This is a preprint of: “Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: general setting”, David Marı́n, Jordi Villadelprat, J. Differential Equations, vol. 275, 684– 732, 2021. DOI: [10.1016/j.jde.2020.11.020] Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: general setting D. Marín and J. Villadelprat BGSMath and Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain Abstract. Given a C ∞ family of planar vector fields {Xµ̂ }µ̂∈Ŵ having a hyperbolic saddle, we study the Dulac map D(s; µ̂) and the Dulac time T (s; µ̂) between two transverse sections located in the separatrices at arbitrary distance from the saddle. We show (Theorems A and B, respectively) that, for any µ̂0 ∈ Ŵ and L > 0, the functions T (s; µ̂) and D(s; µ̂) have an asymptotic expansion at s = 0 for µ̂ ≈ µ̂0 with the remainder being uniformly L-flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so- called Roussarie-Ecalle compensator. The coefficients of these monomials are C ∞ functions “universally” defined, meaning that their existence is established before fixing the flatness L of the remainder and the unfolded parameter µ̂0 . Moreover the flatness L of the remainder is preserved after any derivation with respect to the parameters. We also provide (Theorem C) an explicit upper bound for the number of zeros of T 0 (s; µ̂) bifurcating from s = 0 as µ̂ ≈ µ̂0 . This result enables to tackle finiteness problems for the number of critical periodic orbits along the lines of those theorems on finite cyclicity around Hilbert’s 16th problem. As an application we prove two finiteness results (Corollaries D and E) about the number of critical periodic orbits of polynomial vector fields. Contents 1 Introduction and statement of main results 1 2 A preliminary result 10 3 Proof of the main results 18 4 Applications 28 A Results about the class FLK (W ) 36 2010 AMS Subject Classification: 34C07; 34C20; 34C23. Key words and phrases: Dulac map, Dulac time, asymptotic expansion, uniform flatness, criticality. This work has been partially funded by the Ministry of Science, Innovation and Universities of Spain through the grants MTM2015-66165-P, PGC2018-095998-B-I00 and MTM2017-86795-C3-2-P, the Agency for Management of University and Re- search Grants of Catalonia through the grants 2017SGR1725 and 2017SGR1617, and by the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445). 1 1 Introduction and statement of main results In this paper we study C ∞ unfoldings of planar vector fields having a hyperbolic saddle. The study of the so- called Dulac map of the saddle has attracted the attention of many authors (see for instance [5, 6, 7, 9, 22, 26] and references there in), mainly due to its close connection with Hilbert’s 16th problem (see [10, 28] for details). The Dulac map D( · ; µ̂) of a saddle is the transition map from a transverse section Σσ in its stable separatrix to a transverse section Στ in its unstable separatrix, whereas the Dulac time T ( · ; µ̂) is the time that spends the flow to do this transition, see Figure 1. In a previous paper, see [19], we proved a local version of some results presented here. By local we mean that Σσ and Στ cannot be at arbitrary distance from the saddle but close enough in order that a suitable (local) normal form for the saddle unfolding can be used. As we will see, those local results constitute a basic building block for the more general ones that we will prove here. A polycycle is a graphic Γ with a well-defined return map on one of its sides. The cyclicity of Γ in an unfolding of the vector field is the maximum number of limit cycles that bifurcate from it. If the polycycle is hyperbolic then the return map can be written as the composition of the Dulac maps associated to the passage through each one of its vertices. The limit cycles are fixed points of the return map and to study its number (or even to prove that there are finitely many) a key tool is the asymptotic expansion of the Dulac maps. To this end it is essential that the remainder of the expansion is uniformly flat with respect to the unfolding parameters (see [28, Chapter 5] and the references in the previous paragraph). In case that the return map of Γ is the identity then there is an annulus foliated by periodic orbits where the period function (i.e., the time of the return map) is defined. In this context the object of study are the so-called critical periodic orbits, which are the critical points of the period function. Similarly as with Hilbert’s 16th problem, it arises the notion of criticality of a polycycle Γ, i.e., the maximum number of critical periodic orbits that bifurcate from Γ, see [13, 24]. In the same way as for the cyclicity, an asymptotic expansion of the Dulac time with remainder uniformly flat constitutes a key tool to investigate the criticality of a polycycle. We will consider a C ∞ unfolding of a hyperbolic saddle with poles along its separatrices. The reason why we permit this “polar” factor is because, when dealing with polynomial vector fields, a special attention must be paid to the study of those polycycles with vertices at infinity in the Poincaré disc. The factor can come from the line at infinity in a saddle at infinity or, more generally, appear in a divisor after desingularizing more general singular points at infinity of a polycycle. It is important to remark that (by means of a reparametrization of time) this factor can be neglected to study the Dulac map but, on the contrary, this cannot be done when dealing with the Dulac time. More precisely, setting µ̂ := (λ, µ) ∈ Ŵ := (0, +∞)×W with W an open set of RN , let us take the family of vector fields {Xµ̂ }µ̂∈Ŵ with 1   Xµ̂ (x, y) := xP (x, y; µ̂)∂x + yQ(x, y; µ̂)∂y , (1) xn1 y n2 where • n := (n1 , n2 ) ∈ Z2≥0 , • P and Q belong to C ∞ (V × Ŵ ) for some open set V of R2 containing the origin, • P (x, 0; µ̂) > 0 and Q(0, y; µ̂) < 0 for all (x, 0), (0, y) ∈ V and µ̂ ∈ Ŵ , • λ = − Q(0,0;µ̂) P (0,0;µ̂) . Note that the hyperbolicity ratio of the saddle is an independent parameter although in the applications we will have λ = λ(µ). The hyperbolicity ratio turns out to be the ruling parameter in our study and, besides, having it uncoupled from the rest of parameters simplifies the notation in the computations we shall deal with. Furthermore, see Remark 1.5, we do not lose generality assuming that it is uncoupled. 2 y σ(s) Σσ σ 0 D(s) ϕ( · , σ(s)) τ 0 s τ (D(s)) = ϕ(T (s), σ(s)) x Στ Figure 1: Definition of T ( · ; µ̂) and D( · ; µ̂) in Theorems A and B, respectively. Let σ : (−ε, ε) × Ŵ → Σσ and τ : (−ε, ε) × Ŵ → Στ be two C ∞ transverse sections to Xµ̂ defined by   σ(s; µ̂) = σ1 (s; µ̂), σ2 (s; µ̂) and τ (s; µ̂) = τ1 (s; µ̂), τ2 (s; µ̂) such that σ1 (0; µ̂) = 0 and τ2 (0; µ̂) = 0 for all µ̂ ∈ Ŵ . We denote the Dulac map and Dulac time of Xµ̂ between the transverse sections Σσ and Στ by R( · ; µ̂) and T ( · ; µ̂), respectively. More precisely, see Figure 1, if ϕ(t, p0 ; µ̂) is the solution of Xµ̂ passing through p0 ∈ V at t = 0, for each s ∈ (0, ε) we define R(s; µ̂) and T (s; µ̂) by means of the relation ϕ T (s; µ̂), σ(s); µ̂) = τ (R(s; µ̂); µ̂) Definition 1.1. Consider K ∈ Z≥0 ∪ {+∞} and an open subset U ⊂ Ŵ ⊂ RN +1 . We say that a function ψ(s; µ̂) belongs to the class Cs>0 K (U ), respectively E K (U ), if there exist an open neighbourhood V of {(s, µ̂) ∈ RN +2 ; s = 0, µ̂ ∈ U } = {0} × U  in RN +2 such that (s, µ̂) 7→ ψ(s; µ̂) is C K on V ∩ (0, +∞) × U , respectively V . Finally we denote K E+ (U ) := {ψ(s; µ̂) ∈ E K (U ); ψ(0; µ̂) > 0 for all µ̂ ∈ U }. Here the letter E stands for functions in Cs>0 K (U ) having extension to s = 0.  More formally, the definition of Cs>0 K (U ) and E K (U ) must be thought in terms of germs with respect to relative neighborhoods of {0} × U in (0, +∞) × U . In doing so these sets become rings and we have the inclusions C K (U ) ⊂ E K (U ) ⊂ Cs>0 K (U ). We can now introduce the notion of (finitely) flatness that we shall use in the sequel. Definition 1.2. Consider K ∈ Z≥0 ∪{+∞} and an open subset U ⊂ Ŵ ⊂ RN +1 . Given L ∈ R and µ̂0 ∈ U , we say that ψ(s; µ̂) ∈ Cs>0 K (U ) is (L, K)-flat with respect to s at µ̂0 , and we write ψ ∈ FLK (µ̂0 ), if for each ν = (ν0 , . . . , νN +1 ) ∈ Z≥0 with |ν| = ν0 + · · · + νN +1 6 K there exist a neighbourhood V of µ̂0 and N +2 C, s0 > 0 such that ∂ |ν| ψ(s; µ̂) ν +1 6 Cs L−ν0 for all s ∈ (0, s0 ) and µ̂ ∈ V . ∂sν0 ∂ µ̂ν11 · · · ∂ µ̂NN+1 T If W is a (not necessarily open) subset of U then define FLK (W ) := µ̂0 ∈W FLK (µ̂0 ).  3 j Λk k2 k1 i Figure 2: The filled dots are points (i, j) ∈ Z2≥0 in the set Λk for k = (k1 , k2 ). The principal part of the Dulac map and Dulac time will be expressed in terms of the following defor- mation of the logarithm. Definition 1.3. The function defined for s > 0 and α ∈ R by means of ( −α s −1 α if α 6= 0, ω(s; α) = − log s if α = 0, is called the Ecalle-Roussarie compensator.  Definition 1.4. Given any k = (k1 , k2 ) ∈ Z2≥0 , throughout the paper we shall use the following notation: • Λk := (Z≥k1 × {0}) ∪ (Z≥0 × Z≥k2 ), see Figure 2.  k • Dij := λ > 0 : there exits (i0 , j 0 ) ∈ Λk \ {(i, j)} such that i + λj = i0 + λj 0 for each (i, j) ∈ Λk .  k • Bλ,L := (i, j) ∈ Λk : i + λj 6 L for each L ∈ R and λ > 0, see Figure 4.  k • DL := λ > 0 : there exits (i, j) ∈ Bλ,L k such that λ ∈ Dij k . • For λ = p/q ∈ Q>0 with gcd(p, q) = 1 and (i, j) ∈ Λk , ( k ∅ if (i + rp, j − rq) ∈ Λk for some r ∈ N, Aijλ := {r ∈ Z≥0 : (i − rp, j + rq) ∈ Λk } otherwise. Note that if k2 = 0 then Λk = Z2≥0 = Λ0 regardless of the value of k1 . One can prove on the other hand that Dij k and DL k are discrete subsets of Q>0 , see Remark 3.3.  Let us point out that in the previous definition k stands always for a two-dimensional vector with components in Z≥0 . That being said, if k = (0, 0) then we write Λ0 , Dij 0 , Bλ,L 0 , DL 0 and Aijλ 0 for shortness. 4 We are now in position to state our main results. Let us begin with the one regarding the asymptotic development of the Dulac time. In its statement we use the notation introduced so far and denote ( 0 if n 6= (0, 0), T0 (µ̂) = −1 (2) P (0,0;µ̂) if n = (0, 0), where recall that the components of the vector n = (n1 , n2 ) ∈ Z2≥0 are the orders of the poles of Xµ̂ . Theorem A. Let T (s; µ̂) be the Dulac time of the hyperbolic saddle (1) between  the transverse sections Σσ and Στ . For each (i, j) ∈ Λn there exists Tij ∈ C ∞ ((0, +∞) \ Dij n ) × W such that, for every L > 0 and λ0 > 0, the following properties hold: (a) If λ0 ∈ n / DL then X T (s; µ̂) = T0 (µ̂) log s + Tij (µ̂)si+λj + FL∞ ({λ0 } × W ). n (i,j)∈Bλ 0 ,L (b) If λ0 ∈ DL n then X  T (s; µ̂) = T0 (µ̂) log s + T λij0 ωα (s); µ̂ si+λj + FL∞ ({λ0 } × W ), n (i,j)∈Bλ 0 ,L where λ0 = p/q with gcd(p, q) = 1, α(µ̂) = p − λq and X T λij0 (w; µ̂) := Ti−rp,j+rq (µ̂)(1 + αw)r . n r∈Aijλ 0 Moreover the coefficient of these polynomials in w extend C ∞ to {λ0 } × W . This result is a very significant improvement of the ones that we obtained previously in [15, 18]. Indeed, the main result in [15] can be viewed as an embryonic version of Theorem A that (with the notation introduced here) is addressed to the case n = (0, n2 ) and L = min(1, λ0 n2 ) + ε, so that its principal part contains just two monomials. More important, it is proved in the analytic setting and assuming additionally that family {Xµ̂ }µ̂∈Ŵ is locally equivalent to its linear part. We referred to this assumption as the family linearization property (FLP, in short). Afterwards we extended in [18] that previous result to n = (n1 , n2 ) but still with the same L and the FLP assumption. Next we state our main result about the Dulac map. Before that let us observe that this map is independent of n = (n1 , n2 ) because it does not change after a reparametrization of time in the differential equation. Related with this, and also with regard to the applicability of the theorem, it is important to mention that, thanks to Lemma 4.3, it is not necessary that the separatrices of the saddle are straight lines. Theorem B. Let D(s; µ̂) be the Dulac map of the hyperbolic saddle (1) between  the transverse sections Σσ and Στ . For each (i, j) ∈ Λ0 there exists ∆ij ∈ C ∞ ((0, +∞) \ Dij 0 ) × W such that, for every L > 0 and λ0 > 0, the following properties hold: (a) If λ0 ∈ 0 / DL then X D(s; µ̂) = sλ ∆ij (µ̂)si+λj + FL∞ ({λ0 } × W ). 0 (i,j)∈Bλ 0 ,L−λ0 (b) If λ0 ∈ DL 0 then X  D(s; µ̂) = sλ ∆λij0 ωα (s); µ̂ si+λj + FL∞ ({λ0 } × W ), 0 (i,j)∈Bλ 0 ,L−λ0 5 where λ0 = p/q with gcd(p, q) = 1, α(µ̂) = p − λq and X ∆λij0 (w; µ̂) := ∆i−rp,j+rq (µ̂)(1 + αw)r . 0 r∈Aijλ 0 Moreover the coefficient of these polynomials in w extend C ∞ to {λ0 } × W . Finally, ∆00 (µ̂) > 0 for all µ̂ ∈ Ŵ . Theorem B is closely related with the seminal results by R. Roussarie and his collaborators on the structure of the Dulac map (see [20, 22, 25] and [28, §5.1.3]). Our contribution improves the previous ones in following three aspects (that are also valid for Theorem A with regard to [15, 18]): 1. The monomials appearing in the principal part are completely described, even in the resonant case (i.e., λ0 ∈ Q). Note moreover that max Aijλ k 6 i/p for any k = (k1 , k2 ) ∈ Z2≥0 and that if i < k2 λ then Ai0λk = {0}. This gives a bound on the degrees of T λij0 (w; µ̂) and ∆λij0 (w; µ̂). (We remark in this respect that all over the paper we follow the usual convention that a summation with the index varying in an empty set is equal to zero.) 2. The coefficients Tij and ∆ij are C ∞ functions “universally” defined. More precisely, their existence is established before fixing the flatness L of the remainder and, more important, the parameter λ0 (see the order of quantifiers in the statements). 3. The remainder is given by a function R(s; µ̂) in FL∞ ({λ0 } × W ). The application of Lemma A.1 shows that if we take any K ∈ Z≥0 with K < L then R extends to a C K -function R̂ defined in some open neighbourhood of {0}×{λ0 }×W inside R×R×RN and satisfying ∂ ν R̂(0; µ̂) ≡ 0 for all ν ∈ ZN +2 ≥0 with |ν| 6 K. To the best of our knowledge this constitutes a new result. The flatness of the remainder in the previous results (see [9, §3] and [25] for the Dulac map) holds for derivation with respect to s, which suffices to bound the number of critical periodic orbits or limit cycles. However it does not enable to conclude that R extends smoothly with respect to the parameters (see [19, Appendix A] for a counterexample), which is crucial to study the smoothness properties of the corresponding bifurcation diagram (see [27] for limit cycles). We refer the interested reader to [19, Remark 1.4] for more details on this issue. With regard to second point above let us comment that in a forthcoming paper we will give the explicit expression of Tij and ∆ij for particular (i, j) in terms only the parametrization of the transversal sections and the functions P and Q in (1). For instance, in the context of Hilbert’s 16th problem, to have these expressions when dealing with a particular polycycle is essential in order to compute its cyclicity and to fully understand the bifurcation diagram of emergence/disappearance of limit cycles. Remark 1.5. As we already mentioned we consider the hyperbolicity ratio λ > 0 of the saddle as an independent parameter in µ̂ = (λ, µ). This is by no means a restriction because if we deal with a family 1   Xν (x, y) = n1 n2 xP (x, y; ν)∂x + yQ(x, y; ν)∂y x y with hyperbolicity ratio λ(ν) = − Q(0,0;ν) P (0,0;ν) then instead we can consider 1   X̄(λ,ν) (x, y) = x P̄ (x, y; λ, ν)∂x + y Q̄(x, y; λ, ν)∂y x n1 y n2 with P̄ (x, y; λ, ν) := P (x, y; ν) and Q̄(x, y; λ, ν) := P (0, 0; ν)(−λ + λ(ν)) + Q(x, y; ν). Clearly {X̄(λ,ν) } is a family with a saddle with hyperbolicity ratio λ that restricted to λ = λ(ν) coincides with {Xν }.  6 Next we make some further considerations about the consequences of Theorem A. Let us point out that although we focus on the Dulac time for simplicity in the exposition, they are also valid for Theorem B on the Dulac with the obvious modifications. On account of the second point above the coefficients Tij (and so the polynomials T λij0 ) are independent of the remainder’s flatness L. This endows the asymptotic development of the Dulac time given in Theorem A with a property similar to the unicity of the Taylor series. More precisely, we have a well defined (formal) series  P   Tij (µ̂)si+λj if λ0 ∈ / Q,  (i,j)∈Λn T0 (µ̂) log s + P  (3)   T λij0 ωα (s); µ̂ si+λj if λ0 ∈ Q,  (i,j)∈Λn which is asymptotic to T (s; µ̂) as (s, µ̂) → (0, µ̂0 ) in the sense established by Theorem A. In this respect it is to be referred the works of Saavedra on the Dulac time of (single) analytic vector fields (see [29, 30]). She proves that the Dulac time T (and its derivative T 0 ) of a hyperbolic saddle is asymptotic to a formal series X T̂ (s) = sνk Pk (log s), (4) k∈N where {νk }k∈N is a strictly increasing unbounded sequence of real numbers and Pk is a real polynomial. In Pk other words, |T (s; µ̂0 ) − i=1 sνi Pi (log s)| < o(sνk ) for all k ∈ N. We note that in her result a finite number of exponents νk may be negative because she contemplates the case n1 , n2 ∈ Z<0 as well. 0 0 Definition 1.6. For λ0 ∈ / Q we write si+λj ≺λ0 si +λj if, and only if, i + λ0 j < i0 + λ0 j 0 . If λ0 ∈ Q then we write  0 0 0  i + λ0 j < i0 + λ0 j 0 si+λj k ω ≺λ0 s i +λj k ω ⇔ or  (i, j) = (i0 , j 0 ) and k > k 0 where as usual, setting λ0 = p/q with gcd(p, q) = 1, ω stands for ωα (s) = ω(s; p − λq).  It is clear that ≺λ0 is a strict partial order among the monomials si+λj ω k with i, j, k ∈ Z≥0 . Moreover, if f and g are two monomials such that f ≺λ0 g then, by applying (a) in Lemma A.3, g(s; λ) lim = 0, (s,λ)→(0,λ0 ) f (s; λ) where the limit is taken with (s, λ) ∈ (0, ε) × (0, +∞). For a fixed µ̂0 = (λ0 , µ0 ) ∈ Ŵ , our main concern is to study the number of zeros of T 0 ( · ; µ̂) that are close to s = 0 when µ̂ ≈ µ̂0 . We shall define precisely this notion below but let us advance that, in this respect, the case n = (0, 0) is trivial. This is so because, by applying Theorem A and (d) in Lemma A.3, lim sT 0 (s; µ̂) = T0 (µ̂0 ), (s,µ̂)→(0,µ̂0 ) which is different from zero due to (2). Clearly this fact prevents the existence of zeros of T 0 (s; µ̂) with (s, µ̂) ≈ (0, µ̂0 ). This is the reason why we will suppose n 6= (0, 0) in the statement of Theorem C, which constitutes our third main result. Before that let us make the following key observation. Remark 1.7. We note that ≺λ0 is a strict total order among the monomials that can appear in the formal series (3) of the Dulac time, namely si+λj for λ0 ∈ / Q and si+λj ω k for λ0 ∈ Q. Indeed, this is obvious for λ0 ∈/ Q because then i + λ0 j 6= i + λ0 j for (i, j) 6= (i0 , j 0 ). In case that λ0 ∈ Q, say λ0 = p/q with 0 0 gcd(p, q) = 1, this is due to the fact that if i+λ0 j = i0 +λ0 j 0 with (i, j) 6= (i0 , j 0 ) then there exists r ∈ Z\{0} such that (i0 , j 0 ) = (i − rp, j + rq) and consequently (see Definition 1.4) either Aijλ n 0 = ∅ or Ain0 j 0 λ0 = ∅, which implies T λij0 ≡ 0 and T λi00j 0 ≡ 0, respectively.  7 On account of the previous remark, for a given λ0 > 0 we denote by ti (µ̂) the coefficient of the monomial in the i-th position (with respect to ≺λ0 ) among all the monomials appearing in the formal series (3). Definition 1.8. Assume n 6= (0, 0) and take µ̂0 = (λ0 , µ0 ) ∈ Ŵ . Let {ti }i∈N be the sequence of coefficients with respect to ≺λ0 in the formal series (3). We define `µ̂0 := inf{i ∈ N : ti (µ̂0 ) 6= 0} − η, where η = 1 if n1 n2 = 0 and η = 0 otherwise.  Notice that if n 6= (0, 0) but n1 n2 = 0 then the first monomial (with respect to the order ≺λ0 ) in the formal series (3) is the constant one, which follows taking (i, j) = (0, 0) ∈ Λn . This monomial does not appear in the asymptotic development of T 0 ( · ; µ̂) and this is the reason why we subtract η in the previous definition. Definition 1.9. Let h(s; µ̂) be a function in Cs>0 ∞ (U ) for some open set U ⊂ Ŵ . Given any µ̂0 ∈ U we define Z0 (h( · ; µ̂), µ̂0 ) to be the smallest integer N having the property that there exist δ > 0 and a neighbourhood V of µ̂0 such that for every µ̂ ∈ V the function h(s; µ̂) has no more than N zeros on (0, δ) counted with multiplicities.  We can now state our third main result, that gives a uniform bound for the number of zeros of T 0 ( · ; µ̂) bifurcating from s = 0 when µ̂ ≈ µ̂0 . Theorem C. Consider the family of vector fields {Xµ̂ }µ̂∈Ŵ in (1) and assume that n 6= (0, 0). Let T ( · ; µ̂) be the Dulac time between the transverse sections Σσ and Στ and fix µ̂0 ∈ Ŵ . Then Z0 (T 0 ( · ; µ̂), µ̂0 ) 6 `µ̂0 . If `µ̂0 < +∞ then Theorem C gives an upper bound for the number of zeros of T 0 ( · ; µ̂) that can emerge/disappear from s = 0 when we perturb µ̂ ≈ µ̂0 . It is important to remark, and this is a key feature, that `µ̂0 depends only on the vector field Xµ̂0 and not in the family {Xµ̂ }µ̂∈Ŵ . Note in this respect, cf. (4), that if T 0 (s; µ̂0 ) = βsν logm s + o(sν logm s) with β 6= 0 then `µ̂0 < +∞. Indeed, Theorem A shows that  n  #Bλ0 ,ν −η  if λ0 ∈ / Q, P  `µ̂0 = max{r ∈ Aijλn } + 1 − m − η if λ0 ∈ Q,   (i,j)∈Bn 0 λ0 ,ν where η = 1 if n1 n2 = 0 and η = 0 otherwise. It is to be referred here the paper by Saavedra and Mardešić [17] because (in the analytic context) they prove that if T 0 ( · ; µ̂0 ) 6≡ 0 then there exist some ν ∈ R and m ∈ Z≥0 such that T 0 (s; µ̂0 ) = βsν logm s + o(sν logm s) with β 6= 0. By analytic context we mean assuming that the functions P ( · ; µ̂0 ) and Q( · ; µ̂0 ) in (1), together with the parametrizations σ( · ; µ̂0 ) and τ ( · ; µ̂0 ) of the transverse sections, are analytic. On a rather different tack, but also in this setting, it is to be mentioned that the application of Proposition 4.2 provides a lower bound for Z0 (T 0 ( · ; µ̂), µ̂0 ). We conclude this section by explaining two applications of the tools and results introduced so far in the context of the study of the period function, which was our initial motivation for considering this kind of problems. To this end some additional definitions are needed. A singular point p of a planar differential system is a center if it has a punctured neighbourhood that consists entirely of periodic orbits surrounding p. The period annulus of the center is the largest punctured neighbourhood with this property and we will denote it by P. We embed P in RP2 and denote its boundary by ∂P. Clearly the center p belongs to ∂P, and in what follows we will call it the inner boundary of the period annulus. We also define the outer boundary of the period annulus to be Π := ∂P \ {p}. We point out that Π is a nonempty compact subset of RP2 . The period function of the center assigns to each periodic orbits in P its period. Since the period function is defined on the set of periodic orbits in P, in order to study its qualitative properties usually the first step is to parametrize this set. This can be done by taking a transverse section to the vector field on P, for instance an orbit of the orthogonal vector field. If {γs }s∈(0,1) is such a parametrization, then s 7→ P (s) := {period of γs } is a function that provides the qualitative properties of the period function that we are interested in. (Note that the function P is as smooth as the vector field and the parametrization used.) 8  The critical periods are the isolated critical points of P , i.e. ŝ ∈ (0, 1) such that P 0 (s) = α(s−ŝ)k +o (s−ŝ)k with α 6= 0 and k > 1. In this case, more geometrically, we shall say that γŝ is a critical periodic orbit of multiplicity k of the center. One can easily see that this definition does not depend on the particular parametrization of the set of periodic orbits used. In the next definition dH stands for the Hausdorff distance between compact sets of RP2 . Definition 1.10. Consider a C ∞ family {Xν }ν∈U of planar vector fields with a center and fix some ν0 ∈ U . Suppose that the outer boundary of the period annulus varies continuously at ν0 ∈ U , meaning that for any ε > 0 there exists δ > 0 such that dH (Πν , Πν0 ) 6 ε for all ν ∈ U with kν − ν0 k 6 δ. Then, setting N (δ, ε) = sup {# critical periodic orbits γ of Xν in Pν with dH (γ, Πν0 ) 6 ε and kν − ν0 k 6 δ} ,  the criticality of (Πν0 , Xν0 ) with respect to the deformation Xν is Crit (Πν0 , Xν0 ), Xν := inf δ,ε N (δ, ε).  Let us stress that the number of critical periodic orbits γ of Xν in Pν is counted with multiplicities. Note furthermore that in this definition each vector field Xν is assumed to be C ∞ . To define the outer boundary Πν of the period annulus Pν of Xν we do not compactify the vector field but only the set Pν and to this end there is no need that Xν is polynomial. Certainly Crit (Πν0 , Xν0 ), Xν may be infinite but, if it is not, then it gives the maximal number of critical periodic orbits of Xν that tend to Πν0 in the Hausdorff sense as ν → ν0 . It is clear on the other hand that, for a given ν0 ∈ U, the contour of the period annulus Pν0 changes as we move ν ≈ ν0 . The assumption that the period annulus varies continuously ensures that these changes do not occur abruptly. In this regard note that Xν = −y∂x + (x + νx3 + x5 )∂y , with ν ∈ R, is a polynomial family of vector fields with a center at the origin for which the outer boundary does not vary continuously at ν = 2. Indeed, the period annulus Pν is the whole plane for ν < 2, whereas is bounded for ν = 2 (see [14] for details). Clearly in this situation the criticality of Πν at ν = ν0 has no point. Definition 1.11. Let X be a vector field on R2 (or S2 ). A graphic Γ for X is a compact, non-empty invariant subset which is a continuous image of S1 and consists of a finite number of isolated singular points {p1 , . . . , pm , pm+1 = p1 } (not necessarily distinct) and compatibly oriented separatrices {s1 , . . . , sm } connecting them (i.e., such that the α-limit set of sj is pj and the ω-limit set of sj is pj+1 ). A graphic is said to be hyperbolic if all its singular points are hyperbolic saddles. A polycycle is a graphic with a return map defined on one of its sides. Consider now a C ∞ family of vector fields {Xν } such that, for ν = ν0 , Γ is a hyperbolic polycycle for Xν0 . Then, in the context of Hilbert’s 16th problem, in order to study the cyclicity of (Xν , Γ) it is sometimes assumed that some connections between the hyperbolic saddles in Γ remain unbroken inside the family {Xν }, see for instance [5, 12, 21]. Here, in case that all the separatrix connections remain unbroken we shall say that the hyperbolic polycycle Γ is persistent inside the family {Xν }. When studying the criticality this will be a non-degeneracy condition for the polycycle at the outer boundary of the period annulus.  In order to study the behaviour of the trajectories of a polynomial vector field Y near infinity we can consider its Poincaré compactification p(Y ), see [1, §5] for details, which is an analytically equivalent vector field defined on the sphere S2 . The points at infinity of R2 are in bijective correspondence with the points of the equator of S2 , that we denote by `∞ . Furthermore the trajectories of p(Y ) in S2 are symmetric with respect to the origin and so it suffices to draw its flow in the closed northern hemisphere only, the so called Poincaré disk. Taking this notation into account we can now state two finiteness results for polynomial vector fields that follow from our main results. Corollary D. Let {Xν }ν∈U be a C ∞ family of planar polynomial vector fields. Assume that, for all ν ∈ U, (a) Xν has a center such that the outer boundary Πν of its period annulus is a hyperbolic polycycle that is persistent and varies continuously inside the family, 9 (b) the infinite line `∞ is invariant for p(Xν ) in case that Πν ∩ `∞ 6= ∅, and (c) not all the singularities of Xν in Πν are in `∞ .  Then Crit (Πν0 , Xν0 ), Xν = 0 for any ν0 ∈ U . Recall that, for a given polynomial vector field Y , it is well known that the infinite line `∞ is invariant for p(Y ) if, and only if, the homogeneous part of highest degree of Y is not a multiple of the radial vector field x∂x + y∂y . On the other hand, if the family {Xν }ν∈U is analytic (instead of C ∞ ) and each center is non degenerated then Corollary D implies the existence of a uniform finite upper bound for the number of critical periodic orbits in the whole period annulus. With regard to this aftermath it is to be referred a paper by Chicone and Dumortier [3] because from their main theorem it follows that a center of a (single) analytic vector field with a bounded period annulus cannot have an infinite number of critical periodic orbits. Turning to Hilbert’s 16th problem, the saddle loop constitutes the first type of polycycle that was proved to have finite cyclicity (see [25, Theorem A] and also [9, Theorem 9]). Corollary D shows that the saddle loop has zero criticality, so the next polycycle to consider is the one having two hyperbolic saddles as vertices. Our last result is addressed to this situation and let us advance that the non-trivial case is when both saddles are at infinity, see Figure 7. Corollary E. Let {Xν }ν∈U be a C ∞ family of planar polynomial vector fields. Assume that, for all ν ∈ U, (a) Xν has a center such that the outer boundary Πν of its period annulus is a hyperbolic polycycle with exactly two singularities that is persistent and varies continuously inside the family, and (b) the infinite line `∞ is invariant for p(Xν ) in case that Πν ∩ `∞ 6= ∅  Then Crit (Πν0 , Xν0 ), Xν < +∞ for any ν0 ∈ U such that the center qν0 is non-isochronous. The paper is organized as follows. In Section 2 we prove Theorem 2.5, which constitutes a fundamental preliminary result addressed to both, the Dulac map and the Dulac time. Briefly, for every µ̂0 ∈ Ŵ , K ∈ N and L > 0, Theorem 2.5 provides an asymptotic expansion of D( · ; µ̂) and T ( · ; µ̂) with the coefficients C K functions in a neighbourhood of µ̂0 and the remainder in FLK (µ̂0 ). Next, relying on this result, Theorems A and B follow by showing that in fact these coefficients do not depend on µ̂0 , K and L. More concretely, that we can change the order of the quantifiers ∀ . . . ∃ in the statement of Theorem 2.5 to ∃ . . . ∀. The formalization of this simple idea is rather long and technical but let us advance that the proof of Theorem A from Theorem 2.5 gives as particular case the proof of Theorem B. That being said, Section 3 is devoted to the proofs of Theorems A, B and C. Next, in Section 4, we prove Corollaries D and E, together with a result, namely Lemma 4.3, that straightens globally the separatrices of a saddle depending on parameters. This result, which is well-known to be true locally, is relevant with regard to the applicability of Theorems A and B because it enables to set aside the condition that the separatrices lay on the coordinate axis. Finally, for reader’s convenience, in Appendix A we state some results from [19] about the class of functions FLK (W ) that we shall use in the present paper. 2 A preliminary result This section is completely devoted to the proof of Theorem 2.5. With this aim in view we begin with two results about the class of functions E K (U ), recall Definition 1.1. Lemma 2.1. If f ∈ E K (U ) with K ∈ N verifies f (0; µ̂) = 0 for all µ̂ ∈ U then f = sg some g ∈ E K−1 (U ). Proof. By definition f (s; µ̂) ∈ C K (V ) for some open neighborhoud V of {0} × U . Fix any (s0 , µ̂0 ) ∈ V and observe that the existence of g is clear in a neighbourhood of any point with s0 6= 0. So assume that s0 = 0 10 and take ε > 0 small enough and a closed disc D centered at µ̂0 with D ⊂ U such that [−ε, ε] × D ⊂ V. R1 Then, by Proposition 2 and Remark 3 in [31, §17.1.3], g(s; µ̂) := 0 ∂s f (st; µ̂)dt is C K−1 on [−ε, ε] × D. Furthermore Z 1 Z s sg(s; µ̂) = s∂s f (st; µ̂)dt = ∂s f (u; µ̂)du = f (s; µ̂) − f (0; µ̂) = f (s; µ̂), 0 0 for all (s, µ̂) ∈ [−ε, ε] × D. This proves the existence of g ∈ C K (V 0 ) with V 0 := {(s, µ̂) ∈ V : µ̂ ∈ U }, which is an open neighborhood of {0} × U , verifying f = sg as desired. Corollary 2.2. If f ∈ E K (U ) and m ∈ N with m 6 K then there exist fi ∈ C K−i (U ), i = 0, 1, . . . , m − 1, Pm−1 and g ∈ E K−m (U ) such that f (s; µ̂) = i=0 fi (µ̂)si + sm g(s; µ̂). In particular, for any L > 0, 0 E K (U ) ⊂ C K (U )[s] + FLK (U ), provided that K 0 > K + L. Proof. The first assertion follows by applying Lemma 2.1 recursively. The second assertion for L = 0 0 0 follows by (c) and (d) in Lemma A.2, which show that E K (U ) ⊂ F0K (U ) ⊂ F0K (U ). The second assertion for L > 0 follows by applying the first one with m = dLe > 1 and using then Lemma A.2 to show that 0 sdLe g(s; µ̂) ∈ FdLe ∞ (U )E K −dLe ∞ (U ) ⊂ FdLe (U )F0K (U ) ⊂ FdLe K (U ) ⊂ FLK (U ), thanks to sdLe ∈ FdLe ∞ (U ) and K 0 − dLe > K. In the previous statement C K (U )[s] stands for the set of functions h(s; µ̂) that are polynomial in s with coefficients in C K (U ), i.e., that can be written as h(s; µ̂) = h0 (µ̂)+h1 (µ̂)s+. . .+hm (µ̂)sm for some n ∈ Z≥0 and hi ∈ C K (U ) for i = 0, 1, . . . , m. In the next result we extend this compact notation in the obvious way, for instance to refer to functions h(s; µ̂) ∈ C K (U )[s] + FLK (U ). The following is a key result in order to study the composition of functions inside these type of sets. For the sake of clarity let us remark that, given h(s; µ̂) and g(s; µ̂), by h ◦ g we mean h g(s; µ̂); µ̂). Lemma 2.3. Let us consider α, β ∈ C K (U ) with K ∈ N and suppose that β(µ) > 0 for all µ ∈ U. Then, for L ∈ R≥0 and p ∈ N, the following holds: K0  (a) sβ ◦ sE+ (U ) ⊂ sβ C K (U )[s] + FLK (U ) ⊂ sβ C K (U )[s] + FLK (U ) if K 0 > K + L. 0 (b) sp ωα ◦ sE+ K (U ) ⊂ C K (U )[s, sp ωα ] + FLK ({α = 0}) if K 0 > K + L. 0  K0 (c) sβ C K (U )[s, sp ωα ] + FLK ({α = 0}) ◦ sE+ (U ) ⊂ sβ C K (U )[s, sp ωα ] + FLK ({α = 0}) if K 0 > K + L. (d) sβ C K (U )[s, sp ωα ] + FLK ({α = 0}) ⊂ F` ({α = 0, β > `}) for every ` 6 L. (e) If L > 0 and g(s; µ̂) ∈ sβ C K (U )[s, sp ωα ] + FLK ({α = 0}) with g(s; µ̂) > 0 for all s > 0 small enough, then  sm C K (U )[s] + FLK0 ({α = 0}) ◦ g ⊂ sβm C K (U )[s, sβ , sp ωα ] + FLK ({α = 0, β > L/L0 }) for every L0 > 1 and m ∈ Z≥0 . Proof. For shortness we will write C K , E K and E+ K instead of C K (U ), E K (U ) and E+ K (U ), respectively. The assertion in (a) follows noting that 0 0 sβ ◦ sE+ K ⊂ sβ E+ K ⊂ sβ (C K [s] + FLK (U )) ⊂ sβ C K [s] + FLK (U ). 11 Here we apply Corollary 2.2 in the second inclusion whereas in the third one we use that sβ ∈ F0 (U ) by (c) in Lemma A.3 (thanks to the asumption β > 0) and that F0 (U ) · FL (U ) ⊂ FL (U ) by (g) in Lemma A.2. This proves (a). Due to {α = 0} ⊂ U, by applying (a) in Lemma A.2, the second inclusion above yields 0 sβ ◦ sE+ K ⊂ sβ (C K [s] + FLK (U )) ⊂ sβ (C K [s, sp ωα ] + FLK ({α = 0})). (5) We claim at this point that C K [s, sp ωα ] + FLK ({α = 0}) is closed under sum and product. That it is closed under the sum follows directly from (e) in Lemma A.2. In order to show that it is also closed with respect to the product we note that sp ωα ∈ F0∞ ({α = 0}) by (d) in Lemma A.3 and that, on the other hand, s ∈ F0∞ (U ) ⊂ F0∞ ({α = 0}). Consequently, by applying (g), (c) and (e) in Lemma A.2, we can assert that C K [s, sp ωα ] ⊂ F0K ({α = 0}). Hence, by (g) in Lemma A.2 again, C K [s, sp ωα ] · FLK ({α = 0}) ⊂ F0K ({α = 0}) · FLK ({α = 0}) ⊂ FLK ({α = 0}). Thus, since FLK ({α = 0}) · FLK ({α = 0}) ⊂ F2L K ({α = 0}) ⊂ FLK ({α = 0}) by (g) and (d) in Lemma A.2, and C [s, s ωα ] · C [s, s ωα ] ⊂ C [s, s ωα ], the claim is true with respect to the product as well. K p K p K p To prove (b) we observe that 0 0 0 0 0 0 0 0 sp ωα ◦ sE+ K ⊂ sp E+ K K (E+ K ωα + ωα ◦ E+ ) ⊂ sp E+ K (ωα + E K ) ⊂ E+ K (sp ωα + E K ) ⊂ (C K [s] + FLK (U ))(sp ωα + C K [s] + FLK (U )) ⊂ C K [s, sp ωα ] + FLK ({α = 0}), 0 0 where we use ωα (su) = u−α ωα (s) + ωα (u) in the first inclusion and in the second one that ωα ◦ E+ K ⊂ EK e−z −1 due to ωα (s) = F (α log s) log s with F (z) = z , which is an entire function. Finally, in the fourth inclusion we apply Corollary 2.2, and the last one follows by taking the claim into account. Let us turn now to the proof of (c). With this aim in view note that, taking assertions (a) and (b) into account together with the claim, we get 0 0 C K [s, sp ωα ] ◦ sE+ K ⊂ C K [s, sp ωα ] + FLK ({α = 0}). Accordingly, using also (5) and applying the claim once again, 0 0 sβ C K [s, sp ωα ] ◦ sE+ K ⊂ sβ (C K [s, sp ωα ] + FLK ({α = 0})) ⊂ sβ C K [s, sp ωα ] + FLK ({α = 0}). In the second inclusion above we use first that sβ ∈ F0∞ (U ) by (c) in Lemma A.3 and β > 0, and then Lemma A.2 to conclude that F0∞ (U ) · FLK ({α = 0}) ⊂ FLK ({α = 0}). On account of the above inclusion K0 and (e) in Lemma A.2, to prove (c) it suffices to verify that FLK ({α = 0}) ◦ sE+ ⊂ FLK ({α = 0}). To show this we firstly note that, by (c), (d), (g) and (a) in Lemma A.2, 0 0 0 K sE+ ⊂ F1∞ (U ) · F0K (U ) ⊂ F1K (U ) ⊂ F1K (U ) ⊂ F1K ({α = 0}), and secondly we apply (h) in Lemma A.2. This proves the validity of (c). In order to prove (d) note first that sβ C K [s, sp ωα ] ⊂ F`K ({α = 0, β > `}) by (c) in Lemma A.3. On the other hand, since ` 6 L by assumption, FLK ({α = 0}) ⊂ F`K ({α = 0}) ⊂ F`K ({α = 0, β > `}) by (d) and (a) in Lemma A.2. Hence sβ C K [s, sp ωα ] + FLK ({α = 0}) ⊂ F`K ({α = 0, β > `}) by (e) in Lemma A.2, and this proves the assertion in (d). Finally let us show (e). To this end observe firstly that, since g(s; µ̂) ∈ sβ C K (U )[s, sp ωα ]+FLK ({α = 0}), (sm C K [s] + FLK0 ({α = 0})) ◦ g ⊂ sm C K [s] ◦ (sβ C K [s, sp ωα ] + FLK ({α = 0})) + FLK0 ({α = 0}) ◦ g. (6) Note next that, thanks to assertion (d) in the present result, g ∈ F`K ({α = 0, β > `}) with ` = L/L0 because L0 > 1 by assumption. Hence, since g > 0 and ` > 0, the application of (a) and (h) in Lemma A.2 yields FLK0 ({α = 0}) ◦ g ⊂ FLK ({α = 0, β > L/L0 }). 12 It is clear on the other hand that the first summand in (6) is a finite linear combination of terms of the form (sβ C K [s, sp ωα ] + FLK ({α = 0}))m+i with coefficients in C K and i ∈ Z≥0 . In this respect note that (sβ C K [s, sp ωα ] + FLK ({α = 0}))m+i ⊂ sβ(m+i) C K [s, sp ωα ] + FLK ({α = 0}) ⊂ sβm C K [s, sβ , sp ωα ] + FLK ({α = 0}), where the first inclusion follows from the fact that sβ C K [s, sp ωα ] ⊂ F0K ({α = 0}), due to β > 0 and (d) in Lemma A.3, and that F0K ({α = 0}) · FLK ({α = 0}) ⊂ FLK ({α = 0}), by (g) in Lemma A.2. Finally, from (6) and due to FLK ({α = 0}) + FLK ({α = 0, β > L/L0 }) ⊂ FLK ({α = 0, β > L/L0 }), the inclusion in (e) follows. This concludes the proof of the result. The next result is addressed to regularity properties of the Poincaré transition map (and its associated time) between two transverse sections. Lemma 2.4. Let U be an open set of RN and consider a family of vector fields {Yν }ν∈U of the form 1 Yν = (∂x + h(x, y; ν)y∂y ), y ` f (x, y; ν) where ` ∈ Z and f, h ∈ C K (V × U ) with V = (a, b) × (−c, c) ⊂ R2 for some a < b and c > 0. Suppose also that f (x, 0; ν) > 0 for all x ∈ (a, b) and ν ∈ U . Consider two C K families of transverse sections ξ( · ; ν) : (−ε, ε) → Π1 and ζ( · ; ν) : (−ε, ε) → Π2 to the straight line {y = 0} with ξ2 (0) = ζ2 (0) = 0 and ξ20 (0)ζ20 (0) > 0. If P (s; ν) and T (s; ν) are, respectively, the Poincaré and time maps from Π1 to Π2 of Yν then the following holds: (a) P ∈ C K ((−ε, ε) × U ), P (0; ν) = 0 and P 0 (0; ν) > 0. (b) T (s; ν) = s` T̄ (s; ν) with T̄ ∈ C K−1 ((−ε, ε) × U ). Proof. Let us consider the vector field ∂x +h(x, y; ν)∂y , which is equivalent to Yν , and denote by φ(t, p0 ; ν) its  solution passing through p0 ∈ R2 at time t = 0. Clearly if p0 = (x0 , y0 ) then φ(t, p0 ; ν) = t+x0 , φ2 (t, p0 ; ν) . Therefore, by definition,  ζ2 (P (s)) = φ2 ζ1 (P (s)) − ξ1 (s), ξ(s) for all s ∈ (−ε, ε). (Here, and in what follows when there is no risk of ambiguity, we omit the parameter dependence for the sake of shortness.) Then the assertion in (a) follows easily by the implicit function theorem, the smooth dependence of solutions with respect to initial conditions and parameters (see [4, Theorem 1.1]) and the assumption ξ20 (0)ζ20 (0) > 0. In order to prove (b) note first that if we define Θ(x, s; ν) := φ2 (x − ξ1 (s), ξ(s)), which is a C K function, then Z ζ1 (P (s)) T (s; ν) = Θ(x, s)` f (x, Θ(x, s))dx. ξ1 (s) Since Θ(x, 0) = 0 due to the invariance of {y = 0}, Lemma 2.1 shows that there exists a C K−1 function Θ̄(x, s; ν) such that Θ(x, s) = sΘ̄(x; s) and then we can write T (s; ν) = s` T̄ (s; ν) with Z ζ1 (P (s)) T̄ (s; ν) := Θ̄(x, s)` f (x, Θ(x, s))dx, ξ1 (s) which is also C K−1 in (−ε, ε) × U for ε > 0 small enough. This shows (b) and completes the proof. 13 We can now state and prove the following fundamental result concerning the Dulac map D( · ; µ̂) and the Dulac time T ( · ; µ̂) of the hyperbolic saddle (1) between the transverse sections Σσ and Στ . Theorem 2.5. For every µ̂0 = (λ0 , µ0 ) ∈ Ŵ , K ∈ N and L > 0 there exists a neighborhood U of µ̂0 so that X D(s; µ̂) = sλ ∆ij (ωα (s); µ̂)si+λj + FLK (µ̂0 ), 0 (i,j)∈Bλ 0 ,L−λ0 and X T (s; µ̂) = T0 (µ̂) log s + T ij (ωα (s); µ̂)si+λj + FLK (µ̂0 ), n (i,j)∈Bλ 0 ,L where T0 is given in (2) and ∆ij (w; µ̂), T ij (w; µ̂) ∈ C K (U )[w]. Furthermore degw ∆ij = degw T ij = 0 if / Q and, otherwise, α = p − λq if λ0 = p/q with gcd(p, q) = 1. In the latter case the following additional λ0 ∈ properties hold: (a) ∆ij ≡ 0 (respectively, T ij ≡ 0) if (i + rp, j − rq) belongs to Λ0 (respectively, Λn ) for some r ∈ N, (b) degw ∆ij 6 i/p and degw T ij 6 i/p, (c) if p/q ∈ 0 / Dij (respectively, p/q ∈ n / Dij ) then degw ∆ij = 0 (respectively, degw T ij = 0), In particular degw ∆00 = 0 and, in case that L > λ0 , ∆00 (0; µ̂) > 0 for all µ̂ ∈ U . Proof. Consider the parameter µ̂0 = (λ0 , µ0 ) ∈ (0, +∞)× W , the integer K and the real number L > 0 given in the statement and let us define K 0 = K + 1 + dmax(L, 2L/λ0 )e. (7) 0 By applying [16, Theorem A], the family {Xµ̂ }µ̂∈Ŵ is C K conjugated by a diffeomorphism of the form Φ(x, y, µ̂) = (Φµ̂ (x, y), µ̂) defined in a neighbourhood of (0, 0, µ̂0 ) ∈ R2 × Ŵ to 1    Yµ̂N F = x∂x + − λ + B(u; µ̂) y∂y , (8) ηxn1 y n2 ` + u A(u; µ̂) where η = P (0,0;µ̂) , 1 ` ∈ N and (i) if λ0 ∈ / Q then A = B = 0, and (ii) if λ0 = p/q with gcd(p, q) = 1, then A(u; µ̂) and B(u; µ̂) are polynomials in the resonant monomial 0 0 u := xp y q with coefficients in C K (Ŵ ), i.e., A, B ∈ C K (Ŵ )[u]. Without lost of generality, we can assume that the normalizing domain of Φ is V × U , where V and U are, respectively, neighbourhoods of (0, 0) ∈ R2 and µ̂0 ∈ Ŵ . During the computation of the development of the Dulac map and Dulac time we shall lose order of differentiability of the involved functions. As it is usual in this kind of study, the idea is that we can take K 0 arbitrarily large and that the lost of differentiability is well controlled. In any case, in order to avoid any ambiguity, we give a specific value of K 0 in (7) which is large enough for our purposes. Fix δ > 0 and ε > 0 small enough so that the points (0, δ) and (ε, 0) are inside V. We define two auxiliary transverse sections Σδ and Σε to Xµ̂ , see Figure 3, parametrized by s 7−→ Φµ̂ (s, δ) and s 7−→ Φµ̂ (ε, s), respec- tively. Let P1 (· ; µ̂), D2 (· ; µ̂) and P3 (· ; µ̂) be respectively the transition maps from Σσ to Σδ , Σδ to Σε and 14 T1 T2 T3 Figure 3: Auxiliary transverse sections in the decomposition of T. Σε to Στ . (Here P stands for Poincaré and D stands for Dulac.) Consider also T1 (· ; µ̂), T2 (· ; µ̂) and T3 (· ; µ̂) the corresponding time functions. Omitting the dependence on µ̂ we have that D(s) = P3 (D2 (P1 (s))) and   T (s) = T1 (s) + T2 P1 (s) + T3 D2 (P1 (s)) . In what follows, for the sake of shortness we shall use the compact notation C K , E K and E+ K instead of C (U ), E (U ) and E+ (U ), respectively. That being said, by Lemma 2.4 and Corollary 2.2 we have that K K K 0 0 K −1 P1 (s; µ̂) ∈ sE+ T1 (s; µ̂) ∈ sn1 E K −1 ⊂ sn1 C K [s] + FLK (µ̂0 ), K 0 −1 0 (9) P3 (s; µ̂) ∈ sE+ ⊂ sC K [s] + FLK0 (µ̂0 ) T3 (s; µ̂) ∈ sn2 E K −1 ⊂ sn2 C K [s] + FLK0 (µ̂0 ), where we set L0 := max(1, 2L/λ0 ) and we take (7) into account. Furthermore, setting n = (n1 , n2 ), if λ0 ∈ / Q then the local Dulac map and the local Dulac time are given by   δn2 sλn2 εn1 −λn2 −sn2 if n = 6 (0, 0), −λ λ P (0,0) n1 −λn2 D2 (s; µ̂) = δε s and T2 (s; µ̂) = (10)  −1 log( s ) if n = (0, 0), P (0,0) ε respectively. (Here local means that we work close enough to the saddle so that the normalizing coordinates provided by the normal form Yµ̂N F in (8) can be used.) Of course these maps cannot be computed explicitly when λ0 ∈ Q, and in this case we apply Theorems A and B in [19] which show, respectively, that if λ0 = p/q with gcd(p, q) = 1 then 0 0 D2 (s; µ̂) ∈ sλ C K [sp , sp ωα ] + FLK (µ̂0 ) (11)   and, setting κ := max n1 n2 p , q , 0 0 0 0 T2 (s; µ̂) ∈ T0 (µ̂)(log s − log ε) + sλn2 C K [sp , sp ωα ] + sn1 C K [sp ] + sκp C K [sp , sp ωα ] + FLK (µ̂0 ). (12) Here T0 is the one given in (2), α = α(µ̂) = p − λq, so that α(µ̂0 ) = 0, and we use the compact notation ωα = ω s; α(µ̂) . There are some remarks to be made about the application of Theorems A and B in [19]: 15 • They provide the local Dulac map and local Dulac time with transversal sections normalized to ε = δ = 1, say DN and TN , respectively. To bypass this technical inconvenience we consider the new local change of coordinates Φ̃ = Φ ◦ h where h(x, y) := (εx, δy), so that D2 (s; µ̂) = δDN (s/ε; µ̂) and T2 (s; µ̂) = TN (s/ε; µ̂). Then the principal parts in (11) and (12) follow noting, firstly, that the pull-back of (8) by h preserves the normal form (it only changes η and the coefficients of A and B) and, secondly, that ωα (s/ε) = εα ωα (s) + ωα (ε). • In both results the coefficients of A and B are treated as independent parameters, i.e., 1  1 N X −1   Yα,β = PM x∂x + −p+ αi+1 ui y∂y , β0 xn1 y n2 + u` i=1 βi ui−1 q i=0 where α1 := p − λq. In particular, they show that the remainder of DN (s; α) and TN (s; α, β) belong to FL∞ (U0 ) and FL∞ (U0 × RM +1 ), respectively, where U0 is a neighbourhood of {α1 = 0} in RN . In our 0 application we have αi = αi (µ̂) and βi = βi (µ̂), which are C K functions, and consequently to obtain the remainder in (11) and (12) we must also use (h) in Lemma A.2. Let us consider the case λ0 = p/q ∈ Q first. Then, from (9) and (11), by applying (c) in Lemma 2.3 and taking (7) also into account, we get (D2 ◦ P1 )(s; µ̂) ∈ sλ C K [s, sp ωα ] + FLK (µ̂0 ). (13) (Here, and in what follows, we take α(µ̂0 ) = 0 also into account.) Moreover, from (9) and (12), the application of (a) and (c) in Lemma 2.3 shows that  (T2 ◦ P1 )(s; µ̂) ∈ T0 log s + C K [s] + sn1 C K [s] + sλn2 C K [s, sp ωα ] + sκp C K [s, sp ωα ] + FLK (µ̂0 ), (14) 0 0 where we also use that log(sE+ K ) = log s + E K ⊂ log s + C K [s] + FLK (µ̂0 ) thanks to Corollary 2.2. Finally, if f (s; µ̂) ∈ s C [s] + FL0 (µ̂0 ) then, from (13) and applying (e) in Lemma 2.3, m K K (f ◦ D2 ◦ P1 )(s; µ̂) ∈ sλm C K [s, sλ , sp ωα ] + FLK (µ̂0 ) due to λ(µ̂0 ) = λ0 > L/L0 = min(L, λ0 /2). Using the above inclusion with {f = T3 , m = n2 } and {f = P3 , m = 1}, from (9) and (14) we get, respectively, T = T1 + T2 ◦ P1 + T3 ◦ D2 ◦ P1 ∈ T0 log s + T0 C K [s] + sn1 C K [s] + sκp C K [s, sp ωα ] + sλn2 C K [s, sλ , sp ωα ] + FLK (µ̂0 ) (15) and D = P3 ◦ D2 ◦ P1 ∈ sλ C K [s, sλ , sp ωα ] + FLK (µ̂0 ). Recall that Bλ,L k = {(i, j) ∈ Λk : i + λj 6 L} with Λk = (Z≥k1 × {0}) ∪ (Z≥0 × Z≥k2 ) by definition (see also Figure 4). On account of this the above two inclusions imply, respectively, that we can write X T (s; µ̂) = T0 (µ̂) log s + T̂ij (ωα (s); µ̂)si+λj + FLK (µ̂0 ) (16) n (i,j)∈Bλ 0 ,L and X D(s; µ̂) = sλ ˆ ij (ωα (s); µ̂)si+λj + FLK (µ̂0 ), ∆ (17) 0 (i,j)∈Bλ 0 ,L−λ0 ˆ ij (w; µ̂) and T̂ij (w; µ̂) are polynomials in w with coefficients C K functions in a neighbourhood of µ̂0 . where ∆ Here we use that, thanks to (c) in Lemma A.3, if i + λ0 j > L then si+λj ωα` ∈ FL∞ (µ̂0 ) for any ` ∈ Z≥0 . Observe moreover that these polynomials can be taken with degw ∆ ˆ ij 6 i/p and degw T̂ij 6 i/p. 16 j i + λ0 j = L n2 n1 i Figure 4: The set Bλn0 ,L (filled dots) with two pairs (in red) yielding to a replacement. Let us focus now on the expression in (16) and consider the summation’s grid where (i, j) varies. For each fixed (i, j) ∈ Bλn0 ,L we define, see Figure 4, r? := max {r ∈ Z≥0 : (i, j) + r(p, −q) ∈ Λn } and (i? , j? ) := (i, j) + r? (p, −q). Accordingly (i? , j? ) ∈ Bλn0 ,L and we can write si+λj = si? −r? p+λ(j? +r? q) = si? +λj? s−r? (p−λq) = si? +λj? (1 + αωα )r? . (Certainly r? depends on the particular pair (i, j) considered as well as on n and λ0 = p/q. We omit this ˆ ij , we dependence for the sake of shortness.) Then, for each pair (i, j) ∈ Bλn0 ,L and ` = 0, 1, . . . , degw ∆ replace the monomial si+λj ` ωα in (16) by s ωα (1 + αωα ) to obtain i? +λj? ` r? X T (s; µ̂) = T0 (µ̂) log s + T ij (ωα (s); µ̂)si+λj + FLK (µ̂0 ), n (i,j)∈Bλ 0 ,L with new polynomials T ij (w; µ̂) such that T ij ≡ 0 if (i + rp, j − rq) ∈ Λn for some r ∈ N, i.e., assertion (a) in the statement. Next we replace similarly the monomials in the grid Bλ0 0 ,L−λ0 for the Dulac map in (17). (Note that in this case if λ0 > L then the grid is empty.) We proceed exactly the same way as before but taking Λ0 instead of Λn in the definition of r? given above. In doing so we get X D(s; µ̂) = sλ ∆ij (ωα (s); µ̂)si+λj + FLK (µ̂0 ) 0 (i,j)∈Bλ 0 ,L−λ0 with new polynomials ∆ij (w; µ̂) satisfying the assertion in (a). Let us turn to the proof of the other assertions. That degw ∆ij 6 i/p and degw T ij 6 i/p follows noting that this is true for the polynomials ∆ˆ ij and T̂ij and that, on account of this, after the replacement of s ωα by s i+λj ` ωα (1 + αωα ) we have r? + ` 6 r? + i/p = (i + r? p)/p = i? /p. This proves (b). i? +λj? ` r? Let us prove the validity of (c). We prove first the assertion with regard to the polynomials T ij by contradiction. Suppose λ0 = p/q ∈ n / Dij , which implies T ij = T̂ij , and ` := degw T ij > 0. We distinguish the cases j = 0 and j > 0: 17 • If j = 0 and n2 > 0 then, due to T ij = T̂ij , the monomial si ωα` in T (s) comes necessarily from the summand sκp C K [s, sp ωα ] in (15), where recall that κ = max np1 , nq2 . Hence i = (κ + `)p + i0 for some i0 ∈ Z≥0 . Then, setting j 0 := (κ + `)q, on account of ` > 0 and κ > n2 /q, it follows that (i0 , j 0 ) ∈ Λn \ {(i, 0)} satisfies i + λ0 j = i0 + λ0 j 0 , which implies λ0 ∈ Di0 n , a contradiction. If j = n2 = 0 then si ωα` can also come from the summand sλn2 C K [s, sλ , sp ωα ] = C K [s, sλ , sp ωα ] in (15). Then i = i0 + `p for some i0 ∈ Z≥0 and setting j 0 = `q it follows that (i0 , j 0 ) ∈ Z2≥0 (which is equal to Λn due to n2 = 0) with (i0 , j 0 ) 6= (i, 0) and, on the other hand, i + λ0 0 = i0 + `p = i0 + `qλ0 = i0 + λ0 j 0 . Consequently λ0 ∈ Di0 n , which contradicts the assumption on λ0 again. • Consider finally the case j > 0, which implies j > n2 (see Figure 4). Then, due to T ij = T̂ij , the monomial si+λj ωα` in T (s) comes unavoidably from the summand sλn2 C K [s, sλ , sp ωα ] in (15). Thus i = i0 + `p for some i0 ∈ Z≥0 . In this case, setting j 0 := j + `q ∈ Z≥n2 , we have (i0 , j 0 ) ∈ Λn \ {(i, j)}, due to ` > 0, and i + λ0 j = i0 + λ0 j 0 , which implies λ0 ∈ Dij n , a contradiction. We show next that if p/q ∈ 0 / Dij then ` := degw ∆ij = 0. To this end note first that ∆ij = ∆ ˆ ij by construction. Moreover, due to D(s; µ̂) ∈ s C [s, s , s ωα ]+FL (µ̂0 ), the monomial s λ K λ p K ωα in (17) verifies that i = i0 +p` i+λj ` for some i ∈ Z≥0 . Thus, setting j := j +`q, we have (i , j ) ∈ Λ0 and i+λ0 j = i0 +λ0 j 0 . Since (i, j) 6= (i0 , j 0 ) 0 0 0 0 if and only if ` > 0, this implies that ` = 0, otherwise λ0 ∈ Dij 0 , a contradiction. This concludes the proof for the case λ0 ∈ Q. Let us consider next the case λ0 ∈ / Q. Note first that, on account of (9) and (10), the preceding expressions are valid without using the compensator, i.e., D2 ◦ P1 ∈ sλ C K [s] + FLK (µ̂0 ) and T2 ◦ P1 ∈ τ0 (log s + C K [s]) + sn1 C K [s] + sλn2 C K [s] + FLK (µ̂0 ). Then, exactly as before, using that if f ∈ sm C K [s] + FLK0 (µ̂0 ) then f ◦ D2 ◦ P1 ∈ sλm C K [s, sλ ] + FLK (µ̂0 ), we can conclude that D = P3 ◦ D2 ◦ P1 ∈ sλ C K [s, sλ ] + FLK (µ̂0 ) and T = T1 + T2 ◦ P1 + T3 ◦ D2 ◦ P1 ∈ τ0 log s + τ0 C K [s] + sn1 C K [s] + sλn2 C K [s, sλ ] + FLK (µ̂0 ). ˆ ij = degw T̂ij = 0. This shows the validity of the result write as (17) and (16), respectively, but with degw ∆ for the case λ0 ∈ / Q. With regard to the last assertion in the statement we can already assert that degw ∆00 = 0 in both cases, λ0 ∈ Q and λ0 ∈ / Q. Recall on the other hand that D(s) = P3 (D2 (P1 (s))) with P1 and P3 regular transitions maps, verifying Pi (0; µ̂) = 0 and Pi0 (0; µ̂) > 0 by Lemma 2.4, and D2 (s; µ̂) = δDN (s/ε; µ̂), where DN is the local Dulac map with transversal sections normalized to ε = δ = 1. Since lims→0+ DNs(s;µ̂) λ =1 thanks to [19, Theorem A], we conclude that λ ∆00 (0; µ̂) = ∆00 (µ̂) = P30 (0; µ̂)δ (P10 (0; µ̂)/ε) > 0 and so the last assertion is true. This completes the proof of the result. 3 Proof of the main results This section is devoted to the proofs of Theorems A, B and C. In order to show the first one we shall apply the following two lemmas. Lemma 3.1. Let U be an open subset of RN and C a closed subset inside U . Assume that f ∈ C ∞ (U \ C) and that, for each K ∈ Z≥0 , there exist aScollection of functions {gK m }m∈N and a collection of open subsets {VK }m∈N with gK ∈ C (VK ) and C ⊂ m∈N VK verifying that m m K m m 18 0 0 0 in VK ∩ VK 0 ∩ C, m m m m (a) gK = gK m (b) f = gK in (U \ C) ∩ VKm . Then the function ( f (p) if p ∈ U \ C, fˆ(p) := m gK (p) if p ∈ C ∩ VKm , is well defined and belongs to C ∞ (U ). Proof. We will prove that fˆ ∈ C K (U ) for all K ∈ Z≥0 by induction on K. The base case K = 0 is clear. For the induction step assume that fˆ ∈ C K (U ). Let us fix any p0 ∈ C and take m such that p0 ∈ C ∩ VK+1 m . ˆ Then the partial derivatives of f of order K + 1 at p0 exist and are continuous because   ∂vK gK+1 m (p0 + hw) − ∂vK gK+1 m (p0 ) ∂w ∂vK fˆ (p0 ) = lim = ∂w ∂vK gK+1 m K+1 m (p0 ) = ∂v+w gK+1 (p0 ) h→0 h for any v, w ∈ ZN ≥0 with |v| = K and |w| = 1. This proves the validity of the result. Lemma 3.2. For any k ∈ Z2≥0 and (i? , j? ) ∈ Λk , the set  D̂ik? j? := λ > 0 : there exist (i, j), (i0 , j 0 ) ∈ Λk with (i, j) 6= (i0 , j 0 ) and i + λj = i0 + λj 0 6 i? + λj? is discrete in (0, +∞). Proof. Suppose that limm→∞ λm = λ? ∈ (0, +∞) with λm ∈ D̂ik? j? for all m ∈ N. Then, for each m ∈ N there exist two different pairs (im , jm ) and (i0m , jm 0 ) in Λn such that im + λm jm = i0m + λm jm 0 6 i ? + λ m j? . i0 −i Therefore, since jm − jm0 6= 0, λm = jm m 0 . In addition, due to λ? ∈ (0, +∞), it follows that the sequences m −jm (jm )m and (jm )m are bounded. Thus (jm − jm 0 0 )m is a bounded sequence of integers and, consequently, λm belongs to the set of rational numbers with bounded denominator, which is discrete. Hence λm = λ? for all m ∈ N large enough and the result follows. Remark 3.3. By Lemma 3.2 we can assert that Dij k is a discrete subset of (0, +∞) because Dik? j? ⊂ D̂ik? j? , see Definition 1.4. Exactly the same proof shows that DL k is a discrete set in (0, +∞) as well.  We are now ready to tackle the proof Theorem A, that follows fundamentally by applying Theorem 2.5. Notice that basic difference between Theorem 2.5 and Theorem A is the order of quantifiers in the statements, ∀ . . . ∃ in the first one and ∃ . . . ∀ in the second one. The key idea to prove that the coefficients Tij in the asymptotic expansion of T ( · ; µ̂) are C ∞ is to reverse the order of these quantifiers. Before starting the proof let us make the following easy observation that we shall use several times. Pm Remark 3.4. If i=1 ai s λi + f (s) = 0 for all s ∈ (0, ε), where λi ∈ R with λ1 < λ2 < · · · < λm , a1 , a2 , . . . , am ∈ R and f (s) = o(sλm ) then a1 = a2 = · · · = am = 0.  Proof of Theorem A. The application of Theorem 2.5 shows that, for every µ̂0 = (λ0 , µ0 ) ∈ Ŵ , K ∈ N and L > 0, there exists a neighborhood U = Uµ̂0 KL of µ̂0 such that the Dulac time can be written as X  T (s; µ̂) = T0 (µ̂) log s + T µ̂ij0 KL ωα (s); µ̂ si+λj + R(s; µ̂), (18) n (i,j)∈Bλ 0 ,L 19 where T0 is given in (2), T µ̂ij0 KL (w; µ̂) ∈ C K (U )[w] and R ∈ FLK (µ̂0 ). Furthermore we know that if λ0 ∈ n / Dij µ̂0 KL then degw T ij = 0. Taking this property (and notation) into account, we shall prove the following claim, where recall that Dij n ⊂ Q>0 is a discrete set in (0, +∞), see Definition 1.4.  Key assertion: For each (i, j) ∈ Λn there exists Tij ∈ C ∞ ((0, +∞) \ Dij n ) × W such that, for every µ̂0 ∈ ((0, +∞) \ Dijn ) × W, K ∈ N and L > 0 large enough, we have Tij (µ̂) = T µ̂ij0 KL (0; µ̂) for all µ̂ = (λ, µ) ∈ Uµ̂0 KL with λ ∈ n / Dij . (19) The proof of this is rather long and technical but the theorem will follow almost right away once we prove it. With this end in mind let us fix any (i? , j? ) ∈ Λn and recall that, by Lemma 3.2, D̂in? j? is a discrete set in (0, +∞) that contains Din? j? . We write (0, +∞) \ D̂in? j? as a disjoint union G (0, +∞) \ D̂in? j? = Jir? j? , (20) r∈N where each Jir? j? is a open interval in (0, +∞). Since sup D̂in? j? < ∞ if and only if j? = 0, it turns out that there exists an unbounded interval Jir? j? if and only if j? = 0. Therefore Mir? j? := sup{i? + λj? : λ ∈ Jir? j? } is well defined and finite for all r ∈ N. Furthermore, for each r ∈ N such that Jir? j? 6= ∅ we also define  Zir? j? := (i, j) ∈ Λn : i + λj 6 i? + λj? for all λ ∈ Jir? j? . (21) Observe that this is a finite subset that contains the pair (i? , j? ) by definition. Moreover if we say that Zir? j? = {(i1 , j1 ), (i2 , j2 ), . . . , (im , jm ), (i? , j? )} then, on account of (20) and the definition of D̂in? j? , it can be indexed so that i1 + λj1 < i2 + λj2 < · · · < im + λjm < i? + λj? for all λ ∈ Jir? j? . (22) We claim at this point that i + λj > i? + λj? for all (i, j) ∈ Λn \ Zir? j? and λ ∈ Jir? j? . (23) Let us show this by contradiction. If there exists λ1 ∈ Jir? j? such that i + λ1 j 6 i? + λ1 j? then the fact that (i, j) ∈ / Zir? j? implies the existence of some λ2 ∈ Jir? j? such that i + λ2 j > i? + λ2 j? . Accordingly, by continuity, there exists λ3 ∈ Jir? j? verifying i + λ3 j = i? + λ3 j? and then λ3 ∈ D̂in? j? by definition, which contradicts (20). This proves the validity of the claim. Let us fix now r ∈ N and take any λ0 ∈ Jir? j? . Then, by applying Theorem 2.5 with any µ̂0 = (λ0 , µ0 ), K ∈ N and L > Mir? j? (which in particular implies Zir? j? ⊂ Bλn0 ,L ), we can assert the existence of an open µ̂0 KL neighbourhood U of µ̂0 such that T (s; µ̂) can be written as in (18) with T ij (w; µ̂) ∈ C K (U )[w] for all (i, j) ∈ Bλ0 ,L and R ∈ FL (µ̂0 ). On account of this there exist a neighborhood V of µ̂0 and constants n K C, s0 > 0 such that |R(s; µ̂)| 6 CsL for all µ̂ ∈ V and s ∈ (0, s0 ). At this point we replace U by a smaller open neighbourhood of µ̂0 given by the finite intersection \ {(λ, µ) ∈ U ∩ V : i + λj − i? − λj? − α+ (λ) > 0}, (24) n (i,j)∈Bλ \Zir? j? 0 ,L where we use the notation x+ = max(x, 0) for shortness. Note that µ̂0 belongs to the above set due to (23) and the fact that α(µ̂0 ) = 0. 20 Next we show that degw T µ̂ij0 KL = 0 for all (i, j) ∈ Zir? j? . By contradiction again, if degw T µ̂ij0 KL > 0 then λ0 ∈ Dij n by assertion (c) in Theorem 2.5, which implies the existence of some pair (i0 , j 0 ) ∈ Λn \ {(i, j)} such that i + λ0 j 0 = i + λ0 j 6 i? + λ0 j? , where the inequality holds thanks to (22). Accordingly λ0 ∈ D̂in? j? , 0 that contradicts (20). Thus the assertion is true and, from (18), we can write X X T (s; µ̂) = T0 (µ̂) log s + Tijµ̂0 KL (µ̂)si+λj + T µ̂ij0 KL (ωα (s); µ̂)si+λj + R(s; µ̂), (i,j)∈Zir? j? n (i,j)∈Bλ \Zir? j? 0 ,L where Tijµ̂0 KL (µ̂) := T µ̂ij0 KL (0; µ̂) is a C K function in U. Note that the two last summands above belong to o(si? +λj? ) for each fixed (λ, µ) ∈ U . Indeed, the assertion with regard to the third summand is a consequence of (24) using that, by applying (a) in Lemma A.3 with ν = (0, 0), for each fixed α there exists C > 0 such + that |ω(s; α)| 6 Cs−α | log s| for all s ∈ (0, 1/e). On the other hand, the fact that R ∈ o(si? +λj? ) is clear because R ∈ O(sL ) and, by construction, L > Mir? j? > i? + λj? for all λ ∈ Jir? j? . Therefore X T (s; µ̂) = T0 (µ̂) log s + Tijµ̂0 KL (µ̂)si+λj + o(si? +λj? ). (25) (i,j)∈Zir? j? Now we take any other λ00 ∈ Jir? j? and apply Theorem 2.5 with µ̂00 = (λ00 , µ00 ), K 0 > K and L0 > Mir? j? . µ̂0 K 0 L0 0 In doing so, exactly as before, we obtain an open neighborhood U 0 of µ̂00 and Tij0 ∈ C K (U 0 ) so that X µ̂0 K 0 L0 T (s; µ̂) = T0 (µ̂) log s + Tij0 (µ̂)si+λj + o(si? +λj? ). (i,j)∈Zir? j? Consequently, for each fixed µ̂ = (λ, µ) ∈ U ∩ U 0 with λ ∈ Jir? j? , we get that X   µ̂0 K 0 L0 0 = T (s; µ̂) − T (s; µ̂) = Tijµ̂0 KL (µ̂) − Tij0 (µ̂) si+λj + o(si? +λj? ) (i,j)∈Zir? j? µ̂0 K 0 L0 for all s > 0 small enough. This equality, on account of (22) and Remark 3.4, shows that Tijµ̂0 KL = Tij0 µ̂0 K 0 L0 on U ∩ U 0 for each (i, j) ∈ Zir? j? . Hence, since (i? , j? ) ∈ Zir? j? , all the local functions Tiµ̂?0jKL ? and Ti?0j? glue together and provide a well defined function Ti? j? on Ji? j? × W , which is of class C for arbitrarily r K K ∈ N, i.e., Ti? j? ∈ C ∞ (Jir? j? × W ). In addition, due to (20) and the fact that r is arbitrary, we get that  Ti? j? ∈ C ∞ ((0, +∞) \ D̂in? j? ) × W . It is clear then that Ti? j? does not depend on K, and we remark that it does not depend on µ̂0 or L neither.  So far we have proved the existence of Ti? j? ∈ C ∞ ((0, +∞) \ D̂in? j? ) × W verifying the equality in (19) for every µ̂0 ∈ ((0, +∞) \ D̂in? j? ) × W, K ∈ N and L > 0. In other words, that the key assertion is true replacing Din? j? by D̂in? j? . Our next goal (recall that Din? j? ⊂ D̂in? j? ) is to prove that Ti? j? extends smoothly to (D̂in? j? \ Din? j? ) × W . With this aim in view we fix λ0 ∈ D̂in? j? \ Din? j? and apply Theorem 2.5 with any µ̂0 = (λ0 , µ0 ), K ∈ N and L > 0 large enough to obtain a neighbourhood U of µ̂0 such that the Dulac time T (s; µ̂) can be written as in (18), where T0 is given in (2), T µ̂ij0 KL (w; µ̂) ∈ C K (U )[w] and R ∈ FLK (µ̂0 ). Let us note that degw T µ̂i?0jKL ? / Din? j? , by (c) in Theorem 2.5, and define Tiµ̂?0jKL = 0 due to λ0 ∈ ? (µ̂) := T iµ̂?0jKL ? (0; µ̂), which is a C function in U. Take an open interval I ⊂ (0, +∞) verifying I ∩ D̂i? j? = {λ0 }, which exists because D̂i0? j? K 0 is a discrete set in (0, +∞), and define V := U ∩ (I × W ). We claim that Ti? j? (µ̂) = Tiµ̂?0jKL ? (µ̂) for all µ̂ ∈ V \ {λ = λ0 }. (26) (Note that Ti? j? ∈ C ∞ (V \ {λ = λ0 }), whereas Tiµ̂?0jKL ? ∈ C K (V ). This will be the key point later on.) With regard to the proof of this claim we stress that although we have already proved the key assertion replacing 21 / ((0, +∞) \ D̂in? j? ) × W . That being Din? j? by D̂in? j? , we can neither apply this weaker version because µ̂0 ∈ said, in order to prove (26) we observe first that, taking s = 1+αωα (s) into account and working on α 6= 0, −α we can “deconstruct” any polynomial in ωα (s) and write it as a polynomial in s−α . More generally, one P` P` can readily show using the Newton’s binomial formula that the identity k=0 Ak ω k (s; α) = r=0 Br s−αr holds on α 6= 0 if, and only if, X̀  r  X̀   k k −k Ak = α Br and Br = α (−1)k−r Ak . (27) k r r=k k=r In this way, since α = 0 if and only if λ = λ0 (recall that α := p − λq and λ0 = p/q) and, on the other hand, degw Tijµ̂0 KL 6 bi/pc by (b) in Theorem 2.5, we can write bi/pc  i+λj X µ̂ KL T µ̂ij0 KL ωα (s); µ̂ s = Tei−kp,j+kq 0 (µ̂)si−kp+λ(j+kq) for all µ̂ ∈ V \ {λ = λ0 }, (28) k=0 where the functions Tei−kp,j+kq µ̂0 KL for k = 0, 1, . . . , bi/pc are defined univocally in terms of the coefficients of the polynomial T µ̂ij0 KL ∈ C K (V )[w] by means of the second equality in (27) with ` = bi/pc. Thus, in particular, Tei−kp,j+kq µ̂0 KL ∈ C K (V \ {λ = λ0 }). In doing so for all the polynomials in (18) we obtain that X T (s; µ̂) = T0 (µ̂) log s + Teijµ̂0 KL (µ̂)si+λj + FLK (µ̂0 ) for all µ̂ ∈ V \ {λ = λ0 }. (29) 0 (i,j)∈Bλ 0 ,L (We remark that the summation grid is Bλ0 0 ,L instead of Bλn0 ,L .) In this respect we observe that Teiµ̂?0jKL ? = Tiµ̂?0jKL ? on V \ {λ = λ0 }, i.e., the coefficient of si? +λj? remains unchanged after the “deconstruction process” that brings (18) to (29). This equality is a consequence of the following facts: (a) degw T iµ̂?0jKL ? = 0 due to λ0 ∈ / Din? j? , by (c) in Theorem 2.5, and (b) the values of the exponents in si+λj and si−kp+λ(j+kq) in (28) are the same at λ = λ0 = p/q. Thus, on account of the above equality and setting I \ {λ0 } = I− t I+ , the claim in (26) will follow once we show that Ti? j? = Teiµ̂?0jKL ? on V ∩ (I+ × W ) and V ∩ (I− × W ). (30) r r r In order to prove this let Ji?−j? and Ji?+j? be the intervals of (0, +∞) \ D̂in? j? with I± ⊂ Ji?±j? and note that if r r r λ ∈ Ji?±j? then λ ∈ n / D̂ij for all (i, j) ∈ Zi?±j? . (This is so because otherwise there would exist (i, j) ∈ Zi?±j? and two different pairs (i1 , j1 ), (i2 , j2 ) ∈ Λn such that i1 + λj1 = i2 + λj2 6 i + λj 6 i? + λj? , which would imply r r r λ ∈ D̂in? j? , contradicting that λ ∈ Ji?±j? .) Taking L > 0 large enough, specifically L > max{Mi?−j? , Mi?+j? }, this implies that X T (s; µ̂) = T0 (µ̂) log s + Tij (µ̂)si+λj + o(si? +λj? ) for all µ̂ ∈ V ∩ (I± × W ). (31) r (i,j)∈Zi?±j? r Indeed, to verify this at some µ̂1 = (λ1 , µ1 ) ∈ V with λ1 ∈ I± ⊂ Ji?±j? we use first the identity in (25) taking µ̂1 instead of µ̂0 (we can do this due to the fact that λ1 ∈ / D̂in? j? ) and then the equality follows by applying r± the key assertion in (19) to each (i, j) ∈ Zi? j? . (For this effect recall that the key assertion has already been 22 r r proved on ((0, +∞) \ D̂ij n ) × W and that λ1 ∈ Ji?±j? implies λ1 ∈ n / D̂ij for all (i, j) ∈ Zi?±j? .) Let us define in addition the sets Z˜± := {(i, j) ∈ Λ0 : i + λj 6 i? + λj? for all λ ∈ I± }. Note that if (i, j) ∈ Bλ0 0 ,L \ Z˜± then i + λj > i? + λj? for all λ ∈ I± . This is so because if there exists λ1 ∈ I± such that i + λ1 j 6 i? + λ1 j? then the fact that (i, j) ∈ / Z˜± implies the existence of some λ2 ∈ I± such that i + λ2 j > i? + λ2 j? . Then, by continuity, there would exist λ3 ∈ I± such that i + λ3 j = i? + λ3 j? and so, by definition, λ3 ∈ D̂i0? j? which contradicts that I ∩ D̂i0? j? = {λ0 }. Accordingly, from (29), X T (s; µ̂) = T0 (µ̂) log s + T̃ijµ̂0 KL (µ̂)si+λj + o(si? +λj? ) for all µ̂ ∈ V ∩ (I± × W ). (32) (i,j)∈Z˜± r r Notice that Zi?±j? ⊂ Z˜± due to, recall (21), I± ⊂ Ji?±j? and Λn ⊂ Λ0 . Consequently, from (31) and (32), X   X T̃ijµ̂0 KL (µ̂) − Tij (µ̂) si+λj + T̃ijµ̂0 KL (µ̂)si+λj + o(si? +λj? ) = 0 (33) r r (i,j)∈Zi?±j? (i,j)∈Z˜± \Zi?±j? for all µ̂ ∈ V ∩ (I± × W ). Therefore, since i + λj 6= i0 + λj 0 for all λ ∈ I± and any two different pairs r (i, j), (i0 , j 0 ) ∈ Z˜± , by Remark 3.4 we deduce that T̃ijµ̂0 KL = Tij on V ∩ (I± × W ) for any (i, j) ∈ Zi?±j? . r− r+ This, on account of (i? , j? ) ∈ Zi? j? ∩ Zi? j? , shows (30) and proves that the claim in (26) is true. At this point we are in position to apply Lemma 3.1. To this end, for reader’s convenience, let us denote C = {λ0 } × W and observe that Ti? j? ∈ C ∞ ((I × W ) \ C). Furthermore the claim in (26) shows that for each µ̂0 ∈ C and K ∈ N there exist a neighbourhood VKµ̂0 of µ̂0 and a function Tiµ̂?0jKL ? ∈ C K (VKµ̂0 ) such that Ti? j? = Tiµ̂?0jKL on VKµ̂0 \ C. We take a sequence {µ̂r }r∈N with µ̂r ∈ C = {λ0 } × W for all r ∈ N such that S ? C ⊂ r∈N VKµ̂r and we apply Lemma 3.1 with f = Ti? j? and gK r = Tiµ̂?rjKL ? in order to conclude that Ti? j? extends smoothly to C = {λ0 } × W . Since λ0 ∈ D̂i? j? \ Di? j? is arbitrary and we have previously proved n n  that Ti? j? ∈ C ∞ ((0, +∞) \ D̂in? j? ) × W , we can assert that Ti? j? ∈ C ∞ (((0, +∞) \ Din? j? ) × W ). So far we have proved the key assertion in (19) with regard to the functions Tij with (i, j) ∈ Λn is true. Let us fix now λ0 > 0 and L > 0 and, in order to prove (a), suppose that λ0 ∈ / DL n , i.e., λ0 ∈ n / Dij for all (i, j) ∈ Bλ0 ,L . Then, on account of (19), the application of Theorem 2.5 to any K ∈ N and µ0 ∈ W yields n X \ \ T (s; µ̂) − T0 (µ̂) log s − Tij (µ̂)si+λj ∈ FLK ((λ0 , µ0 )). n (i,j)∈Bλ K∈N µ0 ∈W 0 ,L This proves the assertion in (a) because from Definition 1.2 one can readily verify that \ \ \ FLK ((λ0 , µ0 )) = FLK ({λ0 } × W ) = FL∞ ({λ0 } × W ). K∈N µ0 ∈W K∈N Let us proceed with the proof of (b). So assume λ0 ∈ DL n and take any µ0 ∈ W, K ∈ N and L > 0 large enough. We claim that if (i0 , j0 ) ∈ Λ0 \ Λn then there exists a neighborhood V of µ̂0 = (λ0 , µ0 ) such that T̃iµ̂00j0KL (µ̂) = 0 for all µ̂ ∈ V \ {λ = λ0 }. Indeed, this follows from (33) particularized with (i? , j? ) := (i0 , j0 + n2 ) ∈ Λn because in this case one r can easily verify that (i0 , j0 ) ∈ Z˜± \ Zi?±j? and, exactly as before, by applying Remark 3.4 we deduce that T̃iµ̂00j0KL = 0 on V ∩ (I± × W ). (Let us remark in this respect that, regardless of λ0 ∈ Din? j? , to obtain (33) we only need to work in an interval I such that I ∩ D̂i0? j? = {λ0 }.) Consequently, together with (30), this shows that for all µ0 ∈ W , K ∈ N and L > 0 large enough, there exists a neighbourhood V of µ̂0 such that  e µ̂0 KL Tij (µ̂) if (i, j) ∈ Λn , Tij (µ̂) = 0 if (i, j) ∈ Λ0 \ Λn , 23 for all µ̂ ∈ V \ {λ = λ0 }. Accordingly, if µ̂ ∈ V \ {λ = λ0 } then we can assert that bi/pc X X µ̂0 KL T λij0 (w; µ̂) := Ti−rp,j+rq (µ̂)(1 + αw) = r T̃i−rp,j+rq (µ̂)(1 + αw)r = T µ̂ij0 KL (w; µ̂) ∈ C K (U )[w], n r∈Aijλ r=0 0 where the third equality follows from (28) and si−kp+λ(j+kq) = si+λj s−αk = si+λj (1 + αωα (s))k . Thus, the coefficients of the w-polynomial T λij0 (w; µ̂), which are C ∞ on the open set n o n µ̂ = (λ, µ) ∈ Ŵ : λ ∈ / ∪r∈Aijλ n Di−rp,j+rq , 0 are C K on a neighborhood of C = {λ0 } × W . Exactly as before, the application of Lemma 3.1 shows that these coefficients extend smoothly to C. The above equality also shows the application of Theorem 2.5 to any K ∈ N and µ0 ∈ W gives X \ \ T (s; µ̂) − T0 (µ̂) log s − T λij0 (ωα (s); µ̂)si+λj ∈ FLK ((λ0 , µ0 )) = FL∞ ({λ0 } × W ). n (i,j)∈Bλ K∈N µ0 ∈W 0 ,L This shows the validity of the assertion in (b) and completes the proof of the result. Proof of Theorem B. The application of Theorem 2.5 shows that, for every µ̂0 = (λ0 , µ0 ) ∈ Ŵ , K ∈ N and L > 0, there exists a neighborhood U = Uµ̂0 KL of µ̂0 such that the Dulac map can be written as X D(s; µ̂) = sλ ∆µ̂ij0 KL (ωα (s); µ̂)si+λj + FLK (µ̂0 ), 0 (i,j)∈Bλ 0 ,L−λ0 where ∆µ̂ij0 KL (w; µ̂) ∈ C K (U )[w] verify assertions (a), (b) and (c). Moreover degw ∆00 µ̂0 KL = 0 and if L > λ0 µ̂0 KL then ∆00 (0; µ̂) > 0 for all µ̂ ∈ U . The idea now is to put the factor sλ inside the summation in order to get a more convenient expression. With this aim let us fix ε > 0 small enough such that Bλ0 0 ,L+ε = Bλ0 0 ,L and shrink U so that λ − λ0 < ε for all µ̂ = (λ, µ) ∈ U. On account of this, if µ̂ ∈ U and (i, j) ∈ Bλ0 0 ,L−λ0 then i + λ(j + 1) 6 L − λ0 + λ < L + ε and, setting ( µ̂0 KL ∆µ̂i,j−1 0 KL (w; µ̂) if j > 1, Z ij (w; µ̂) := 0 if j = 0, we can write X D(s; µ̂) = Z µ̂ij0 KL (ωα (s); µ̂)si+λj + FLK (µ̂0 ). 0 (i,j)∈Bλ 0 ,L Certainly Z µ̂ij0 KL (w; µ̂) ∈ C K (U )[w] and, more important, one can easily show that these polynomials also verify assertions (a), (b) and (c) in the statement of Theorem 2.5. Taking this into account it is clear from the above expression that the proof of Theorem A from Theorem 2.5 gives as particular case, taking T0 ≡ 0 and n = (0, 0), the proof of the present result. That being said, since it is only a matter of adapting the notation in the obvious way, we do not include it for the sake of shortness. Proof of Theorem C. Suppose that `µ̂0 < +∞, otherwise there is nothing to be proved. To show the result we adapt the derivation-division algorithm used by R. Roussarie to bound the cyclicity of an unfolding of a saddle loop (see Theorem 19 in page 113 of [28]). Here for convenience we will use the derivation D := s∂s instead of the usual one. Let us take the given µ̂0 = (λ0 , µ0 ) and consider the case λ0 ∈ Q first. Suppose that λ0 = p/q with gcd(p, q) = 1 and denote α(λ) = p − λq for the sake of shortness. Note in particular that α(λ0 ) = 0. 24 In addition, when there is not risk of confusion, we simply write ω or ω(s; α) instead of ω(s; p − λq). In what follows the notation ω k + · · · will mean that after the sign + there is an unwritten polynomial a1 (λ)ω k−1 +. . .+ak (λ) with the coefficients ai (λ) continuous in a neighbourhood of λ0 . Taking this notation into account one can readily show the following properties:  (a) D(sa ω k ) = (a − kα)sa ω k − ksa ω k−1 = sa ω k−1 (a − kα)ω − k for any continuous function a(λ) in a neighbourhood of λ0 . In particular, D(ω) = −(1 + αω) = −s−α and D(sa ω k ) = sa (∗ω k + · · · ) in case that a(λ0 ) 6= 0, where here (and in what follows) we use the symbol ∗ to replace any continuous function of λ which is non-zero at λ = λ0 . (b) On account of the chain rule, D(R(ω)) = R0 (ω)D(ω) = −R0 (ω)s−α for any function R. Hence  k  k+r−1 ω + ··· −α (k − r)ω + ··· D = s . ωr + · · · (ω r + · · · )2 (c) If a(λ) is a continuous function with a(λ0 ) 6= 0 then, from (b),  k   k k+r−1  aω + ··· a ω + ··· −α (k − r)ω + ··· D s r =s a r +s ω + ··· ω + ··· (ω r + · · · )2  k  a ω + ··· (k − r)ω k+r−1 + · · · =s a r + (1 + αω) ω + ··· (ω r + · · · )2 (a + α(k − r))ω k+r + · · · ∗ω k+r + · · · = sa r 2 = sa r . (ω + · · · ) (ω + · · · )2 That being said, for any L > 0, by applying Theorem A we can write T (s; µ̂) = β00 + si1 +λj1 (β01 ω k1 +β11 ω k1 −1 + . . . + βk11 ) + . . . + sim +λjm (β0m ω km + β1m ω km −1 + . . . + βkmm ) + R(s; µ̂), where ki ∈ Z≥0 for i = 1, 2, . . . , m and R ∈ FL∞ (µ̂0 ). Here we do not have logarithmic term since the assumption n = (n1 , n2 ) 6= (0, 0) implies T0 = 0, see (2). Note on the other hand that the constant term β00 only appears if n1 n2 = 0 because otherwise (0, 0) ∈ / Λn . Furthermore (recall Remark 1.7) • 0 < i1 + λ0 j1 < i2 + λ0 j2 < · · · < im + λ0 jm 6 L. • The coefficients βrd = βrd (µ̂) are labelled according to the position of its corresponding monomial with respect to the order ≺λ0 , see Definition 1.6. More precisely, βrd is the coefficient of sid +λjd ω kd −r and  0  d < d0 id +λjd kd −r id0 +λjd0 kd0 −r s ω ≺λ0 s ω ⇔ or  d = d0 and r < r0 . In particular, if {ti }i∈N is the sequence as introduced in Definition 1.8 then it turns out that ti = βrd Pd−1 with i = r + d + 1 − η + m=1 km , where recall that η = 1 if n1 n2 = 0 and η = 0 if n1 n2 6= 0. Pb−1 • We fix a, b ∈ Z≥0 with a 6 kb to be the ones that `µ̂0 = a + b + 1 − η + m=1 km , see Definition 1.8. Then, and this is crucial, βab (µ̂0 ) = t`µ̂0 (µ̂0 ) 6= 0. 25 Define ξ0 (s; µ̂) := s−i1 −λj1 DTµ̂ (s) and note that Z0 (T 0 ( · ; µ̂), µ̂0 ) = Z0 (ξ0 , µ̂0 ). Due to id + λ0 jd 6= 0 for all d = 1, 2, . . . , m, by applying (a) we get ξ0 (s; µ̂) =β01 (∗ω k1 + . . .) + β11 (∗ω k1 −1 + · · · ) + . . . + ∗βk11 m X h i + s−i1 −λj1 sid +λjd β0d (∗ω kd + · · · ) + β1d (∗ω kd −1 + · · · ) + . . . + ∗βkdd + R0 (s; µ̂), d=2 where the remainder is given by R0 := s−i1 −λj1 D(R).With regard to the flatness properties of the remainder note first that D(R) = sR0 ∈ FL∞ (µ̂0 ) by applying (f ) and (g) in Lemma A.2. On the other hand, setting L1 := i1 + λ0 j1 , by applying (c) in Lemma A.3 we get s−i1 −λj1 ∈ F−L ∞ 1 −ε (µ̂0 ) for some ε > 0 (that we fix from now on). Accordingly (g) in Lemma A.2 shows that R0 = s −i1 −λj1 D(R) ∈ FL−L ∞ 1 −ε (µ̂0 ). From this point on we will make ` = `µ̂0 steps of a derivation-division algorithm to construct a sequence of functions ξ1 , ξ2 , . . . , ξ` such that the last one is locally non-zero and the bound for Z0 (ξ0 , µ̂0 ) will follow from a recurrent application of Rolle’s Theorem. We gather these steps in several stages. In the first stage we eliminate, one by one and in this order, the coefficients β01 , β11 , . . . , βk11 , in the second stage we get rid of the coefficients β02 , β12 , . . . , βk22 , and so on until the last stage, in which we remove β0b , β1b , . . . , βa−1 b . Since all the steps in each stage are exactly the same, we only explain in detail the first and last stages for the sake of shortness. Certainly the flatness of the remainder will decrease in each step but this will not be a problem because, thanks to Theorem A, the coefficients βji are independent of L and we can take this number arbitrarily large. This will guarantee that the remainder in the last step is flat enough for our purposes. However, but only in the first steps of the algorithm, we will pay attention on the flatness of the remainder for reader’s convenience. First stage: Let u1 (s; λ) = ∗ω k1 + · · · be the function multiplying the coefficient β01 in ξ0 . This function does not vanish in a neighbourhood of (0, µ̂0 ) because u1 (s; λ)/ω k1 tends to some non-zero value as (s, λ) → (0, λ0 ) since lim(s,λ)→(0,λ0 ) ω(s;α) 1 = 0 by (a) in Lemma A.3. Consequently if we define ξ1 := sα (u1 )2 D(ξ0 /u1 ) then, by Rolle’s Theorem, Z0 (ξ0 , µ̂0 ) 6 Z0 (ξ1 , µ̂0 ) + 1. Due to id + λ0 jd − (i1 + λ0 j1 ) 6= 0 for d = 2, 3, . . . , m, taking properties (b) and (c) into account we get ξ1 =sα (u1 )2 D(ξ0 /u1 ) = β11 (∗ω 2(k1 −1) + · · · ) + . . . + βk11 (∗ω k1 −1 + · · · ) m X h i −i1 −λj1 +α +s sid +λjd β0d (∗ω k1 +kd + · · · ) + β1d (∗ω k1 +kd −1 + · · · ) + . . . + βkdd (∗ω k1 + · · · ) d=2 + sα (D(R0 )u1 − R0 D(u1 )). Moreover, since D(ω) = −s−α , the remainder can be written as R1 := sα (D(R0 )u1 − R0 D(u1 )) = sα D(R0 )(∗ω k1 + · · · ) + R0 (∗ω k1 −1 + · · · ). Note that the functions sα , ∗ω k1 + · · · and ∗ω k1 −1 + · · · belong to F−ε ∞ (µ̂0 ) thanks to (c) in Lemma A.3 and, consequently, R1 ∈ FL−L ∞ 1 −3ε (µ̂0 ) by applying Lemma A.2. Exactly for the same reasons as before, if u2 = ∗ω 2(k1 −1) + · · · is the function multiplying the coefficient β1 in ξ1 then Z0 (ξ1 , µ̂0 ) 6 Z0 (ξ2 , µ̂0 ) + 1 1 with ξ2 :=sα (u2 )2 D(ξ1 /u2 ) = β21 (∗ω 4k1 −6 + · · · ) + . . . + βk11 (∗ω 3k1 −4 + · · · ) m X h i + s−i1 −λj1 +2α sid +λjd β0d (∗ω 3k1 +kd −2 + · · · ) + . . . + βkdd (∗ω 3k1 −2 + · · · ) + R2 d=2 26 and where R2 := sα (D(R1 )u2 − R1 D(u2 )) ∈ FL−L ∞ 1 −5ε (µ̂0 ). We get in this way a sequence of functions ξ1 , ξ2 , . . . , ξk1 +1 such that Z0 (ξi−1 , µ̂0 ) 6 Z0 (ξi , µ̂0 ) + 1 and where m X h i ξk1 +1 := s−i1 −λj1 +(k1 +1)α sid +λjd β0d (∗ω r+kd + · · · ) + . . . + βkdd (∗ω r + · · · ) + Rk1 +1 d=2 with Rk1 +1 ∈ ∞ FL−L1 −(2k1 +3)ε Here r is a natural number depending on k1 that we do not specify (µ̂0 ). because its expression is not relevant. Thus, at the end of the first stage we removed the first k1 +1 coefficients of the asymptotic development of ξ0 and we get ξk1 +1 such that Z0 (ξ0 , µ̂0 ) 6 Z0 (ξk1 +1 , µ̂0 ) + k1 + 1. Next, in the second stage, we begin with the function si1 +λj1 −(i2 +λj2 )−(k1 +1)α ξk1 +1 =β02 (∗ω r+k2 + · · · ) + . . . + βk22 (∗ω r + · · · ) m X h i + s−i2 −λj2 sid +λjd β0d (∗ω r+kd + · · · ) + . . . + βkdd (∗ω r + · · · ) d=3 i1 +λj1 −(i2 +λj2 )−(k1 +1)α +s Rk1 +1 . The application of Lemmas A.2 and A.3 shows that the remainder belongs to FL−L ∞ 2 −2(k1 +2)ε (µ̂0 ) where L2 := i2 + λ0 j2 since s i1 +λj1 −(i2 +λj2 )−(k1 +1)α ∈ FL1 −L2 −ε (µ̂0 ). We eliminate, one by one and in this order, the coefficients β02 , β12 , . . . , βk22 following verbatim the steps that we carried out in the first stage. (It is in this stage that we use the inequalities id + λ0 jd 6= i2 + λ0 j2 for d = 3, 4, . . . , m.) In doing so we get a sequence of functions ξk1 +2 , ξk1 +3 , . . . , ξk1 +k2 +2 such that, exactly as before, Z0 (ξi−1 , µ̂0 ) 6 Z0 (ξi , µ̂0 ) + 1. Last stage: Pb−1 In the final stage, since n=1 (kn + 1) = ` − a by construction, we begin with ξ`−a :=β0b (∗ω r+kb + · · · ) + . . . + βab (∗ω r+kb −a + · · · ) + . . . + βkbb (∗ω r + · · · ) m X h i + s−ib −λjb sid +λjd β0d (∗ω r+kd + · · · ) + . . . + βkdd (∗ω r + · · · ) + R`−a , d=b+1 where r is now a natural number that depends on k1 , k2 , . . . , kb−1 . Consequently, with a additional steps of the derivation-division algorithm as we did to ξ0 in the first stage, by using properties (b) and (c) we get ξ` :=βab (∗ω r̂+kb −a + · · · ) + βa+1 b (∗ω r̂+kb −a−1 + · · · ) + . . . + βkbb (∗ω r̂ + · · · ) m X h i + s−ib −λjb +aα sid +λjd β0d (∗ω r̂+kd +a + · · · ) + . . . + βkdd (∗ω r̂+a + · · · ) + R` , d=b+1 where r̂ is once again a natural number and, taking L > 0 large enough, we can guarantee that R` ∈ F1∞ (µ̂0 ). Finally, after dividing by the function multiplying the coefficient βab , ξ` ∗ω r̂+kb −a−1 + · · · ∗ω r̂ + · · · =βab + βa+1 b + . . . + β b k ∗ω r̂+kb −a + · · · ∗ω r̂+kb −a + · · · b ∗ω r̂+kb −a + · · · m X  r̂+kd +a  d ∗ω + ··· ∗ω r̂+a + · · · R` +s−ib −λjb +aα sid +λjd β0 r̂+k −a d + . . . + βkd r̂+k −a + r̂+k −a + · · · . ∗ω b + ··· ∗ω b + ··· ∗ω b d=b+1 Accordingly, on account of id + λjd − ib − λjb + aα |λ=λ0 = id +λ0 jd −(ib +λ0 jb ) > 0 for all d = b+1, . . . , m, by applying (a) in Lemma A.3 with ν = (0, 0) and thanks to the flatness of the remainder, from the above expression we can conclude that ξ` lim = βab (µ̂0 ) 6= 0. (s,µ̂)→(0,µ̂0 ) ∗ω r̂+kb −a + · · · 27 Hence Z0 (ξ` , µ̂0 ) = 0. Consequently, due to Z0 (ξi−1 , µ̂0 ) 6 Z0 (ξi , µ̂0 ) + 1 for all i, this implies that Z0 (T 0 ( · ; µ̂), µ̂0 ) = Z0 (ξ0 , µ̂0 ) 6 `, as desired. Consider finally the case λ0 ∈ / Q. This is an easier situation than the previous one because, by Theorem A, we know that T (s; µ̂) = t1 + t2 si1 +λj1 + t3 si2 +λj2 + . . . + tm sim +λjm + R(s; µ̂), where, once again, R ∈ FL∞ (µ̂0 ) and, on the other hand, 0 < i1 + λ0 j1 < i2 + λ0 j2 < · · · < im + λ0 jm 6 L. Note then that we can treat this case using the previous approach particularised with k1 = . . . = km = 0. We obtain in this way the desired bound Z0 (T 0 ( · ; µ̂), µ̂0 ) 6 `. This concludes the proof of the result. Remark 3.5. It is clear from its proof that Theorem C is valid if {T (s; µ̂)}µ̂∈Ŵ is any family of functions in Cs>0 ∞ (Ŵ ) verifying the conclusion of Theorem A.  4 Applications Theorem C establishes an upper bound for Z0 (T 0 ( · ; µ̂), µ̂0 ). It will be also convenient to have some tool in order to ensure a lower bound. We begin this section with a result that is addressed to this issue. It is in fact an adaptation of a well-known technique used to study the bifurcation of zeros (see [2, Theorem 2.1] or [9, Lemma 15] for instance). Definition 4.1. Consider the functions gi : Ŵ → R for i = 1, 2, . . . , k. The real variety V (g1 , g2 , . . . , gk ) is defined to be the set of µ̂ ∈ Ŵ such that gi (µ̂) = 0 for i = 1, 2, . . . , k. We say that g1 , g2 , . . . , gk are independent at µ̂? ∈ V (g1 , g2 , . . . , gk ) if the following conditions are satisfied: (1) Every neighbourhood of µ̂? contains two points µ̂1 , µ̂2 ∈ V (g1 , . . . , gk−1 ) such that gk (µ̂1 )gk (µ̂2 ) < 0 (if k = 1 then we set V (g1 , . . . , gk−1 ) = V (0) = Ŵ for this to hold). (2) The varieties V (g1 , . . . , gi ), 2 6 i 6 k − 1, are such that if µ̂0 ∈ V (g1 , . . . , gi ) and gi+1 (µ̂0 ) 6= 0, then every neighbourhood of µ̂0 contains a point µ̂ ∈ V (g1 , . . . , gi−1 ) such that gi (µ̂)gi+1 (µ̂0 ) < 0. (3) If µ̂0 ∈ V (g1 ) and g2 (µ̂0 ) 6= 0, then every open neighbourhood of µ̂0 contains a point µ̂ such that g1 (µ̂)g2 (µ̂0 ) < 0. It is clear that if gi ∈ C 1 (Ŵ ) for i = 1, 2, . . . , k then a sufficient condition for g1 , g2 , . . . , gk to be independent at µ̂? is that the gradients ∇g1 (µ̂? ), ∇g2 (µ̂? ) . . . , ∇gk (µ̂? ) are linearly independent vectors of RN +1 .  Pn Proposition 4.2. Consider F (s; µ̂) = i=1 δi (µ̂)fi (s; µ̂) + fn+1 (s; µ̂), where fi ∈ C ((0, ε) × Ŵ ) and ∞ δi ∈ C 0 (Ŵ ). If µ̂? ∈ V (δ1 , δ2 , . . . , δn ) satisfies (a) F (s; µ̂? ) is not identically zero on (0, ρ) for every ρ ∈ (0, ε), (b) fi (s; µ̂) > 0, 1 6 i 6 n, for all (s, µ̂) in a neighbourhood of (0, µ̂? ), fi+1 (s;µ̂) (c) lims→0 fi (s;µ̂) = 0, 1 6 i 6 n, for every µ̂ in a neighbourhood of µ̂? , and (d) δ1 , δ2 , . . . , δn are independent at µ? , then Z0 (F ( · ; µ̂), µ̂? ) > n. 28 Proof. Fix any ρ > 0 and any neighbourhood U of µ̂? . Then, by the assumption (a), there exists s1 ∈ (0, ρ) such that F (s1 ; µ̂? ) = fn+1 (s1 ; µ̂? ) 6= 0. Suppose for instance that F (s1 ; µ̂? ) > 0. Then, on account of (1) in Definition 4.1, we can take µ̂1 ∈ U ∩ V (δ1 , δ2 , . . . , δn−1 ) such that δn (µ̂1 ) < 0 and close enough to µ̂? so that, by continuity, F (s1 ; µ̂1 ) > 0. Observe that F (s; µ̂1 ) = δn (µ̂1 )fn (s; µ̂1 ) + fn+1 (s; µ̂1 ). Thus, by (b) and (c), lims→0 fFn(s;µ̂ 1) (s;µ̂1 ) = δn (µ̂1 ) < 0 and we can take s2 ∈ (0, s1 ) such that F (s2 ; µ̂1 ) < 0. Next, thanks to (2) in Definition 4.1, we can choose µ̂2 ∈ U ∩ V (δ1 , δ2 , . . . , δn−2 ) with δn−1 (µ̂2 ) > 0 and close enough to µ̂1 so that F (s1 ; µ̂2 ) > 0 and F (s2 ; µ̂2 ) < 0. Note that F (s; µ̂2 ) = δn−1 (µ̂2 )fn−1 (s; µ̂2 ) + δn (µ̂2 )fn (s; µ̂2 ) + fn+1 (s; µ̂2 ). F (s;µ̂2 ) Consequently, by (b) and (c), lims→0 fn−1 (s;µ̂2 ) = δn−1 (µ̂2 ) > 0 and we can choose s3 ∈ (0, s2 ) such that F (s3 ; µ̂2 ) > 0. Next we take µ̂3 ∈ U ∩ V (δ1 , δ2 , . . . , δn−3 ) with δn−2 (µ̂3 ) < 0 and close enough to µ̂2 so that F (s1 ; µ̂3 ) > 0, F (s2 ; µ̂3 ) < 0 and F (s3 ; µ̂3 ) > 0. We repeat this process n − 2 times after which we find a parameter µ̂n+1 ∈ U and 0 < sn+1 < sn < . . . < s2 < s1 < ρ, such that (−1)i+1 F (si ; µ̂n+1 ) > 0 for all i = 1, 2, . . . , n + 1. By applying Bolzano’s theorem we can assert the existence of at least n zeros of F ( · ; µ̂n+1 ) inside the interval (0, δ). Accordingly Z0 (F ( · ; µ̂), µ̂? ) > n and this concludes the proof. We prove next an auxiliary result that enables to straighten globally the separatrices of a saddle de- pending on parameters. This result, which is well-known to be true locally, is relevant with regard to the applicability of Theorems A and B. It will be essential, for instance, in the proof of Corollaries D and E, in which we do not have any assumption regarding the separatrices of the saddles. Lemma 4.3. Consider a C ∞ family {Xν }ν∈RN of planar vector fields defined in some open set W of R2 . Let us fix some ν0 ∈ RN and assume that, for all ν ≈ ν0 , Xν has a hyperbolic saddle point at pν ∈ W with (global) stable and unstable separatrices Sν+ and Sν− , respectively. Consider two closed connected arcs `± ⊂ Sν±0 , having both an endpoint at pν0 . In case of a homoclinic connection (i.e., Sν+0 = Sν−0 ) we require additionally that `+ ∩ `− = {pν0 }. Then there exists a neighborhood V of (`+ ∪ `− ) × {ν0 } in R2 × RN and a C ∞ diffeomorphism Φ : V → Φ(V ) ⊂ R2 × RN with Φ(x, y, ν) = (φν (x, y), ν) such that Φ((Sν+ × {ν}) ∩ V ) ⊂ {x = 0} × {ν} and Φ((Sν− × {ν}) ∩ V ) ⊂ {y = 0} × {ν}. In other words, (φν )? (Xν ) = X̂ν where X̂ν (x, y) = xP (x, y; ν)∂x + yQ(x, y; ν)∂y , with P, Q ∈ C ∞ (Φ(V )). Proof. The existence of such a diffeomorphism in a neighbourhood U of (pν0 , ν0 ) ∈ RN +2 is well-known (see for instance [9, p. 11] or [28, p. 92]). The proof of this local result is based on the existence and smoothness of the center-stable and center-unstable manifolds (see [8, 11]) for the system of differential equations in R2 × RN obtained by adding the equation ν̇ = 0 to (ẋ, ẏ) = Xν (x, y), and the fact that (in this context) these manifolds are unique (see [23, p. 165]). Let ΦL : U → ΦL (U ) ⊂ RN +2 be this diffeomorphism that straightens locally the separatrices Sν± . We also denote by X the vector field associated to the above- mentioned system in RN +2 and by ϕ its flow. Furthermore let ψ be the flow of the “straightened” vector field (ΦL )? (X) that leaves the coordinate planes invariant. The idea is to extend ΦL : U → RN +2 taking advantage of the fact that small enough neighbourhoods of ` × {ν0 } and `− × {ν0 } will be mapped by ϕ (in forward and backward time, respectively) inside U . With + this aim in view we take δ > 0 and consider the open set Bν ⊂ R2 in Figure 5. The boundary of Bν consists in four (pieces of) trajectories of the flow ψν together with four segments (inside x = ±δ and y = ±δ) where the straightened vector field is transversal. We define B := {(p, ν) ∈ RN +2 : p ∈ Bν , |ν − ν0 | < δ} ⊂ ΦL (U ), which is an open neighbourhood of (0, 0, ν0 ) by the continuous dependence with respect to initial conditions. 29 W+ T+ p+ ΦL `+ δ M+ ψ L+ ϕ pν0 L− −δ δ T− M− p− `− ΦL (U ) −δ U W− E Bν0 Figure 5: Slice ν = ν0 of the sets involved in the construction of V = M+ ∪ E ∪ M− . Next we take a C ∞ bump function ρ : RN +2 → R such that ρ|B̄ ≡ 1 and ρ|Ac ≡ 0, where A is an open set with B̄ ⊂ A ⊂ ΦL (U ), and define Y := ρ (ΦL )? (X), which is a complete vector field in RN +2 that leaves invariant ν = constant. By abuse of notation we also refer to the flow of this new vector field by ψ. We define E := (ΦL )−1 (B), which is an open neighbourhood of (pν0 , ν0 ) ∈ RN +2 . Since Y |B = (ΦL )? (X), we have that ΦL is a conjugacy between the flows of X|E and Y |B , that we shall denote by ϕE and ψB , respectively. We take open neighbourhoods L± of `± × {ν0 } in RN +2 , see Figure 5, such that L+ ∩ L− ⊂ E, L+ ∩ ∂E ⊂ (ΦL )−1 (B ∩ {y = δ}) and L− ∩ ∂E ⊂ (ΦL )−1 (B ∩ {x = δ}). Similarly we take open neighbourhoods T + and T − of {x = 0, y > 0} × {ν0 } and {y = 0, x > 0} × {ν0 } in RN +2 , respectively, such that T + ∩ T − ⊂ B, T + ∩ ∂B ⊂ {y = δ} and T − ∩ ∂B ⊂ {x = δ}. Hence, in particular, L± ∩ ∂E is a transversal section for X and T ± ∩ ∂B is a transversal section for Y. This will be a key point in the forthcoming reasoning. Let us denote the endpoint of `± not being pν0 by p± and take τ− < 0 < τ+ such that ϕ(τ± , (p± , ν0 )) ∈ E. By the continuous dependence with respect to initial conditions there exists a neighbourhood W± of (p± , ν0 ) such that (a) ϕ(τ± , W± ) ⊂ E. (b) ϕ(< 0, 2τ± >, W± ) ⊂ L± . (c) M± := ϕ(< 0, τ± >, W± ) are disjoint.   (d) ψ < 0, −2τ± >, ΦL ϕ(< 0, 2τ± >, W± ) ∩ E ⊂ T ± . 30 (Here < a, b > stands for the smallest closed interval containing a, b ∈ R.) For a given z ∈ V := M+ ∪ E ∪ M− ⊂ RN +2 we define  Φ(z) := ψ − tz , ΦL (ϕ(tz , z) , where tz is any t ∈< 0, τ± > such that ϕ(t, z) ∈ E for z ∈ M± , whereas tz = 0 for z ∈ E. We must prove that the function Φ : V → RN +2 is well defined, smooth and injective. Let us show first that Φ is well defined. To this end we remark that M± ∩ E 6= ∅. Let us fix for instance any z ∈ M+ and suppose that ϕ(t1 , z) =: w1 ∈ E and ϕ(t2 , z) =: w2 ∈ E with t2 > t1 > 0. Then we must show that ψ(−t1 , ΦL (w1 )) = ψ(−t2 , ΦL (w2 )). Note that w2 = ϕ(t2 − t1 , w1 ). Moreover, thanks to the inclusion in (b), the definition of M+ in (c) and the transversality of X at L+ ∩ ∂E, it turns out that ϕ(t, w1 ) ∈ E for all t ∈ [0, t2 − t1 ]. Likewise, due to (d) and the transversality of Y at T + ∩ ∂B, it follows that ψ(−t, ΦL (ϕ(t, w1 )) ∈ B for all t ∈ [0, t2 − t1 ]. Accordingly, using also that ΦL is a conjugacy between ϕE and ψB ,    ψ t1 − t2 , ΦL (w2 ) = ψ t1 − t2 , ΦL (ϕ(t2 − t1 , w1 ) = ψB t1 − t2 , ΦL (ϕE (t2 − t1 , w1 ) = ΦL (ϕE (t1 − t2 , ϕE (t2 − t1 , w1 )) = ΦL (w1 ), that implies ψ(−t1 , ΦL (w1 )) = ψ(−t2 , ΦL (w2 )), as desired. Let us turn next to the smoothness of Φ. To this end we observe that we can take tz = τ± for any z ∈ M± , which implies that Φ is smooth on M+ ∪ M− because ϕ(τ± , · ), ψ(−τ± , · ) and ΦL are smooth. The smoothness on E is clear since Φ|E = ΦL by definition. With regard to the injectivity we take z1 , z2 ∈ V with z1 6= z2 and we claim that then Φ(z1 ) 6= Φ(z2 ). There are four cases to consider: 1. z1 , z2 ∈ E. The claim is obvious in this case because Φ|E = ΦL . 2. z1 , z2 ∈ M± . Since we can take tz1 = tz2 = τ± , the claim follows from the injectivity of ϕ(τ± , · ), ψ(−τ± , · ) and ΦL . 3. z1 ∈ M± \ E and z2 ∈ E. Due to Φ(z2 ) ∈ B, it suffices to show that Φ(M± \ E) ∩ B = ∅. (34) We take z ∈ M± \ E, so that ϕ(tz , z) ∈ L± ∩ E with tz ∈< 0, τ± > and w := ΦL (ϕ(tz , z)) ∈ T ± ∩ B. By contradiction, if Φ(z) = ψ(−tz , w) ∈ B then, due to (d) and the transversality of Y at T ± ∩ ∂B, the compact set K = {ψ(−t, w) : t ∈< 0, tz >} is inside B. However C = {ϕ(t, z) : t ∈< 0, tz >} intersects ∂E and it is easy to verify that Φ(C) ⊂ K, which contradicts Φ(∂E) ⊂ ∂B. 4. z1 ∈ M+ \ E and z2 ∈ M− \ E. We have Φ(M± ) ⊂ T ± and thus, from (34), Φ(M± \ E) ⊂ T ± \ B. On account of T + ∩ T − ⊂ B, the claim follows in this case as well. Finally we remark that, by construction, if z = (p, ν) ∈ R2 × RN then Φ(z) = (φν (p), ν). This completes the proof. Proof of Corollary D. Consider the given ν0 ∈ U ⊂ RN and let us fix that the outer boundary Πν has k hyperbolic saddles, p1ν , p2ν , . . . , pkν , that for convenience we label according to the sense of the flow with the first one not being in `∞ . (Here we use the assumption that Πν is a persistent polycycle, see Definition 1.11). Denote the hyperbolicity ratio of piν by λi (ν) and set λi0 := λi (ν0 ). Let us suppose first that k > 2. (We postpone the case k = 1 because the proof is slightly different.) 31 p2ν u V3 p3ν v p3ν Σ2 Σ3 V2 p4ν `∞ p1ν V4 p2ν p5ν p4ν y Σ1 x Σ4 V1 p5ν V5 Σ5 p1ν `∞ Figure 6: Following the proof of Corollary D, polycycle Πν for k = 5 in the Poincaré disc (left) and a ν-slice of the sections Σi and sets Vi (right). Since by assumption the infinite line `∞ is invariant for the flow of p(Xν ), if piν ∈ `∞ then exactly one of its separatrices is contained in `∞ . (This easily implies that Πν has an even number of singularities at infinity, assembled in pairs, see Figure 6.) For each i = 1, 2, . . . , k we place a transversal section Σi in the heteroclinic connection between piν and pi+1 ν . In addition we denote the Dulac map and the Dulac time of Xν from Σi−1 to Σi by Di ( · ; ν) and Ti ( · ; ν), respectively. Thus, setting D̂i := Di ◦ Di−1 ◦ · · · ◦ D1 , the return time from Σ0 := Σk to Σ0 is given by   T (s) = T1 (s) + T2 ◦ D̂1 (s) + . . . + Tk ◦ D̂k−1 (s). (35) In order to study each Dulac time it is more convenient to compactify the vector field Xν by means of a projective change of coordinates rather than to use the Poincaré compactification p(Xν ). To this end we note that, since the center is not global, there exists a straight line `a := {α1 x + α2 y = β} not intersecting Pν for all ν ≈ ν0 and such that piν ∈ / `a ∩ `∞ for all i = 1, 2, . . . , k. Without lost of generality, by means of a rotation and a translation, we can assume that `a = {y = 0}. That being said we perform the projective change of coordinates {u = xy , v = y1 }, which brings Xν (x, y) to 1   X̂ν (u, v) = P̂ (u, v; ν)∂u + v Q̂(u, v; ν)∂v . v d−1 Here d > 2 is the degree of Xν , whereas P̂ and Q̂ are polynomials with the coefficients depending C ∞ on ν. (Let us remark that Xν and X̂ν are conjugated, which is essential to study the return time, whereas Xν and p(Xν ) are only equivalent.) Note that in doing so the infinite line is mapped to {v = 0}. On account of this, by applying Lemma 4.3, for each i = 1, 2, . . . , k there exists a neighbourhood Vi of (piν0 , ν0 ) and a C ∞ diffeomorphism Φi : Vi → Φi (Vk ) with Φi (x, y, ν) = (φiν (x, y), ν) such that (φiν )? (X̂ν |Vi ) = X̄νi , where ( 1  d − 1 if piν ∈ `∞ , i X̄ν (u, v) = κi uP̄i (u, v; ν)∂u + v Q̄i (u, v; ν)∂y with κi = v 0 if piν ∈ / `∞ , and P̄i , Q̄i ∈ C ∞ (Φi (Vi )). We choose ε > 0 small enough so that 2ε < min(1, λi0 ) for all i = 1, 2, . . . , k, which in particular implies that Bλ0 i ,2ε = {(0, 0)}. Then, for each i = 1, 2, . . . , k, 0 32 • we apply Theorem B to the family {X̄νi , ν ≈ ν0 } with L = λi0 + 2ε to conclude that the Dulac map Di of the hyperbolic saddle piν from Σi−1 to Σi can be written as  Di (s; ν) = ∆i00 sλi (ν) + Fλ∞i +2ε (ν0 ) = sλi (ν) ∆i00 + Fε∞ (ν0 ) , 0 where ∆i00 (ν) > 0. (In the second equality above we use Lemmas A.2 and A.3.) Note in particular that Di ∈ Fλ∞i −ε (ν0 ) ⊂ Fε∞ (ν0 ) due to λi0 > 2ε. 0 • we apply Theorem A with L = ε and n = (n1 , n2 ) = (0, κi ) to show that the Dulac time Ti from Σi−1 to Σi writes as Ti (s; ν) = T0i log s + T00 i + Fε∞ (ν0 ), (36) where T0i (ν) = −1 P̄i (0,0;ν) / `∞ and, due to d > 2, T0i (ν) ≡ 0 if piν ∈ `∞ . > 0 if piν ∈ Since D1 ∈ Fε∞ (ν0 ), an easy application of Lemma A.2 shows that λ2  D̂2 (s) = sλ1 (∆100 + Fε∞ (ν0 ) ∆200 + Fε∞ 2 1 λ2 λ1 λ2 2 (ν0 ) = ∆00 (∆00 ) s (1 + Fε∞ (ν0 ))λ2 (1 + Fε∞ 2 (ν0 ))  = ∆200 (∆100 )λ2 sλ1 λ2 (1 + Fε∞ (ν0 ))(1 + Fε∞ 2 (ν0 )) = s λ1 λ2 ∆200 (∆100 )λ2 + Fε∞ 2 (ν0 ) , where in the third equality we use that the map (s, ν) 7→ (1 + s)λ2 (ν) − 1 belongs to F1∞ (ν0 ). Similarly, ˆ i00 + F ∞ D̂i (s) = sλ1 ···λi (∆ ˆi εi (ν0 )) for some ∆00 > 0, so that D̂i ∈ Fε∞ i (ν0 ). Hence, by applying Lemma A.2 again, from (36) we get that if i ∈ {1, 2, . . . , k} then  Qi−1 ˆ i + F∞ εi+1 (ν0 ) with T̂i := T0 `=1 λ` > 0 i Ti ◦ D̂i−1 (s) = T̂i log s + T00 + T0i log ∆ 00 i and where we set D̂0 = Id for the sake of convenience. This equality, together with (35), shows that Pk k+1 (ν0 ) with T̄0 := T (s; ν) = T̄0 (ν) log s + T̄00 (ν) + Fε∞ i=1 T̂i . Observe that, and this is the key point, T̄0 (ν) > T̂1 (ν) = T01 (ν) > 0 due to p1ν ∈ / `∞ by construction. Hence, from (f ) and (g) in Lemma A.2, s∂s T (s; ν) = T̄0 (ν) + sFε∞ ∞ k+1 −1 (ν0 ) = T̄0 (ν) + Fεk+1 (ν0 ) → T̄0 (ν0 ) 6= 0 as (s, ν) → (0, ν0 ). Therefore we can assert the existence of some δ > 0 such that if s ∈ (0, δ) and kν − ν0 k 6 δ then ∂s T (s; ν) 6= 0. Consequently Z0 (T 0 ( · ; ν), ν0 ) = 0 and we claim that this implies Crit (Πν0 , Xν0 ), Xν = 0. By contradiction, if the criticality is not zero then there exists a sequence {γνi }i∈N , where each γνi is a critical periodic orbit of Xνi , such that νi → ν0 and dH (γνi , Πν0 ) → 0 as i → +∞. Then, since Πν varies continuously at ν0 and dH (γνi , Πνi ) 6 dH (γνi , Πν0 ) + dH (Πνi , Πν0 ), we have dH (γνi , Πνi ) → 0 as i → +∞, which contradicts that Z0 (T 0 ( · ; ν), ν0 ) = 0. (Let us remark that this last implication is not true without the assumption that Πν varies continuously at ν0 because the parametrization τ (s; ν) of the transversal section Σ0 is taken such that τ (0; ν) ∈ Πν for all ν.) This proves the claim and so the result for the case k > 2 is true. Let us consider finally the case k = 1, i.e., assume that Πν is a (finite) saddle loop. In this case we place two transversal sections Σ1 and Σ2 in the heteroclinic connection. By applying Lemma 4.3 and Theorem A exactly as we did before we can assert that the Dulac time from Σ1 to Σ2 is TD (s; ν) = T0 log s+T00 +Fε∞ (ν0 ) with T0 = T0 (ν) > 0. Then the return time from Σ1 to Σ1 is given by T = TD + TR , where TR (s; ν) is the regular time of −Xν from Σ1 to Σ2 , which is a C ∞ function in a neighbourhood of (s, ν) = (0, ν0 ) by Lemma 2.4. Hence s∂s T (s; ν) = T0 (ν) + Fε∞ (ν0 ) + s∂s TR (s; ν) and lim(s,ν)→(0,ν0 ) s∂s T (s; ν) = T0 (ν0 ) 6= 0. Then, exactly as before, this shows the validity of the result for k = 1 as well and completes the proof. 33 u p2ν v Σ2 V2 qν p1ν y Σ1 V1 x Figure 7: Following the proof of Corollary E, phase portrait of Xν in the Poincaré disc with the shape of the period annulus of the center at qν and the placement of the transversal sections in the outer boundary Πν . (To be more precise V1 and V2 are in fact open sets in RP2 × U.) Proof of Corollary E. Note first of all that we can assume that both hyperbolic saddles are at the infinite line `∞ , because otherwise the result follows by applying Corollary D. For each ν ∈ U, let Pν be the period annulus of the center qν and denote the two saddles at its outer boundary Πν by p1ν and p2ν . (Here we use the assumption that Πν is a persistent polycycle, see Definition 1.11). Let us also denote the respective hyperbolicity ratios by λ1 (ν) and λ2 (ν). It is well known that λ1 (ν)λ2 (ν) = 1 for all ν ∈ U , otherwise the return map defined near the polycycle Πν cannot be the identity. Moreover the center is not global since the infinite line `∞ is invariant for the flow of p(Xν ) by hypothesis and the two singularities in Πν are hyperbolic saddles. Thus there exists a straight line `a := {α1 x + α2 y = β} not intersecting Pν for all ν ≈ ν0 and such that pkν ∈ / `a ∩ `∞ for k = 1, 2. Without lost of generality, by means of a rotation and a translation, we can assume that `a = {y = 0}, so that the shape of the period annulus Pν in the Poincaré disc is as we draw in Figure 7. We place a transversal section Σ1 in the finite heteroclinic connection between the saddles and denote by TR ( · ; ν) the corresponding return time for Xν . Note that TR is a parametrization of the period function of the center qν near the outer boundary of Pν . In order to study TR we take an auxiliary transversal section Σ2 in `∞ so that T = T1 + T2 , where T1 is the Dulac time of Xν from Σ1 to Σ2 and T2 is the Dulac time of −Xν from Σ1 to Σ2 . Our goal is to apply Theorem A to obtain the asymptotic development of T1 and T2 . With this aim in view we perform firstly the projective change of coordinates {u = xy , v = y1 }, that brings Xν (x, y) to 1   X̂ν (u, v) = P̂ (u, v; ν)∂u + v Q̂(u, v; ν)∂v . v d−1 Here d > 2 is the degree of Xν , whereas P̂ and Q̂ are polynomials with the coefficients depending C ∞ on ν. By an abuse of notation we still denote the two hyperbolic saddles of X̂ν at v = 0 coming from the two vertices of Πν at `∞ by p1ν and p2ν . Secondly, by Lemma 4.3, for k = 1, 2 there exists a neighbourhood Vk of (pkν0 , ν0 ) and a C ∞ diffeomorphism Φk : Vk → Φk (Vk ) with Φk (x, y, ν) = (φkν (x, y), ν) such that (φkν )? (X̂ν |Vk ) = X̄νk , where 1  X̄νk (u, v) = d−1 uP̄k (u, v; ν)∂u + v Q̄k (u, v; ν)∂y v with P̄k , Q̄k ∈ C (Φk (Vk )). Note in addition that each X̄νk has a hyperbolic saddle at the origin with ∞ 34 hyperbolicity ratio λk (ν). It is important to observe at this point that T2 is the Dulac map of −X̄ν2 and that, for this vector field, the hyperbolicity ratio of the saddle is 1/λ2 (ν) = λ1 (ν). Therefore, and this is crucial, we will apply Theorem A to two different families of vector fields that have saddles with equal hyperbolicity ratio λ(ν) := λ1 (ν) = 1/λ2 (ν). In doing so for X̄ν1 , setting λ0 := λ(ν0 ) and taking n = (0, d−1), we have a well defined formal series  P   T 1 (ν)si+λ(ν)j if λ0 ∈ / Q,  (i,j)∈Λn ij 1 T̂1 (s; ν) := T0 (ν) log s + P    T 1ij ωα (s); ν si+λ(ν)j if λ0 ∈ Q,  (i,j)∈Λn which is asymptotic to T1 (s; ν) as (s, ν) → (0, ν0 ). Note that T01 ≡ 0 because n 6= (0, 0) due to d − 1 > 0 (otherwise the given vector field Xν would be linear). Similarly, by applying Theorem A to −X̄ν2 we have a well defined formal series  P   T 2 (ν)si+λ(ν)j if λ0 ∈ / Q,  (i,j)∈Λn ij T̂2 (s; ν) := T02 (ν) log s + P    T 2ij ωα (s); ν si+λ(ν)j if λ0 ∈ Q,  (i,j)∈Λn which is asymptotic to T2 (s; ν) as (s, ν) → (0, ν0 ) and where T02 ≡ 0 again. Note that if λ0 ∈ Q then X T kij (w; ν) = k Ti−rp,j+rq (ν)(1 + αw)r for k = 1, 2, n r∈Aijλ 0 where α(ν) = p − λ(ν)q and λ0 = p/q with gcd(p, q) = 1. Thus, setting Tij3 := Tij1 + Tij2 and T 3ij := T 1ij + T 2ij ,  P   Tij3 (ν)si+λ(ν)j if λ0 ∈ / Q,  (i,j)∈Λn T̂R (s; ν) := T̂1 (s; ν) + T̂2 (s; ν) = P    T 3ij ωα (s); ν si+λ(ν)j if λ0 ∈ Q,  (i,j)∈Λn is a well defined formal series that is asymptotic to TR (s; ν) = T1 (s; ν) + T2 (s; ν) as (s, ν) → (0, ν0 ) in the sense established by Theorem A. We use at this point the assumption that the center qν0 is non-isochronous. On account of this, by applying the result of Saavedra and Mardešić in [17] we can assert that the formal series T̂R (s; ν0 ) is not constant. This implies that `ν0 , computed as explained in Definition 1.8 with respect to the formal series T̂R (s; ν), is finite (i.e, `ν0 ∈ N). Then the application of Theorem C (see Remark 3.5) shows that Z0 (TR0 ( · ; ν), ν0 ) 6 `ν0 < +∞  and, since Πν varies continuously at ν0 , Crit (Πν0 , Xν0 ), Xν 6 `ν0 . This shows the validity of the result in case that the period annulus is unbounded and completes the proof because, as we already mentioned, the bounded case follows by Corollary D. We conclude this section by pointing out that, even in an unfolding of polynomial centers, the fact that the outer boundary of its period annulus is a hyperbolic polycycle varying continuously does not imply its persistence as required in Corollaries D and E. Indeed, let us consider the 1-parametric family of quadratic differential systems  ẋ = −y + xy, Xν ẏ = x + (ν − 2)x2 + 2y 2 , which has a center at the origin for all ν ∈ R. Figure 8 displays its phase portrait in the Poincaré disc for ν ≈ 0. (The reader is referred to [15] for the complete bifurcation diagram of the phase portrait of the 35 ν<0 ν=0 ν>0 Figure 8: Phase portrait of the unfolding {Xν ; ν ≈ 0} in the Poincaré disc, where the origin is shifted to the left for convenience. quadratic centers.) One can verify that the outer boundary Πν of its period annulus varies continuously at ν0 = 0. Following the notation in Definition 1.11, the hyperbolic polycycle Γ is the triangle with an edge at infinity and it occurs that both separatrix connections with the finite saddle are broken for ν 6= 0. The outer boundary becomes a saddle loop for ν < 0 and an unbounded 2-cycle for ν > 0. Our goal for further research is to develop tools to study the criticality of this type of unfolding. A Results about the class FLK (W ) In this appendix, for reader’s convenience, we collect some technical results from [19] about the class of functions FLK (W ) that we use in the present paper. Lemma A.1. Let U be an open set of RN , K ∈ Z≥0 and g(s; µ) ∈ Cs>0 K (U ) such that, for some W ⊂ U and L ∈ R, g(s; µ) ∈ FL (W ). If L > K then g extends to a C -function ĝ, defined in some open neighbourhood K K of {0} × W in RN +1 , and satisfying ∂ ν ĝ(0; µ) = 0 for all µ ∈ W and ν ∈ ZN +1 ≥0 with |ν| 6 K. 0 Lemma A.2. Let U and U 0 be open sets of RN and RN respectively and consider W ⊂ U and W 0 ⊂ U 0 . Then the following holds: T S (a) FLK (W ) ⊂ FLK (Ŵ ) for any Ŵ ⊂ W and n FLK (Wn ) = FLK ( n Wn ). (b) FLK (W ) ⊂ FLK (W × W 0 ). (c) C K (U ) ⊂ E K (U ) ⊂ F0K (W ). 0 (d) If K > K 0 and L > L0 then FLK (W ) ⊂ FLK0 (W ). (e) FLK (W ) is closed under addition. K−|ν| (f ) If f ∈ FLK (W ) and ν ∈ ZN ≥0 +1 with |ν| 6 K then ∂ ν f ∈ FL−ν0 (W ). (g) FLK (W ) · FLK0 (W ) ⊂ FL+L K 0 (W ). (h) Assume that φ : U 0 → U is a C K function with φ(W 0 ) ⊂ W and let us take g ∈ FLK0 (W 0 ) with L0 > 0 and verifying g(s; η) > 0 for all η ∈ W 0 and s > 0 small enough. Consider also any f ∈ FLK (W ). Then h(s; η) := f (g(s; η); φ(η)) is a well-defined function that belongs to FLL K 0 0 (W ). 36 Next result gathers some interesting properties of the Ecalle-Roussarie compensator. In the statement we use the notation x+ := max(x, 0) and x− := max(−x, 0) for, respectively, the positive and negative part of a given x ∈ R. Note in particular that then x = x+ − x− and |x| = x+ + x− . Lemma A.3. The following assertions hold: (a) For each compact set I ⊂ R and ν ∈ Z2≥0 there exists a constant C > 0 such that + |∂ ν ω(s; α)| 6 Cs−α −ν0 | log s||ν|+1 for all α ∈ I and s ∈ (0, 1/e). Moreover lims→0+ 1 ω(s;α) = α− uniformly on α ∈ R so that, in particular, lim(s,α)→(0+ ,0) 1 ω(s;α) = 0. 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