preprint of: “Vanishing set of inverse Jacobi multipliers and attractor/repulsor sets”, Isaac A. Garcı́a, Jaume Giné, Jaume Llibre, Susanna Maza, Chaos, vol. 31, 013113:1–9, 2021. DOI: [10.1063/5.0020360] VANISHING SET OF INVERSE JACOBI MULTIPLIERS AND ATTRACTOR/REPULSOR SETS ISAAC A. GARCÍA, JAUME GINÉ, JAUME LLIBRE, AND SUSANNA MAZA Abstract. In this paper we study conditions under which the zero- set of the inverse Jacobi multiplier of a smooth vector field contains its attractor/repulsor compact sets. The work generalizes previous re- sults focusing on sink singularities, orbitally asymptotic limit cycles and monodromic attractor graphics. Taking different flows on the torus and the sphere as canonical examples of attractor/repulsor sets with differ- ent topologies, several examples are constructed illustrating the results presented. 1. Introduction and main results We consider C 1 autonomous ordinary differential equations ẋ = f (x) de- in Rn where x = (x1 , . . . , xn ) ∈ Rn and its associated vector field fined P X = ni=1 fi (x)∂xi where f (x) = (f1 (x), . . . , fn (x)). P We denote by sing(X ) the set of singular points of X and by divX = ni=1 ∂fi (x)/∂xi its diver- gence. Let φt be the flow associated to X with φ0 the identity function. A set Γ ⊂ Rn is invariant if φt (Γ) = Γ for any t for which the flow is defined. We define the distance from a point p ∈ Rn to the set Γ as d(p; Γ) = inf{d(p, q) : q ∈ Γ} where d(p, q) is the Euclidean distance between the points p and q. If d(φt (p); Γ) → 0 when t → ∞ we will write φt (p) → Γ as t → ∞. The basin of attraction of a set Γ is the set {p ∈ Rn : φt (p) → Γ}, that is the set of initial conditions that approaches to Γ asymptotically under the forward flow. The basin of attraction is an open set. There are several equivalent ways to define an attractor set. In [12] and [13] Milnor introduced a definition of attractor. A set Γ ⊂ Rn is an at- tractor set if it possesses a compact neighborhood U ⊂ Rn and there is a positive increasing time sequence {tn }n∈N ⊂ R+ with tn → ∞ such that Γ = ∩n∈N φtn (U ) where the nested sequence of sets U ) φt1 (U ) ) φt2 (U ) ) · · · holds, this U is called a trapping neighborhood of Γ. This intersection Γ is always invariant, φt (Γ) = Γ. 2010 Mathematics Subject Classification. 34C05, 34C25, 34C45, 37C10, 37C70. Key words and phrases. Attractor set, repulsor set, Jacobi multiplier, vector fields, flows, ordinary differential equations. 1 2 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA We recall that a non-compact set Γ such that φt (p) → Γ as t → ∞ for any point p ∈ Rn is not necessarily an invariant set. An example with n = 2 is given by the curve Γ = {(x, y) ∈ R2 : y = exp(−x) sin(x)} and the flow φt (x, y) = (x + t, y exp(−t)) associated to the system ẋ = 1, ẏ = −y. There is a simple sufficient condition to guarantee that a compact invari- ant set Γ is indeed an attractor. It is based on the Liouville divergence-flow relation Z d |φt (U )| = div(X )dx. dt t=t0 φt0 (U ) for any Lebesgue measurable subset U ⊂ Rn , see a proof in Proposition 14 of the Appendix. Here |A| denotes the volume of the set A. From this expression it follows easily that, if φt (U ) ⊆ U and div(X )|U < 0 then the volume of U contracts to zero under the flow, that is, |φt (U )| → 0 as t → ∞. In particular, if U is compact then it is a trapping neighborhood of an attractor set Γ ⊂ U with |Γ| = 0. A neighborhood U of a subset Γ ⊂ Rn is a subset of Rn which contains an open set containing Γ. The ε-neighborhood of Γ is the set Uε (Γ) = ∪p∈Γ Uε (p), where for any point p ∈ Rn , Uε (p) is the open ball centered at p with radius ε. We say that the orbit through a point p of an invariant set Γ is dense if, for any ε > 0 and any q ∈ Γ, there is a time τ such that φτ (p) ∈ Uε (q). An attractor is an attractor set which contains a dense orbit, that is, the flow is transitive. This extra condition is included to remove the possibility that an attractor be the union of smaller attractor sets. An attractor is minimal if all its orbits are dense. A repulsor, a repulsor set and a bassin of repulsion are an attractor, an attractor set and a bassin of attraction for t → −∞, respectively. An important tool in the analysis of the dynamics of vector fields X in Rn is the determination of manifolds embedded in Rn which are invariant under the flow of X . An invariant manifold defined by the zero-set of a C 1 function F : Rn → R has associated a continuous cofactor K if X (F ) = KF holds. Sign defined cofactors of attractor invariant sets are good tools to localize their basin of attraction as the following theorem shows. Our main results are the following two theorems. Theorem 1. Let X be a C 1 vector field in Rn generating a flow φt (.) and having a C 1 invariant manifold {F = 0} ⊂ Rn with associated continuous cofactor K. Let Ω ⊂ Rn be an open set such that Ω ∩ {F = 0} = ∅, φt (Ω) ⊆ Ω, the restriction F |Ω is well defined and the restriction K|Ω is sign defined and uniformly bounded. Then Ω\sing(X ) belongs to the basin of attraction or repulsion of {F = 0} if K|Ω < 0 or K|Ω > 0, respectively. Theorem 1 is proved in section 2. VANISHING SET OF INVERSE JACOBI MULTIPLIERS 3 A C 1 function V : N ⊂ Rn → R non locally null in N is said to be an inverse Jacobi multiplier for the vector field X in N if V is a solution of the linear first order partial differential equation X V = V divX . This kind of functions 1/V were used by Jacobi to find the last additional first integral needed to get complete integrability of X although currently they have been used to analyze several dynamical properties of X . The special case n = 2 has been treated in many papers, see [7] and references therein and some applications in the higher dimensional case n ≥ 3 have been reported in [2, 3]. The next theorem relates the zero-set of inverse Jacobi multipliers and the compact attractor/repulsor sets and it is a generalization of some results in [8, 4] for n = 2 and in [1] when n ≥ 2. Theorem 2. Consider a C 1 vector field X in Rn and let Γ ⊂ Rn be any zero measure compact attractor/repulsor set for its associated flow φt . Let V be any inverse Jacobi multiplier of X defined in some neighborhood of Γ. The following statements hold. (a) There exists a point p ∈ Γ such that V (p) = 0, hence V (φt (p)) = 0 for all t ∈ R. (b) If Γ is a minimal attractor/repulsor then Γ ⊂ V −1 (0). Theorem 2 is proved in section 2. We remark that the proof of Theorem 2 is essentially different than the one given in Theorem 17 of [1] for the special case when Γ is an asymptotically orbitally stable limit cycle. Our method is more general because it is based on the saturation under the flow of a (global) trapping neighborhood of Γ, while the proof of [1] needs the saturation of just a neighborhood of a point of Γ under the extra assumption that |Γ| = 0. The reader may consult [11] to see how different stability definition are related between them. We recall that an invariant set Γ is orbitally stable if for any neighborhood W ⊂ Rn of Γ there exists another neighborhood U ⊂ W of Γ such that φt (U ) ⊂ W . Moreover Γ is orbitally asymptotically stable if it is orbitally stable and there is a neighborhood U of Γ such that for any point p ∈ U it follows that d(φt (p); Γ) → 0 when t → ∞. The next proposition proves why our approach is more general than the one given in [1]. Proposition 3. Let Γ be a compact invariant set orbitally asymptotically stable for a flow φt . Then Γ possesses a trapping neighborhood. Proposition 3 is proved in section 2. The paper is organized as follows. The second section is dedicated to prove our main results. In section 3 we show some nontrivial applications of the presented theory focusing on flows with invariant tori and spheres, while in section 4 we consider other examples. 4 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA 2. Proof of the main results Proof of Theorem 1. The first step furnishes an expression (see equation (1)) for F along the orbits of the system. This formula, although simple, will provide the main tool to prove the theorem. Using the method of character- istics we can integrate the expression X (F ) = KF along the regular orbit of X through any point p ∈ Rn with p 6∈ sing(X ) in some time interval. This can be achieved by realizing that d ((F ◦ φt )(p)) = (X (F ) ◦ φt )(p) = (KF ) ◦ φt )(p). dt The outcome is that we get Z t  (1) (F ◦ φt )(p) = F (p) exp (K ◦ φτ )(p) dτ . 0 We consider any point p ∈ Ω\sing(X ) and let κ be the uniform bound of the cofactor K in Ω, that is, |K(x)| ≤ κ for any x ∈ Ω. From (1) and the fact that φt (Ω) ⊆ Ω it follows that Z t   |F (p)| exp (κt) if K|Ω > 0, |F (φt (p))| = |F (p)| exp K(φτ (p)) dτ ≤ 0 |F (p)| exp (−κt) if K|Ω < 0. Therefore, limt→±∞ |F (φt (p))| = 0 according to whether K|Ω < 0 or K|Ω > 0, respectively. This proves the theorem.  Remark 4. Theorem 1 in the particular case that Ω = Rn \{F = 0} implies that {F = 0} is a global attractor or repulsor set according to whether K|Ω < 0 or K|Ω > 0, respectively. Remark 5. The Cauchy-Kovalevskaya Theorem [5] guarantees the existence of a local analytic solution F around any given non-singular point of an analytic vector field X of the partial differential equation X (F ) = KF with a fixed K. But Theorem 1 needs that F and K be globally defined on Ω, see for example (1). A very important particular case consists in the class of polynomial vector fields of degree m and its eventual invariant algebraic variety {F = 0}. Note that in this particular case the associated cofactors K are also polynomials and its degree is bounded by m − 1, hence the constant cofactors are natural candidates to apply Theorem 1. Proof of Theorem 2. We prove the theorem assuming that Γ is a compact attractor set, a similar proof can be done if Γ is a repulsor set. Given a Lebesgue measurable set D ⊂ Rn , we denote by |D| its volume. Given any fixed measurable set D ⊂ N \ V −1 (0), we define the integral invariant as the volume integral Z 1 I(t; D) = dx. φt (D) V VANISHING SET OF INVERSE JACOBI MULTIPLIERS 5 If V is an inverse Jacobi multiplier of the vector field X in N ⊂ Rn then its zero-set V −1 (0) = {p ∈ N | V (p) = 0} is composed of orbits of X in N . Indeed for any point p ∈ N , denoting by φt the flow of X , it follows as in (1) that Z t  (2) V (φt (p)) = V (p) exp divX ◦ φs (p) ds . 0 In particular, if V (p) = 0 then V (φt (p)) = 0 for all t. Let N ⊂ Rn be a neighborhood of Γ such that V |N is defined. Let U be trapping neighborhood of Γ, hence U is compact and there is a positive increasing time sequence {tn }n∈N ⊂ R+ with tn → ∞ such that \ Γ= φtn (U ) n∈N with U ) φt1 (U ) ) φt2 (U ) ) · · · . In particular, Γ is an invariant set. Without loss of generality we may assume that U ⊂ N . If V vanishes at any point q ∈ U \Γ then from (2) V vanishes on all the orbit φt (q) for any t > 0 and by the continuity of V , V vanishes in at least one point of Γ, and consequently if Γ is a minimal attractor V vanishes in all Γ. So we assume the above does not happen, that is, that V |U \Γ 6= 0. Under our assumptions the volume |φt (U )| is defined for all t ≥ 0 and, since Γ is compact and has zero measure, |φt (U )| → 0 as t → ∞. We consider the compact set D = {φt (p) | p ∈ ∂U and 0 ≤ t ≤ t1 } ⊂ U and the integral invariant Z 1 I(t; D) = dx. φt (D) V For any fixed t ≥ 0, by the Mean Value Theorem for Integrals there exists a point p(t) ∈ φt (D) such that 1 I(t; D) = |φt (D)|. V (p(t)) Clearly |φt (D)| is defined for all t ≥ 0 and |φt (D)| → 0 as t → ∞, hence 1/V (p(t)) → ∞ as t → ∞, because I(t; D) is independent of t. We conclude that V (p(t)) → 0 and since p(t) → p ∈ Γ, by continuity of V , V (p) = 0 proving statement (a). Under the hypothesis of (b) Γ is moreover a minimal attractor. Then considering again the point p ∈ Γ with V (p) = 0 predicted by statement (a), its associated orbit γp = {φt (p) : t ∈ R} is dense on Γ. Recall that γp ⊂ V −1 (0) from (2), so by the continuity of V we claim that Γ ⊂ V −1 (0). This last assertion follows taking into account the topological considerations that the zero-set of a continuous function is a closed set (the pre-image of 6 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA the closed set {0}) and that the topological closure of a dense subset is the whole set.  Proof of Proposition 3. Let U1 ⊂ Rn be a sufficiently small compact neigh- borhood of the orbitally asymptotically stable set Γ. Then for any point p ∈ ∂U1 on the boundary of U1 there is a point p∗ (p) ∈ ∂U1 and a time τ (p) ≥ 0 such that φτ (p) (p) = p∗ (p) ∈ ∂U1 and φt (p∗ ) ⊂ U1 \∂U1 for all t > 0. Since U1 is compact, the function τ : ∂U1 → R is bounded and more- over it is a continuous function by the theorem of continuous dependence of solutions on initial conditions. Hence defining 0 ≤ T1 = maxp∈∂U1 {τ (p)} it follows that Γ ⊂ φt (U1 ) ( U1 for any t > T1 . We define U2 = φτ1 (U1 ) for some fixed τ1 > T1 . Repeating the above arguments we construct a sequence {Uj }j∈N ⊂ Rn of compact neighborhoods of Γ and a sequence {τj }j∈N ⊂ R of positive times such that Γ ⊂ Uj and φτj (Uj ) := Uj+1 ( Uj for any j. As P a result, defining the time sequence {tj }j∈N ⊂ R as tj = ji=1 τi and taking U = U1 we may rewrite the former as U ) φt1 (U ) ) φt2 (U ) ) · · · , which implies that U is a trapping neighborhood of Γ.  3. Some applications 3.1. Flows with invariant torus. We consider (embedded in R3 ) a 2- dimensional torus T2 = S1 × S1 . In cartesian coordinates we describe the centered at the origin by T2 = {F = 0} ⊂ R3 with F (x, y, x) = torus p (R − x2 + y 2 )2 + z 2 − r2 , where R is the radius from the origin to the center of the generating torus, and r is the radius of the torus with r < R, see Figure 1. Figure 1. The two generating radii of T2 . A local curvilinear coordinate system (ρ, θ, ϕ) well adapted to the solid torus with boundary T2 is defined by (3) x = (R + ρ cos θ) cos ϕ, y = (R + ρ cos θ) sin ϕ, z = ρ sin θ VANISHING SET OF INVERSE JACOBI MULTIPLIERS 7 with 0 ≤ ρ ≤ r and (θ, ϕ) ∈ S1 ×S1 being S1 = R/2πZ. In these coordinates, T2 = {ρ = r} and the points of the torus are parameterized by the angles (θ, ϕ). Therefore a vector field generating a flow on T2 is given by θ̇ = A(θ, ϕ), ϕ̇ = B(θ, ϕ), where A and B are C 1 functions defined on T2 , hence 2π-periodic in both variables. Using these coordinates we analyze the zero-set of the inverse Jacobi mul- tipliers of a family of decoupled 3-dimensional analytic vector fields X having invariant attractor T2 and such that the restricted vector field X |T2 satisfies the hypothesis of Kolmogorov’s theorem (see [14]) for the flows on the torus. Proposition 6. Consider the decoupled analytic system ρ̇ = (r − ρ)f (ρ, θ, ϕ), (4) θ̇ = g(θ, ϕ), ϕ̇ = h(θ, ϕ), with f (r, θ, ϕ) > 0, g 2 +h2 > 0 and where the restricted system to the invari- ant torus T2 = {ρ = r} possesses an analytic invariant Lebesgue measure with positive density. Then T2 is an attractor set and if there is one analytic inverse Jacobi multiplier V of (4) such that T2 ⊂ V −1 (0), then the zero-set of any other analytic inverse Jacobi multiplier (4) defined in a neighborhood of T2 must contain T2 . Proof. System (4) possesses the invariant torus T2 = {ρ = r} with associated cofactor −f (ρ, θ, ϕ). Hence, if f (r, θ, ϕ) > 0 for any (θ, ϕ) ∈ T2 then T2 is an attractor by simple inspection of the first component of the field (4) or by Theorem 1. The restricted vector field X |T2 = g(θ, ϕ)∂θ + h(θ, ϕ)∂ϕ satisfies the hypothesis of Kolmogorov’s theorem for flows on the torus, that is, is analytic, fixed-point-free (g 2 + h2 > 0) and possesses an analytic invariant measure with positive density. Then Kolmogorov’s theorem (see [14]) assures the existence of an analytic diffeomorphism ξ on T2 which transforms X |T2 into the orbitally linearizable vector field ξ∗ (X |T2 ) = A(θ, ϕ)(∂θ + ω∂ϕ ) with A(θ, ϕ) > 0 and ω ∈ R. In particular, the diffeomorphism (ρ, θ, ϕ) 7→ (ρ, ξ(θ, ϕ)) and the time rescaling dividing the field by A(θ, ϕ) transforms (4) into (5) ρ̇ = (r − ρ)fˆ(ρ, θ, ϕ), θ̇ = 1, ϕ̇ = ω where fˆ(ρ, θ, ϕ) = f (ρ, ξ(θ, ϕ))/A(θ, ϕ). By hypothesis there is an analytic inverse Jacobi multiplier V of (5) such that T2 ⊂ V −1 (0). We will prove (by contradiction) that then there is no analytic inverse Jacobi multiplier V̂ such that T2 6⊂ V̂ −1 (0). The case ω 6∈ Q is trivial by Theorem 2(b) because T2 is a minimal attractor of (5) so we restrict ourselves to the remaining case ω ∈ Q where T2 is foliated by periodic orbits. We define Σ = T2 ∩ V̂ −1 (0) which is either empty or composed by a union of periodic orbits of (5) on T2 . We recall that a real analytic function of 8 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA several variables that vanishes on a set of positive measure must be identi- cally zero. Then, if Σ has positive measure and the function V̂ |T2 is analytic then it must be identically zero and therefore we reach the contradiction T2 ⊂ V̂ −1 (0). Hence Σ must be a zero-measure set on T2 . Notice that if V̂ exists then H = V /V̂ would be a non-trivial first integral of (5) well defined in T2 \Σ and satisfying (6) T2 \Σ ⊂ H −1 (0). Since T2 6⊂ V̂ −1 (0), there exists at least one periodic orbit C ⊂ T2 of (5) such that C 6⊂ Σ. The flow on the torus is given by θ(t; θ0 ) = θ0 + t( mod 2π), ϕ(t; ϕ0 ) = ϕ0 + ωt( mod 2π). Then C = {(r, θ(t; θ0 ), ϕ(t; ϕ0 )) : 0 ≤ t < T } where T = 2πq is the minimal period when ω = p/q is irreducible. We will see that there is no (non-trivial) first integral of (5) well defined in T2 \Σ and satisfying (6). We denote by ρ(t; (ρ0 , θ0 , ϕ0 )) the first component of the flow associated to (5), there is r∗ > 0 such that ρ(t; (ρ0 , θ0 , ϕ0 )) → r as t → ∞ for any r − r∗ < ρ0 < r + r∗ , because T2 is an attractor. Hence the stable manifold WCs ⊂ R3 of C satisfies WCs ⊃ MC = {(ρ0 , θ(t; θ0 ), ϕ(t; ϕ0 )) : r − r∗ < ρ0 < r + r∗ , 0 ≤ t < T } where MC is a two-dimensional manifold such that MC ∩T2 = C. The planar restricted vector field X |MC has the limit cycle C. If there was one first integral H of (5) well defined in T2 \Σ then its restriction H|MC would be either a constant or a (nontrivial) first integral of X |MC near C. The last case produce clearly a contradiction because there is no first integral near a limit cycle in a planar system. So we continue assuming the first option, that is, H|MC is a constant. Then, for any C 6⊂ Σ, H|MC = H|C = 0 where the last equality follows from (6), hence H|T2 \Σ = 0. In summary, defining [ M= MC ⊂ R3 , C⊂T2 \Σ we have proved that H|M = 0. We conclude that H ≡ 0 in a neighborhood of T2 because the measure of Σ is zero and therefore M has positive measure in R3 . This implies that V̂ cannot exists.  Remark 7. In the particular case that fˆ only depends on ρ, system (5) has the inverse Jacobi multiplier V = (r − ρ)fˆ(ρ) where clearly T2 ⊂ V −1 (0) holds. If moreover we take fˆ(ρ) ≡ 1 then the linear vector field (5) has the first integral H = cos(ϕ − ωθ) well defined in R3 when ω ∈ Z although condition (6) does not hold. Given a C 1 vector field X , we say that a compact attractor set Γ ⊂ Rn is monodromic if there is a compact transversal section Σ to the flow with Σ ∩ Γ 6= ∅ and Σ ⊂ U , where U is a trapping neighborhood of Γ, where a Poincaré map Π : Σ → Σ is defined. Moreover if (7) Π(Σ) ⊆ Int(Σ), VANISHING SET OF INVERSE JACOBI MULTIPLIERS 9 where Int(Σ) denotes the set of all interior points of Σ, then Σ is called a trapping region for Π. Applying Π to both sides of (7) and taking into account that Π is a diffeomorphism, hence Π(Int(Σ)) = Int(Π(Σ)), and it follows that the iterated of Π satisfy Πn (Σ) ⊆ Int(Πn−1 (Σ)) for any n ∈ N. This means that \ Σ∩Γ= Πn (Σ). n∈N We concentrate on the particular case that Γ = T2 (diffeomorphically) and that Σ ∩ Γ = S1 (diffeomorphically) is a transversal section of the flow on T2 generated by the vector field Y = XT2 = A(θ, ϕ)∂θ + B(θ, ϕ)∂ϕ . Notice that for those vector fields Y having no singular points and such that B > 0 we may consider the differential equation dθ (8) = F(θ, ϕ) dϕ with F = A/B and Σ = {(θ, ϕ) : ϕ = 0} ⊂ T2 is the parameterization of the cross section. Thus we consider its Poincaré map Π̂ = Π|T2 : S1 → S1 defined as Π̂(θ0 ) = θ(2π; θ0 ) being θ(ϕ; θ0 ) the solution of (8) with initial condition θ(0; θ0 ) = θ0 . Clearly Π̂ is an orientation preserving diffeomorphism on the circle because orbits on T2 do not intersect. Consequently, without loss of generality we have (9) Π̂0 (θ) > 0 for all θ ∈ S1 , where the prime means derivative with respect to θ. Any periodic orbit φt (p) with p = (θ∗ , 0) ∈ S1 ⊂ T2 will correspond with a fixed point of some iterate of Π̂, that is, there is k ∈ N such that Π̂k (θ∗ ) = θ∗ . If we deal with a family of vector fields Xλ depending on some parameters λ and T2 persists as invariant manifold under small parameter variations (for example under normally hyperbolicity conditions), then the resulting Poincaré map Π̂λ changes its orbit structure. To understand these bifurcations the rotation number of Π̂λ is introduced. Recall that (12) restricted to T2 has a periodic orbit if and only if the rotation number Π̃n (θ) − θ 1 a(θ) + a(Π̂(θ)) + · · · + a(Π̂k−1 (θ)) Rot(Π̂) = lim = lim n→∞ n k→∞ 2π k of Π̂ is a rational number, see for example [14], where Π̃ : R → R is a lift of Π̂ and Π̂(θ) = θ + a(θ). We can prove the following general result. Theorem 8. Consider a C 1 vector field Y on a manifold diffeomorphic to a torus T2 having a cross section diffeomorphic to S1 and a rational rotation number. Then any inverse Jacobi multiplier of Y must vanish on all the periodic orbits of Y. 10 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA Proof. If a periodic orbit of Y = A(θ, ϕ)∂θ +B(θ, ϕ)∂ϕ passes trough a point (θ, ϕ) = (θ∗ , 0) ∈ S1 ⊂ T2 , then there is some k ∈ N with Π̂k (θ∗ ) = θ∗ . Let v̂(θ, ϕ) be the inverse Jacobi multiplier of Y, that is, Y(v̂) = v̂div(Y). Then the function v(θ, ϕ) = v̂(θ, ϕ)/B(θ, ϕ) is also an inverse Jacobi multiplier of (8), that is, it satisfies the partial differential equation ∂v ∂v ∂F + F(θ, ϕ) = (θ, ϕ) v(θ, ϕ). ∂ϕ ∂θ ∂θ Adapting Theorem 3 of the work [6] to the differential equations (8) on T2 it follows the fundamental relation between the Poincaré map Π̂ associated to the flow of Y with cross section S1 and the inverse Jacobi multiplier v(θ, ϕ) of (8) which is v(Π̂(θ), 0) = v(θ, 0) Π̂0 (θ) for any θ ∈ S1 . Clearly this equation can be rewriting as v(Π̂j (θ), 0) = v(Π̂j−1 (θ), 0) Π̂0 (Πj−1 (θ)) for any j ∈ N. Using this fact, if Π̂k (θ∗ ) = θ∗ for some k then v(θ∗ , 0) = v(Π̂k (θ∗ ), 0) = v(Π̂k−1 (θ∗ ), 0) Π̂0 (Πk−1 (θ∗ )) = v(Π̂k−2 (θ∗ ), 0) Π̂0 (Πk−2 (θ∗ )) Π̂0 (Πk−1 (θ∗ )) .. (10) . k−1 Y = v(θ∗ , 0) (Π̂0 ◦ Π̂j )(θ∗ ). j=0 So equation (10) implies that v(θ∗ , 0) = 0 because Π̂0 ◦ Π̂j > 0 for all j ∈ N by (9). Hence v (and also v̂) vanishes on the periodic orbit through the point (θ∗ , 0).  3.2. The restriction of inverse Jacobi multipliers to invariant tori or spheres. We consider a 2-dimensional manifold Γ embedded in R3 that can be a torus T2 or a sphere S2 . In any case there are local coordinates (ρ, θ, ϕ) with ρ ∈ R+ and (θ, ϕ) ∈ T2 or (θ, ϕ) ∈ S2 such that the equation of Γ is given by Γ = {ρ = r}. In the case of a torus these coordinates are (3) while for the sphere they are the spherical coordinates x = ρ cos ϕ sin θ, y = ρ sin ϕ sin θ and z = ρ cos θ. Proposition 9. Any analytic vector field X in R3 with the invariant man- ifold Γ = {ρ = r} written in the local curvilinear coordinates (ρ, θ, ϕ) is expressed as ρ̇ = F κ (ρ)f1 (ρ, θ, ϕ), (11) θ̇ = f2 (ρ, θ, ϕ), ϕ̇ = f3 (ρ, θ, ϕ), with F (ρ) = r−ρ, fi analytic functions near Γ, κ ∈ Z+ , and f1 |Γ 6≡ 0. Under the extra hypothesis that f1 |Γ be sign-defined, Γ is attractor or repelling when VANISHING SET OF INVERSE JACOBI MULTIPLIERS 11 κ is odd and f1 |Γ is positive or negative, respectively. Otherwise Γ is semi- stable when κ is even. Let V be an analytic inverse Jacobi multiplier of (11) with Γ ⊂ V −1 (0) having vanishing multiplicity m ∈ Z+ , that is, V = F m W with W |Γ 6≡ 0. Then W |Γ is an inverse Jacobi multiplier of the restricted vector field X |Γ provided that either κ = m = 1 or κ ≥ 2. Proof. The invariance of Γ and the analyticity of system (11) implies that ρ̇ = F κ (ρ)f1 (ρ, θ, ϕ) and that there is κ ∈ Z+ such that f1 |Γ 6≡ 0. Imposing the restriction f1 |Γ to be sign-defined we reach the conclusions on the at- tractivity, repulsivity or semi-stability of Γ just by simple inspection of the first component of system (11). Imposing that V = F m W satisfies the partial differential equation X (V ) = V div(X ) and taking the restriction of this equation to Γ it is straightforward to see that Y(v) = v div(Y) + G, where Y = X |Γ , v = W |Γ and G = (κ − m)f1 |Γ F κ−1 Therefore G|Γ = 0 when either κ = m = 1 or κ ≥ 2 and in that case v is an inverse Jacobi multiplier of Y.  Remark 10. The case m = 0 can also be added in the proof of Proposition 9. The outcome is that if V is an inverse Jacobi multiplier of the vector field X associated to system (11) with Γ 6⊂ V −1 (0), then V |Γ is an inverse Jacobi multiplier of the restricted vector field X |Γ provided that κ ≥ 2. Remark 11. In the proof of Proposition 9 we check that X (F ) = KF with cofactor K = −F κ−1 f1 . Therefore K|Γ < 0 if κ = 1 and f1 |Γ > 0, in which case, from the continuity of K, K is negative in a neighborhood U ⊂ R3 of Γ. So only under these restrictions we can apply Theorem 1 with the open set Ω = U \Γ to conclude that Γ is an attractor. As a corollary of Theorem 8, Proposition 9 and Remark 10 we obtain a complementary result to Proposition 6. Corollary 12. Consider an analytic vector field X in R3 having the in- variant torus T2 = {ρ = r} expressed in local curvilinear coordinates (3) like (11) with f1 |T2 sign-defined. Assume X |T2 has a cross section S1 and a rational rotation number. Let V be an analytic inverse Jacobi multiplier of system (11). (i) If T2 ⊂ V −1 (0) with vanishing multiplicity m ∈ Z+ and either κ = m = 1 or κ ≥ 2, then W = V /(r − ρ)m restricted to T2 is an inverse Jacobi multiplier of X |T2 with W |T2 6≡ 0, and W |T2 vanishes in all the periodic orbits on T2 . (ii) If T2 6⊂ V −1 (0) and κ ≥ 2, then V |T2 is an inverse Jacobi multiplier of X |T2 , and V |T2 vanishes in all the periodic orbits on T2 . 12 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA 4. Examples A simple method to construct vector fields X in Rn possessing a prescribed invariant manifold {F = 0} ⊂ Rn is given by X = XF + F Y where Y is an arbitrary vector field and XF is a vector field having F as first integral. For example, in R3 we can take XF = Z × ∇F , where Z is an arbitrary vector field in R3 . Since XF (F ) ≡ 0 we have X (F ) = KF , where the cofactor K of F for X is just K = Y(F ). Since Y is arbitrary, we may choose it in order that K be semi-definite negative with the simple selection Y = −∇F which produces K = −k∇F k2 , hence K restricted to Rn \crit(F ) is negative. Here crit(F ) denotes the set of critical points of the function F . This construction was used in [9] for example. If additionally we take XF divergence free, that is, div(XF ) ≡ 0, then it is easy to check that F is an inverse Jacobi last multiplier if and only if it is a harmonic function, that is, it is a solution of the Laplace equation ∇2 F = 0. Of course, the topology of the manifold {F = 0} imposes some restrictions on the behavior of the flow on it. In 1964 Schwartz [15], generalizing Denjoy’s theorem, proved that the only minimal sets that a C 2 - flow on a compact surface can have are either equilibria, periodic orbits or the whole surface (in which case the surface is a torus). 4.1. The invariant sphere. Example 1. We consider the sphere S2 of radius R embedded in R3 with equation {F (x, y, z) = x2 + y 2 + z 2 − R2 = 0}. Taking the polynomial vector field in R3 given by X = Z ×∇F −F ∇F with a nonzero constant vector field Z = Z1 ∂x + Z2 ∂y + Z3 ∂z . The cofactor of F is K(x, y, z) = −4(x2 + y 2 + z 2 ) which is negative in R3 \{(0, 0, 0)}. The singularities of X are located at the origin with eigenvalues 2R2 and 2(R2 ± ikZk), and hence it is a source of flow, and at the points ±RZ/kZk on S2 . We claim that S2 is an attractor set with basin of attraction R3 \{S2 ∪ {(0, 0, 0)}}. To see this we consider a neighborhood U of S2 whose boundary is given by two spheres {Gj = 0} centered at the origin of arbitrary radius R1 and R2 satisfying 0 < R1 < R < R2 , that is, Gj = x2 + y 2 + z 2 − Rj2 . We have X (Gj )(x, y, z) = −4(x2 + y 2 + z 2 )F (x, y, z) so that X (Gj )|{Gj =0} = −4Rj2 (Rj2 − R2 ), hence X (G1 )|{G1 =0} > 0 and X (G2 )|{G2 =0} < 0 implying that U is a trapping neighborhood of S2 and the claim follows. Notice that we can apply Theorem 1 with Ω = U \S2 . On the other hand it is straightforward to see that V (x, y, z) = (x2 + y 2 + is an inverse Jacobi multiplier of X and S2 ⊂ V −1 (0). The z 2 )3/2 F (x, y, z) restricted vector field X |S2 = Z × ∇F is linear and has two polynomial inde- pendent first integrals: F and Z(F ). Therefore X induces on S2 a foliation VANISHING SET OF INVERSE JACOBI MULTIPLIERS 13 of periodic orbits which are indeed circles around an axis in the direction Z. Example 2. Take the vector field X = Z ×∇F −F ∇F with div(Z ×∇F ) ≡ 0 and consider the harmonic function 1 1 F (x, y, z) = p − , x2 + y2 + z2 R both defined in D = R3 \{(0, 0, 0)} and such that S2 = F −1 (0). By con- struction F is an inverse Jacobi multiplier of the vector field X in D and S2 is an invariant set. To be more precise we can take Z = Z1 ∂x +Z2 ∂y +Z3 (z)∂z with constants Z1 and Z2 in which case div(X ) = −(x2 + y 2 + z 2 )−2 is negative in D but S2 is not an attractor set because D is not simply connected and the origin acts like a sink under the flow of X . Compare this example in the light of Remark 4 on Theorem 1 with the selection Ω = R3 \{S2 ∪ {0, 0, 0}} that satisfies φt (Ω) = Ω but the cofactor K = div(X ) of F is not uniformly bounded in Ω. Proceeding in the same way as in the former example we consider the same two spheres {Gj = 0} of the previous Example 1 and we obtain   2 1 1 X (Gj )|{Gj =0} = − , Rj Rj R where 0 < R1 < R < R2 . Therefore X (G1 )|{G1 =0} > 0 and X (G2 )|{G2 =0} < 0, implying that S2 is semi-stable (repelling the interior and attracting the exterior). See the flow near the sphere in Figure 2 for the particular case Z = ∂x +∂y . Figure 2. Two orbits attracted by two different periodic orbits on the sphere. 14 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA 4.2. An attractor torus with limit cycles. Using the curvilinear coor- dinates (ρ, θ, ϕ) defined by (3), we consider the differential system ρ̇ = r − ρ, (12) θ̇ = 1 − k sin(θ − ϕ), ϕ̇ = ω + k sin(θ − ϕ), which arises in the studies of linearly coupled limit cycle oscillators with parameters 0 ≤ ω ≤ 1 and k ≥ 0, see [10]. From the first component we see that (12) possesses T2 as invariant attractor set. The last two components of (12) define a flow on T2 whose behaviour is easily obtained from the associated equation Ω̇ = 1 − ω − 2k sin Ω, where Ω = θ − ϕ. The equilibria of this equation produces phase-locked periodic orbits in (12). Since it can be solved explicitely it is easy to show that, under the parameter constraint 2k > 1 − ω, there are two isolated hyperbolic periodic orbits (one stable and the other unstable) of system (12) with θ̇ = ϕ̇ = ω ∗ , being the constant frequency ω ∗ = (ω + 1)/2. Both phase-locked periodic orbits lie on the curve C = {(θ, ϕ) ∈ T2 : F(θ, ϕ) = ω − 1 + 2k sin(θ − ϕ) = 0}. A saddle-node bifurcation occurs at 2k = 1 − ω when both periodic orbits coalesce and disappear for 2k < 1 − ω giving (structurally unstable) either quasiperiodic or rational flow depending on the parameters, see the forthcoming analysis using rotation numbers to discern between these two possibilities. Let Σ = {(θ, ϕ) : θ = 0} ⊂ T2 be a cross section for the flow of family (12) on T2 and consider its Poincaré map Π : S1 → S1 . In the book [10] the rotation number Π is computed when ω ∈ [0, 1] is fixed and the parameter k ≥ 0. The outcome is that Rot(Π) = 1 for 2k ≥ 1 − ω whereas 1 + ω − δ(ω, k) p Rot(Π) = , with δ(ω, k) = (1 − ω 2 )2 − 4k 2 1 + ω + δ(ω, k) when 2k < 1−ω. The function Rot(Π) is a continuous increasing function of k whose values are in the interval [ω, 1]. In summary when 2k > 1 − ω there are exactly two isolated hyperbolic periodic orbits to which all the other orbits on T2 approach in the future or in the past. At 2k = 1 − ω a saddle- node bifurcation occurs and the former two periodic orbits coalesce in a semistable periodic orbit. Finally, in the complementary case 2k < 1 − ω, all the orbits on T2 are periodic or asymptotically periodic if Rot(Π) ∈ Q, while the dynamics is quasi-periodic where the orbits are dense and nonperiodic for Rot(Π) 6∈ Q. 4.2.1. The case ω = 1. For any k ≥ 0 there are two hyperbolic periodic orbits on C = {(θ, ϕ) ∈ T2 : F(θ, ϕ) = sin(θ − ϕ) = 0}. Let UT2 be a suf- ficiently small neighborhood of T2 and consider the following two functions Vi : UT2 → R with Vi ∈ C 1 (UT2 ) for i = 1, 2: V1 (ρ, θ, ϕ) = (r − ρ)F(θ, ϕ) VANISHING SET OF INVERSE JACOBI MULTIPLIERS 15 and    F (θ,ϕ) q if θ − ϕ 6∈ {0, π}, V2 (ρ, θ, ϕ) = tan( θ−ϕ 2 )   0 if θ − ϕ ∈ {0, π}, where the bars denote the modulus of the complex number inside. We claim that V1 and V2 are both inverse Jacobi multipliers (this may be checked by direct computations) defined on UT2 such that T2 ⊂ V1−1 (0), T2 6⊂ V2−1 (0) but the condition (T2 ∩ C) ⊂ V2−1 (0) holds. We draw the flow near the torus in Figure 3 for the parameters ω = k = 1 and the torus with radius R = 2 and r = 1/2. Figure 3. Two limit cycles (red and green orbits) on the in- variant attractor torus and the other two orbits approaching them. 4.3. A non-attractor invariant torus in V −1 (0) ⊂ R3 . The torus T2 can be embedded in R3 as follows: the points (x, y, z) ∈ T2 are parameterized by x = (R + r cos θ) cos ϕ, y = (R + r cos θ) sin ϕ, z = r sin θ. Assuming a dynamics (θ(t), ϕ(t)) on T2 we have differentiating that ẋ = −r cos ϕ sin θ θ̇ − (R + r cos θ) sin ϕ ϕ̇, ẏ = −r sin ϕ sin θ θ̇ − (R + r cos θ) cos ϕ ϕ̇, ż = r cos θ θ̇, which can be rewritten as 2Rxz ẋ = −y ϕ̇ − θ̇, R 2 − r 2 + x2 + y 2 + z 2 2Ryz ẏ = xϕ̇ − 2 θ̇, R − r 2 + x2 + y 2 + z 2 1 ż = (−r2 − R2 + x2 + y 2 + z 2 )θ̇. 2R 16 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA Since R2 − r2 + x2 + y 2 + z 2 > 0 in all R3 we perform a time rescaling and obtain the vector field ẋ = −y(R2 − r2 + x2 + y 2 + z 2 )ϕ̇ − 2Rxz θ̇, (13) ẏ = x(R2 − r2 + x2 + y 2 + z 2 )ϕ̇ − 2Ryz θ̇, 1 ż = (R2 − r2 + x2 + y 2 + z 2 )(−r2 − R2 + x2 + y 2 + z 2 )θ̇, 2R which has, by construction, the invariant torus T2 . Taking a linear flow on the torus (θ̇, ϕ̇) = (ω1 , ω2 ) with rationally independent frequencies ω1 /ω2 6∈ Q implies that all the orbits on T2 are dense. Otherwise, when ω1 /ω2 ∈ Q all the orbits on T2 are periodic. One can easily check that neither the cofactor K of F , nor div(X ) is sign defined in any neighborhood of T2 . Consider anyway a neighborhood U (r∗ ) ⊂ R3 of T2 whose boundary is formed by the concentric tori centered at the origin with common generating radius R and tube radius r∗ − r and r∗ + r. Since U (0) = T2 we have that U (r∗ ) is as small as we want by taking r∗ > 0 sufficiently close to zero. We can parameterize U (r∗ ) as U (r∗ ) = {((R + ρ cos θ) cos ϕ, (R + ρ cos θ) sin ϕ, ρ sin θ) ∈ R3 : (θ, ϕ) ∈ S1 × S1 , r − r∗ ≤ ρ ≤ r + r∗ }, and the volume element is dxdydz = Jdρ dθ dϕ with Jacobian J = ρ(R + ρ cos θ), hence the restriction J|T2 6= 0. The divergence of (13) is 2ω1 2ω1 div(X ) = z(x2 +y 2 +z 2 −r2 −2R2 ) = ρ(ρ2 +2Rρ cos θ−r2 −R2 ) sin θ. R R Then the volume integral of div(X ) on U (r∗ ) is given by Z Z 2π Z 2π Z r+r∗ div(X )dxdydz = dϕ dθ div(X ) ρ(R + ρ cos θ)dρ = 0, U (r∗ ) 0 0 r−r∗ for any r∗ ≥ 0. Therefore, by (1), |φt (U (r∗ ))| is constant for all t and any r∗ which implies that T2 is not an attractor set. Taking squares to remove the involved square root, the equation F = 0 can be rewritten into the polynomial form F(x, y, z) = 0 with F(x, y, z) = 4R2 (x2 + y 2 ) − (−r2 + R2 + x2 + y 2 + z 2 )2 and obviously T2 ⊂ {F = 0}, which is an invariant algebraic surface of (13). Moreover, the invariance of the z-axis for the flow of (13) is reflected by the fact that {x2 + y 2 = 0} is also invariant. Now we can construct the following inverse Jacobi multiplier V (x, y, z) = (x2 + y 2 )1/2 F(x, y, z) and T2 ⊂ V −1 (0). 4.4. Higher dimensional flows. We cannot use n degree of freedom inte- grable Hamiltonian system giving rise to flows on n-dimensional tori since they are not attractor sets. Instead we will consider the flow generated by uncoupled nonconservative oscillators possessing attracting limit cycles: (14) ẋi = −ωi yi − xi f (xi , yi ), ẏi = ωi xi − yi f (xi , yi ), i = 1, 2, VANISHING SET OF INVERSE JACOBI MULTIPLIERS 17 with f (xi , yi ) = x2i +yi2 −1 and constant frequencies ωi . The product S1 ×S1 of the two limit cycles (circles {f (xi , yi ) = 0} of radius 1) in the (x1 , y1 ) and (x2 , y2 ) planes is an attractor two-torus T2 ⊂ R4 . Taking polar coordinates xi = ri cos ϕi , yi = ri sin ϕi , system (14) writes (15) ṙi = ri (1 − ri2 ), ϕ̇i = ωi , i = 1, 2. Thus T2 = {r1 = r2 = 1} and its points are parameterized by the angles Q (ϕ1 , ϕ2 ). It is easy to see that V (r1 , ϕ1 , r2 , ϕ2 ) = 2i=1 ri (ri2 −1) is an inverse Jacobi multiplier of (15) and that T2 ⊂ V −1 (0). Moreover we claim that (15) does not admit any C 1 function on T2 as first integral, because (14) cannot have a first integral in a neighborhood of the limit cycles f (xi , yi ) = 0. In particular this implies that V is, up to multiplicative constants, the only inverse Jacobi multiplier of (15). Remark 13. Since (15) has separated variables one can easily obtain two functions Hi (ri , ϕi ) = ri−2 (ri2 − 1) exp (2ϕi /ωi ) with i = 1, 2 that satisfy X (Hi ) ≡ 0, being X the vector field associated to (15), but they are not periodic functions on ϕi so do not define functions on T2 . 5. Appendix The following Liouville’s result is a well known result about the divergence- flow relation. We give a proof for the sake of completeness. Proposition 14. Let U ⊂ Rn be any measurable subset and φt the flow of a C 1 differential system with associated vector field X defined in Rn . Then Z d (16) |φt (U )| = div(X )dx. dt t=t0 φt0 (U ) Proof. We will prove (16) for t0 = 0 without loss of generality. Let ẋ = f (x) be the differential system associated to the vector field X . Expanding the flow by Taylor series at t = 0 we have φt (x) = φ0 (x)+t∂t (φt (x))|t=0 +O(t2 ) = x + tf (x) + O(t2 ). Taking derivatives with respect to x in this expression we have Dx (φt (x)) = I + tDx (f (x)) + O(t2 ) with I the identity matrix. On the other hand, by the formula of the change of variables in integrals we know that Z |φt (U )| = det(Dx (φt (x)))dx. φ0 (U ) Therefore Z |φt (U )| = det(I + t Dx (f (x)) + O(t2 ))dx Zφ0 (U ) = (1 + t Tr(Dx (f (x))) + O(t2 ))dx Zφ0 (U ) = (1 + t div(X ) + O(t2 ))dx, φ0 (U ) 18 I.A. GARCÍA, J. GINÉ, J. LLIBRE, AND S. MAZA and the derivative of this expression with respect to the variable t gives (16) with t0 = 0.  Acknowledgements The first, second and fourth authors are partially supported by a Ministe- rio de Ciencia, Innovación y Universidades grant number MTM2017-84383-P and by an AGAUR grant number 2017SGR-1276. The third author is par- tially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] L.R. Berrone and H. Giacomini, Inverse Jacobi multipliers, Rend. Circ. Mat. Palermo (2) 52 (2003), 77–130. [2] A. Buică and I.A. Garcı́a, Inverse Jacobi multipliers and first integrals for nonau- tonomous differential systems, Z. Angew. Math. Phys. 66 (2015), 573-–585. [3] A. Buică, I.A. Garcı́a and S. Maza, Inverse Jacobi multipliers: recent applications in dynamical systems, Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat. 54 (2013) 127–141. [4] A. Enciso and D. Peralta-Salas, Existence and vanishing set of inverse integrat- ing factors for analytic vector fields, Bull. Lond. Math. Soc. 41 (2009), 1112–1124. [5] G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1995. [6] I.A. Garcı́a, H. Giacomini and M. Grau, The inverse integrating factor and the Poincaré map, Trans. Amer. Math. Soc. 362 (2010), 3591–3612. [7] I.A. Garcı́a and M. Grau, A survey on the inverse integrating factor, Qual. Theory Dyn. Syst. 9 (2010), 115-–166. [8] H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence, and uniqueness of limit cycles, Nonlinearity 9 (1996), 501–516. [9] F. González-Gascón and D. Peralta-Salas, Attractors and symmetries of vector fields: the inverse problem, J. Math. Anal. Appl. 335 (2007), 789-–807. [10] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences 42. Springer-Verlag, New York, 1990 [11] G. Leonov, Strange attractors and classical stability theory, Nonlinear Dyn. Syst. Theory 8 (2008), 49-–96. [12] J. Milnor, On the concept of attractor, Commun. Math. Phys. 99 (1985), 177–195. [13] J. Milnor, On the concept of attractor: Correction and remarks, Commun. Math. Phys. 102 (1985), 517–519. [14] I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds. An overview, Lecture Notes in Mathematics 1705. Springer-Verlag, Berlin, 1999. [15] A. Schwartz, A generalization of the Poincaré-Bendixson theorem to closed two- dimensional manifolds, Amer. J. Math. 85 (1963), 453–458. VANISHING SET OF INVERSE JACOBI MULTIPLIERS 19 Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain Email address: garcia@matematica.udl.cat Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain Email address: gine@matematica.udl.cat Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain Email address: smaza@matematica.udl.cat