This is a preprint of: “Tips of tongues in the double standard family”, Kuntal Banerjee, Xavier Buff, Jordi Canela, Adam Epstein, Nonlinearity, vol. 34(12), 8174, 2021. DOI: [10.1088/1361-6544/ac2d80] TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY KUNTAL BANERJEE, XAVIER BUFF, JORDI CANELA, AND ADAM EPSTEIN Abstract. We answer a question raised by Misiurewicz and Rodrigues con- cerning the family of degree 2 circle maps Fλ : R/Z → R/Z defined by b Fλ (x) := 2x + a + sin(2πx) with λ := (a, b) ∈ R/Z × (0, 1). π We prove that if Fλ◦n − id has a zero of multiplicity 3 in R/Z, then there is a system of local coordinates (α, β) : W → R2 defined in a neighborhood W of λ, such that α(λ) = β(λ) = 0 and Fµ◦n − id has a multiple zero with µ ∈ W if and only if β 3 (µ) = α2 (µ). This shows that the tips of tongues are regular cusps. Introduction Following Misiurewicz and Rodrigues [MR07], we consider the family of double standard maps of the circle Fλ : R/Z → R/Z defined by b Fλ (x) := 2x + a + sin(2πx) with λ := (a, b) ∈ R/Z × [0, 1]. π If b ∈ [0, 1/2), then Fλ : R/Z → R/Z is expanding and all periodic cycles of Fλ in R/Z are repelling. If b ∈ [1/2, 1], it may happen that Fλ : R/Z → R/Z has a non-repelling cycle. The multiplier of such a cycle belongs to [0, 1]. There is at most one such cycle. Connected components of the open sets of parameters λ ∈ (a, b) ∈ R/Z × [0, 1] for which Fλ has an attracting cycle are called tongues (see [MR07] and [D10]). The period of the attracting cycle remains constant in each tongue, and is called the period of the tongue These tongues can be understood as an analogue to Arnold tongues of the stan- dard maps of the circle. The family of standard maps Aa,b : R/Z → R/Z is given by b Aa,b (x) := x + a + sin(2πx) with (a, b) ∈ R/Z × [0, 1] 2π and was introduced by Arnold in [A65] as a family of perturbations of rotation maps of the circle. If b ∈ [0, 1], each map Aa,b is a homeomorphism of the circle and has a rotation number associated to it. In this case, tongues are connected sets of parameters for which the corresponding map has a given rational rotation number p/q ∈ Q/Z. Notice that double standard maps do not have a rotation number associated to them since they are not homeomorphisms of the circle. Instead, The first author was supported by by FRPDF allotment 2018-19 of Presidency University. The second and third authors were supported by ANR grant Lambda ANR-13-BS01-0002. The third author was also supported by Spanish Ministry of Economy and Competitiveness, through the Marı́a de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), and by BGSMath Banco de Santander Postdoctoral 2017. 1 2 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN tongues in the double standard family can be classified in terms of the type of the corresponding attracting cycle (see [MR07]). Let T be a tongue of period p ≥ 1 of the double standard family. The boundary of T consists of two smooth curves which are graphs with respect to b and intersect tangentially at the tip λT ∈ R/Z × (0, 1) (see [MR07, MR08] and Figure 1). If λ ∈ ∂T , then Fλ has a cycle of period p and multiplier 1. On the one hand, if λ ∈ ∂T r{λT }, then the points of the cycle are double zeros of Fλ◦p − id. On the other hand, for the tip parameter λT the points of the cycle are triple zeros of Fλ◦pT − id. Moreover, there is a cusp bifurcation which takes place around λT (see [HK91] for an introduction to cusp bifurcations). 1 0.75 0.5 0 0.25 0.5 0.75 1 Figure 1. The tongues of the family Fλ . The horizontal axis corresponds to the parameter a and the vertical axis to b. We draw in black the boundary of the tongue of period 1. It arises as a relevant topic to understand the shape of a given tongue T near its tip λT . This can be studied in terms of the order of contact of its boundary curves. Let B1 (b) and B2 (b) be the parametrizations of the boundary curves of T with respect to b. If λT = (a0 , b0 ), then we say that the order of contact of the two curves at the tip λT is r if the limit |B1 (b) − B2 (b)| lim b→b0 |b − b0 |r+1 is positive and finite. In the case of the standard family, the order of contact of the boundary curves at the tip of Arnold tongues depends on the rotation number. If a tongue has rotation number p/q, with p and q coprime, the order of contact is q − 1 (see [A65]). However, this situation is very different for the double standard family due to the cusp bifurcation at the tip of its tongues. Misiurewicz and Rodrigues [MR07] proved that the order of contact of the boundary curves is 1/2 for the unique tongue of period 1. This is equivalent to saying that the cusp bifurcation which takes place around λT for the tongue of period 1 is generic (see [MR11]). In [MR08] they asked whether this property holds for all tongues of the family Fλ . In this article, we answer positively to this question. More precisely, we prove that TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 3 near the tip of any tongue of the double standard family, the two boundary curves form an ordinary cusp. Theorem 1. Assume Fλ◦n − id has a zero of multiplicity 3 in R/Z. Then there is a system of local coordinates (α, β) : W → R2 defined a neighborhood W of λ in R/Z × (0, 1), such that α(λ) = β(λ) = 0 and Fµ◦n − id has a multiple zero with µ ∈ W if and only if β 3 (µ) = α2 (µ). Our proof relies on a transversality result due to Adam Epstein for families of finite type analytic maps, which itself relies on an injectivity result of a linear map acting on an appropriate space of quadratic differentials. Even though the proof we present is done specifically for the tongues of the family of double standard maps Fλ , it may be adapted to study cusp bifurcations of other families of holo- morphic maps. Cusp bifurcations are a common phenomenon in the parameter planes of real-analytically parametrized families of holomorphic maps (see for in- stance [Mi92, CFG15, CFG16, NS03]). However, the non-holomorphic dependence on the parameter hinders the study of the parameter planes of such families. In this respect, this paper aims to provide a strategy to analyze the genericity of cusp bifurcations. The proof of Theorem 1 is structured as follows. In §1, we prove that the maps Fλ are finite type analytic maps. In §2, we define the functions α and β. In §3, we identify the derivatives of those functions at λ. In §4, we state and prove the injectivity result. In §5, we prove that (α, β) is a system of local coordinates. Some classical results on quadratic differentials are collected in Appendix A. Notation If U is a complex manifold, we denote by TU the tangent bundle of U and for z ∈ U , we denote by Tz U the tangent space to U at z. If φ : U → C is a holomorphic function, we denote by dφ : TU → C the exterior derivative of φ (this is a holomorphic 1-form on U ). If F : U → V is a holomorphic map between complex manifolds U and V , we denote by DF : TU → TV the bundle map Tz U 3 v 7→ Dz F (v) ∈ TF (z) V . Assume f : U → V is a holomorphic map between Riemann surfaces. If ω is a holomorphic 1-form on V , then f ∗ ω := ω ◦ Df is a holomorphic 1-form on U . If ϑ is a holomorphic vector field on V , then there is a meromorphic vector field f ∗ ϑ on U satisfying Df ◦ f ∗ ϑ = ϑ ◦ f . We will consider various holomorphic families t 7→ γt defined near 0 in C. We will employ the notation dγt γ := γ0 and γ̇ := . dt t=0 1. Finite type analytic maps The notion of finite type analytic maps originates in [E1]. Let f : X → Y be an analytic map of complex 1-manifolds, possibly disconnected. An open set V ⊆ Y is evenly covered by f if f|U : U → V is a homeomorphism for each component U of f −1 (V ); we say that y ∈ Y is a regular value for f if some neighborhood V 3 y is evenly covered, and a singular value for f otherwise. Note that the set Sf of singular values is closed. Recall that x ∈ X is a critical point if the derivative of f at x vanishes, and then f (x) ∈ Y is a critical value. We say that y ∈ Y is an 4 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN asymptotic value if f approaches y along some path tending to infinity relative to X. It follows from elementary covering space theory that the critical values together with the asymptotic values form a dense subset of Sf . In particular, every isolated point of Sf is a critical or asymptotic value. An analytic map f : X → Y of complex 1-manifolds is of finite type if • f is nowhere locally constant, • f has no isolated removable singularities, • Y is a finite union of compact Riemann surfaces, and • Sf is finite.  If Y is connected, we define degf as the finite or infinite number card f −1 (y) which is independent of y ∈ YrSf . When f : X → Y is a finite type analytic map with X ⊆ Y, we say that f is a finite type analytic map on Y. We first prove that the maps Fλ extend to finite type analytic maps. 1.1. Preliminaries. Set T := C/Z and Λ := T × C∗ . Let F : Λ × T → T be the holomorphic map defined by b F (λ, z) = 2z + a + sin(2πz) with λ := (a, b) ∈ Λ. π For λ ∈ Λ, let Fλ : T → T be the holomorphic map defined by Fλ (z) := F (λ, z). It will be convenient to consider the global coordinate w : T → C∗ defined by w(z) = e2πiz . Note that it is an isomorphism. Thus, adding two points denoted z = +i∞ (or w = 0) and z = −i∞ (or w = ∞), T may be compactified into a Riemann surface Tb isomorphic to the Riemann sphere. We will prove that for λ ∈ Λ, the map Fλ : T → T b is a finite type analytic map b on T. b Note that 1.2. The singular set. Fix λ := (a, b) ∈ Λ and set f := Fa,b : T → T. w ◦ f = e2πia w2 eb(w−1/w) and f ∗ (dw) = e2πia eb(w−1/w) (bw2 + 2w + b) dw. In particular, f has two critical points counting multiplicities: the solutions of bw2 + 2w + b = 0, i.e., the points c± ∈ T such that √ ± −1 ± 1 − b2 w(c ) = . b If b 6= 1, those are simple critical points of f . We denote by Cf = {c+ , c− } ⊂ T the set of critical points of f and by Vf := f (Cf ) ⊂ T the set of critical values of f . Lemma 2. The singular set Sf is equal to Vf ∪ {±i∞}. Proof. We already identified the set of critical values of f . Note that ±i∞ are singular values since those points are omitted values. It is therefore enough to show that f does not have any asymptotic value in T. If v ∈ T is an asymptotic value, then there exists a curve γ : [0, 1) → T, such that γ(t) → ±i∞ and f ◦ γ(t) → v as t → 1. We assume that γ(t) → +i∞. The proof for the case γ(t) → −i∞ is analogous. TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 5 It is convenient to lift via the canonical covering π : C → T := C/Z. Choose A ∈ C such that π(A) = a. Let f˜ : C → C be defined by b f˜(Z) = 2Z + A + sin(2πZ) so that π ◦ f˜ = f ◦ π. π Let Γ : [0, 1) → C be a lift of γ : [0, 1) → T, i.e., satisfying π ◦ Γ = γ. Then, f˜ ◦ Γ is a lift of f ◦ γ, thus f˜ ◦ Γ(t) converges in C as t → 1. Set X := Re(Γ) : [0, 1) → R and Y := Im(Γ) : [0, 1) → R. Then, b e−2πY e2πiX − e2πY e−2πiX f˜ ◦ Γ = 2(X + iY ) + A + sin(2πΓ), sin(2πΓ) = π 2i and Y (t) → +∞ as t → 1. It follows that as t → 1, b 2πY (t) −2πiX(t) f˜ ◦ Γ(t) ∼ 2X(t) − e e . 4πi  can distinguish 2 cases. If there exists a sequence {tk }k∈N converging to 1 with We X(tk ) k∈N bounded, then b 2πY (tk ) f˜ ◦ Γ(tk ) ∼ e −→ +∞. 4π k→+∞ Otherwise, X(t) → ±∞ as t → 1 and there exists a sequence {tk }k∈N converging to 1 with X(tk ) ∈ Z for all k ∈ N, so that b f˜ ◦ Γ(tk ) ∼ 2X(tk ) + i e2πY (tk ) −→ ∞. 4π k→+∞  ˜ In both cases, the sequence f ◦ Γ(tk ) k∈N cannot converge in C.  Corollary 3. The map f : T → T b is a finite type analytic map on T. b More precisely, f : Trf −1 (Vf ) → TrVf is a covering map. 2. Splitting triple zeros In the remainder of the article, we fix a parameter λ := (a, b) ∈ R/Z × (0, 1) such that Fλ◦n − id has a triple zero x ∈ R/Z. We set f := Fλ : T → T. The point x is periodic for f with period p dividing n. For k ≥ 0, we set xk := f ◦k (x) and we denote by hxi := {x0 , x1 , . . . , xp−1 } the cycle of x. Since f : R/Z → R/Z preserves the orientation, the multiplier of f ◦p at x is necessarily 1 and there is a local coordinate ζ : (T, x) → (C, 0) vanishing at x satisfying (1) ζ(z̄) = ζ̄(z) and ζ ◦ f ◦p = ζ + ζ 3 + O(ζ 5 ). According to the Weierstrass Preparation Theorem, there exist a neighborhood W1 ⊂ Λ of λ, a neighborhood W2 ⊂ C of 0 and analytic functions A : W1 → C, B : W1 → C, C : W1 → C and g : W1 × W2 → C such that for µ ∈ W1 , (2) ζ ◦ Fµ◦p − ζ = Pµ (ζ) · g(µ, ζ) with (3) A(λ) = B(λ) = C(λ) = 0, g(λ, ζ) = 1 + O(ζ 2 ) and (4) Pµ (ζ) := A(µ) + B(µ)ζ + C(µ)ζ 2 + ζ 3 . 6 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN The polynomial Pλ has a zero of multiplicity 3 at 0, and as µ varies in W1 , this zero splits in three zeros (counting multiplicities) of Pµ . When µ ∈ R/Z × (0, 1), the map Fµ◦n −id commutes with z 7→ z̄, so that the polynomial Pµ has real coefficients. For such a parameter µ, a multiple zero of Pµ is necessarily real. For any µ ∈ Λ, the function ζ ◦ Fµ◦p − ζ vanishes at the periodic points of Fµ of period dividing p, and so, divides ζ ◦ Fµ◦n − ζ which vanishes at the periodic points of period dividing n. In addition, if n = mp, then ζ ◦ f ◦n − ζ = mζ 3 + O(ζ 5 ). So, there is an analytic function h : W1 × W2 → C such that for µ ∈ W1 , ζ ◦ Fµ◦n − ζ = Pµ (ζ) · h(µ, ζ) with h(λ, ζ) = m + O(ζ 2 ) Since f only has two critical points in T, it has a single non-repelling cycle, that is, the cycle hxi. All other cycles of f in R/Z are repelling. Shrinking W1 if necessary, it follows that for µ ∈ W1 , the function ζ ◦ Fµ◦n − ζ has a multiple zero in R/Z if and only if the polynomial Pµ has a multiple zero in R/Z. According to the previous discussion, this is the case if and only if Pµ has a multiple zero. Let α : W1 → C and β : W1 → C be defined by C3 BC A C2 B α := − + and β := − . 27 6 2 9 3 Then, discriminant(Pµ ) = 108β 3 (µ) − 108α2 (µ). So, if µ ∈ W1 , the polynomial Pµ has a multiple zero if and only if β 3 (µ) = α2 (µ). In order to prove Theorem 1, it is therefore enough to show that (α, β) is a system of local coordinates near λ. For this purpose, we shall show that the restrictions of dα and dβ to Tλ Λ are linearly independent. Since A, B and C vanish at λ, 1 1 dα|Tλ Λ = dA|Tλ Λ and dβ|Tλ Λ = − dB|Tλ Λ . 2 3 It is therefore enough to show that the forms dA|Tλ Λ and dB|Tλ Λ are linearly independent. 3. Identifying the derivatives Here, we identify dA(v) and dB(v) for v ∈ Tλ Λ. First, to each v ∈ Tλ Λ, we shall associate a meromorphic vector field ϑv on T having simple poles along Cf ∪{±i∞}, such that for all z ∈ TrCf , Df ◦ ϑv (z) := Dλ,z F (v, 0). Second, for k ∈ [1, p], let ζk : (T, xk ) → (C, 0) be the local coordinate vanishing at xk defined by ζk := ζ ◦ f ◦(p−k) . Our identification goes as follows. Proposition 4. Let qA and qB be quadratic differentials, defined and meromorphic near hxi, such that qA − (dζk )2 /ζk and qB − (dζk )2 /ζk2 are holomorphic at xk for all k ∈ [1, p]. Then, for all v ∈ Tλ Λ, p X p X dA(v) = residue(qA ⊗ ϑv , xk ) and dB(v) = residue(qB ⊗ ϑv , xk ). k=1 k=1 In the remaining parts of this section we prove Proposition 4. TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 7 3.1. Meromorphic vector fields. Assume v ∈ Tλ Λ and z ∈ TrCf . Then, the derivative Dz f : Tz T → Tf (z) T is an isomorphism and Dλ,z F (v, 0) ∈ Tf (z) T. Let ϑv be the vector field defined on TrCf by  ϑv (z) := (Dz f )−1 Dλ,z F (v, 0) ∈ Tz T. Lemma 5. For all v ∈ Tλ Λ, the vector field ϑv is holomorphic on TrCf , mero- b vanishes at z = ±i∞ and has at worst simple poles along Cf . morphic on T, Proof. The map v 7→ ϑv is linear. So, it is enough to prove the result for va := d/da and vb := d/db. We have 2πie2πia w2 eb(w−1/w) d 2πiw2 d ϑv a = = 2 e2πia eb(w−1/w) (bw2+ 2w + b) dw bw + 2w + b dw and e2πia w2 (w − 1/w)eb(w−1/w) d w3 − w d ϑv b = 2πia b(w−1/w) 2 = 2 e e (bw + 2w + b) dw bw + 2w + b dw Those two vector fields have the required properties.  Denote by Tf the space of meromorphic vector fields on T b which are holomorphic on TrCf , vanish at ±i∞ and have at worst simple poles along Cf . In other words,   c3 w3 + c2 w2 + c1 w d 3 Tf := ; (c , c 1 2 3, c ) ∈ C . bw2 + 2w + b dw Let Θf : Tλ Λ → Tf be the linear map defined by Θf (v) := ϑv . Let τ ∈ Tf be the radial vector field d τ := w . dw Note that τ − f ∗ τ belongs to Tf . Indeed, bw3 + w2 + bw d τ − f ∗τ = ∈ Tf . bw2 + 2w + b dw Lemma 6. The space Tf is the direct sum of the image of Θf and the line spanned by τ − f ∗ τ : Tf = Im(Θf ) ⊕ Vect(τ − f ∗ τ ). Proof. The dimension of Tf is 3. Thus, it is enough to show that the three vector fields ϑva , ϑvb and τ − f ∗ τ are linearly independent. Equivalently, it is enough to show that the three functions w2 , w3 − w and bw3 + w2 + bw are linearly independent. This is true since b 6= 0.  Assume now v ∈ Tλ Λ and let t 7→ λt ∈ Λ be a curve such that λ̇ = v. Let t 7→ ft be the family of maps defined by ft := Fλt : T → T. Then, for each z ∈ T, f˙(z) = Dλ,z F (v, 0) = Dz f ◦ ϑv (z) with ϑv := Θf (v) ∈ Tf . 8 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN Lemma 7. For all k ≥ 1, dft◦k ∗ = Df ◦k ◦ ϑkv with ϑkv := ϑv + f ∗ ϑv + · · · + f ◦(k−1) ϑv . dt t=0 Proof. The proof follows from an elementary induction on k ≥ 1 using the following fact: if ht = gt ◦ ft with f˙ = Df ◦ ϑ and ġ = Dg ◦ τ , then ḣ = ġ ◦ f + Dg ◦ f˙ = Dg ◦ τ ◦ f + Dg ◦ Df ◦ ϑ = Dh ◦ (f ∗ τ + ϑ).  Note that the poles of ϑnv are the critical points of f and their iterated preimages (up to order n − 1). The two critical points of f are in TrR/Z, and so are all their preimages. Therefore, ϑnv is holomorphic in a neighborhood of R/Z. In particular, it is holomorphic near the parabolic periodic point x ∈ R/Z. 3.2. Polar parts of quadratic differentials. Our identification of the derivatives dA|Tλ Λ and dB|Tλ Λ relies on the use of quadratic differentials (see Appendix A for basics regarding quadratic differentials). Recall that ζ : (T, x) → (C, 0) is a local coordinate vanishing at x such that ζ ◦ f ◦p = ζ + ζ 3 + O(ζ 5 ). We shall use the quadratic differential (dζ)2 /ζ and (dζ)2 /ζ 2 which are defined and meromorphic near x in T. Following §A.7, if Z ⊂ T is a finite set, if q is a quadratic differential, defined and meromorphic near Z, and if ϑ is a vector field, defined and meromorphic near Z, we shall use the notation X hq, ϑiZ := residue(q ⊗ ϑ, z). z∈Z If q has at worst simple poles along Z and if θ is defined on Z with θ(z) ∈ Tz T for z ∈ Z, we shall use the notation hq, θiZ := hq, ϑiZ where ϑ is any vector field, defined and holomorphic near Z, with ϑ(z) = θ(z) for z ∈ Z. The result does not depend on the choice of extension. Lemma 8. For all v ∈ Tλ Λ,     (dζ)2 p (dζ)2 p dA(v) = , ϑv and dB(v) = , ϑv . ζ x ζ2 x Proof. According to Equations (2), (3) and (4),   ζ ◦ ft◦p − ζ = A(λt ) + B(λt )ζ + O(ζ 2 ) · 1 + O(ζ 2 ) . Taking the derivative with respect to t and evaluating at t = 0 yields dζ ◦ Df ◦p ◦ ϑpv = dA(v) + dB(v)ζ + O(ζ 2 ). According to Equation (1),  ζ ◦ f ◦p = ζ + O(ζ 3 ) so that dζ ◦ Df ◦p = 1 + O(ζ 2 ) dζ. As a consequence dζ(ϑpv ) = dζ ◦ Df ◦p ◦ ϑpv + O(ζ 2 ) = dA(v) + dB(v)ζ + O(ζ 2 ). TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 9 Thus,     dζ(ϑpv ) (dζ)2 p dA(v) = residue dζ, x = , ϑv ζ ζ x and similarly     dζ(ϑpv ) (dζ)2 p dB(v) = residue dζ, x = , ϑv .  ζ2 ζ2 x Rather than working near x with the vector field ϑpv , it will be convenient to work along the cycle hxi with the vector field ϑv . Recall that for k ∈ [1, p], the local coordinate ζk : (T, xk ) → (C, 0) vanishes at xk and is defined by ζk := ζ ◦ f ◦(p−k) . Lemma 9. For all k ∈ Z/pZ,     (dζk+1 )2 (dζk )2 (dζk+1 )2 (dζk )2 f∗ − and f∗ 2 − ζk+1 ζk ζk+1 ζk2 are holomorphic near xk . Proof. If k ∈ [1, p − 1], then ζk = ζk+1 ◦ f , so that     (dζk+1 )2 (dζk )2 (dζk+1 )2 (dζk )2 f∗ = and f ∗ 2 = . ζk+1 ζk ζk+1 ζk2  If k = p, then ζp = ζ and ζ1 ◦ f = ζ ◦ f ◦p = 1 + O(ζp2 ) ζp . As a consequence,  f ∗ (dζ1 ) = 1 + O(ζp2 ) dζp ,     (dζ1 )2  2 (dζp ) 2 (dζ1 )2  (dζp )2 f ∗ = 1 + O(ζp ) and f ∗ 2 = 1 + O(ζp2 ) .  ζ1 ζp ζ1 ζp2 Proof of Proposition 4. Recall that ζp = ζ. According to the previous lemma, for all k ∈ Z/pZ, (dζk )2 (dζ)2 (f ◦k )∗ − ζk ζ is holomorphic near x. By assumption, qA − (dζk )2 /ζk is holomorphic at xk . It follows that (f ◦k )∗ qA − (dζ)2 /ζ is holomorphic near x. Since (f ◦k )∗ ϑv is holomorphic near x, we therefore have   (dζ)2 , (f ◦k )∗ ϑv = (f ◦k )∗ qA , (f ◦k )∗ ϑv x = hqA , ϑv ixk . ζ x As a consequence * p−1 + p−1 (dζ)2 X ◦k ∗ X dA(v) = , (f ) ϑv = hqA , ϑv ixk = hqA , ϑv ihxi . ζ k=0 x k=0 This proves Proposition 4 for dA. The proof for dB is similar.  10 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN 4. Injectivity of ∇f In order to prove Theorem 1, we need to use the global properties of the map f . Up to now, we only used the local properties near the cycle. For this purpose, it is important that the quadratic differentials qA and qB which appear in Proposition b Here, we define such quadratic differentials qA 4 are globally meromorphic on T. and qB and we prove that the linear map ∇f := id − f∗ is well defined and injective on the vector space Vect(qA , qB ) spanned by qA and qB . 4.1. A space of quadratic differentials. Denote by Q(T) the space of mero- morphic quadratic differentials on Tb which have at worst simple poles at z = ±i∞. Given Z ⊂ T, denote by Q(T; Z) ⊂ Q(T) the subspace of quadratic differentials which are holomorphic outside Z. Finally, we denote by Q1 (T; Z) ⊂ Q(T; Z) the subspace of quadratic differentials having at worst simple poles. Lemma 10. Any polar part of quadratic differential along  hxi may be realized as the polar part of a quadratic differential in Q T; hxi∪{c+ } having at worst a simple pole at c+ . Proof. For all k ∈ [0, p − 1], the quadratic differentials (dw)2 (dw)2 (dw)2   , 2 and j for j ≥ 3 w − w(xk ) w − w(c+ ) w w − w(xk ) w w − w(xk )  belong to Q T; hxi ∪ {c+ } . The first has a simple pole at xk , the second has a double pole at xk , and the third has a pole of order j ≥ 3 at xk . Thus, they generate the space of polar parts at xk .    From now on, we assume that qA ∈ Q T; hxi ∪ {c+ } and qB ∈ Q T; hxi ∪ {c+ } have at worst simple poles at c+ and that qA − (dζk )2 /ζk and qB − (dζk )2 /ζk2 are holomorphic at xk for all k ∈ [0, p − 1]. We set Qf := Vect(qA , qB ). 4.2. Pairing quadratic differentials in Qf with vector fields in Tf . Recall that d τ := w and τ − f ∗ (τ ) ∈ Tf . dw Lemma 11. For all q ∈ Qf , hq, τ − f ∗ τ ihxi = 0. Proof. Assume q ∈ Qf . According to Lemma 9, q − f ∗ q is holomorphic near hxi. Since τ is also holomorphic near hxi, and since f is a local isomorphism near hxi, hq, f ∗ τ ihxi = hf ∗ q, f ∗ τ ihxi = hq, τ ihxi .  TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 11 4.3. Pushing forward quadratic differentials in Qf . According to Corollary 3, f : Trf −1 (Vf ) → TrVf is a covering map. Here, we show that for all q ∈ Qf , b the following series defines a meromorphic quadratic differential on T: X (5) f∗ q = g ∗ q. g inverse branch of f The (minor) difficulty is that the degree of the covering map is not finite, and that q may fail to be integrable on T b since it may have multiple poles along hxi. So, we cannot apply directly the results presented in Appendix A. The reason why the series in Equation (5) converges is that q is locally integrable near the essential singularities of f , i.e., the points ±i∞. Lemma 12. If q ∈ Qf , the series in Equation (5) converges locally uniformly in  Tr Vf ∪ hxi . Its sum f∗ q is a meromorphic quadratic differential on T. b  Proof. Assume q ∈ Qf . Let V ⊂ Tr Vf ∪ hxi is compactly contained in Tr b hxi. Then, U := f (V ) is compactly contained in Tr −1 b hxi. In particular, q is integrable on U . In addition, f : U → V is a covering map. It follows that the series in Equation (5) converges uniformly on V and  that f∗ q is integrable on V . This shows f∗ q is holomorphic on Tr Vf ∪ hxi and has at worst simple poles at ±i∞ and on Vf . To see that f∗ q is meromorphic near xk , k ∈ [1, p], let V ⊂ TrVf be a topological disk containing xk . Then, U := f −1 (V ) is the disjoint union of a topological disk U 0 containing xk−1 and a open set U 00 compactly contained in Tr b hxi. Then, (f |U 00 )∗ q 0 is holomorphic. In addition, f : U → U is an isomorphism so that (f |U 0 )∗ q – and thus f∗ q = (f |U 0 )∗ q + (f |U 0 )∗ q – is meromorphic near xk .  We may therefore consider the linear map ∇f := id − f∗ : Qf → Q(T). It will be convenient to set Y := {c+ } ∪ Vf . Lemma 13. We have the inclusion ∇f (Qf ) ⊆ Q1 (T; Y ). Proof. Assume q ∈ Qf . As mentioned  in the proof of the previous lemma, f∗ q is holomorphic on Tr Vf ∪ hxi and has at worst simple poles at ±i∞ and on Vf . In addition, for k ∈ [1, p], the polar part of f∗ q at xk is equal to the polar part of g ∗ q where g is the inverse branch of f sending xk to xk−1 . According to Lemma 9, q − f∗ q is therefore holomorphic near hxi. It follows that q − f∗ q ∈ Q1 (T; Y ).  4.4. Injectivity of ∇f . An observation due to the fourth author [E2] is that the linear map ∇f is injective on Qf , and that this is the key to the proof of Theorem 1. Proposition 14. The linear map ∇f : Qf → Q(T) is injective. Proof. We must prove that f∗ q 6= q for q ∈ Qf r{0}. If q were integrable on T, the result would follow immediately from Proposition 21, since we would have kf∗ qkL1 (T) < kqkL1 (T) . Since q may have double poles near hxi, it may fail to be integrable on T. In that case, we may proceed as follows. 12 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN Assume q ∈ Qf r{0}. For ε > 0 small, let Vε ⊂ T be the union of topological disks p [  Vε := |ζk | < ε . k=1 Set Uε := f −1 (Vε ) ⊂ T. Then, hxi ⊂ Uε and so, q is integrable on TrUε . As a consequence, kf∗ qkL1 (TrVε ) < kqkL1 (TrUε ) . Similarly, for ε0 < ε, kf∗ qkL1 (Vε rVε0 ) < kqkL1 (Uε rUε0 ) . As a consequence, the function ε 7→ kqkL1 (TrUε ) − kf∗ qkL1 (TrVε ) is positive and decreasing. In particular, it has a positive limit. Note that kqkL1 (TrUε ) − kqkL1 (TrVε ) = kqkL1 (Vε rUε ) − kqkL1 (Uε rVε ) ≤ kqkL1 (Vε rUε ) . We deduce from the following lemma that f∗ q 6= q.  Lemma 15. For any q ∈ Qf , lim kqkL1 (Vε rUε ) = 0. ε→0 Proof. Since ζ ◦ f ◦p = ζ + O(ζ 3 ), there is a constant κ1 such that |ζ ◦ f ◦p | ≥ |ζ| − κ1 |ζ|3 . Since q has at worst a double pole at x, there is a constant κ2 such that for |ζ| small enough |dζ 2 | |q| ≤ κ2 . |ζ|2 Note that for ε > 0 small enough,    Vε rUε = |ζ| < ε r |ζ ◦ f ◦p | < ε ⊂ ε − κ1 ε3 ≤ |ζ| < ε . Thus, Z 2π Z ε rdrdt 1 0 ≤ kqk L1 (V ε rUε ) ≤ κ2 = 2πκ2 ln −→ 0.  0 ε−κ1 ε3 r2 1 − κ1 ε2 ε→0 5. Linear independence We may now complete the proof that dA|Tλ Λ and dB|Tλ Λ are linearly indepen- dent. According to Proposition 4, for all v ∈ Tλ Λ, dA(v) = qA , Θf (v) hxi and dB(v) = qB , Θf (v) hxi . According to Lemma 6, Tf = Im(Θf ) ⊕ Vect(τ − f ∗ τ ). Showing that dA|Tλ Λ and dB|Tλ Λ are linearly independent therefore amounts to proving that for all q ∈ Qf r{0}, there exists ϑ ∈ Tf such that hq, ϑihxi 6= 0. TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 13 5.1. Guiding vector fields. Set Z := Cf ∪ Vf and denote by TZ the space of maps ξ : Z → TT satisfying ξ(z) ∈ Tz T for all z ∈ Z. Lemma 16. For any ϑ ∈ Tf , there exists a unique ξϑ ∈ TZ such that for any vector field ξ, defined and holomorphic near Z with ξ(z) = ξϑ (z) for z ∈ Z, the vector field ϑ + ξ − f ∗ ξ is holomorphic and vanishes along Cf . Proof. Fix ϑ ∈ Tf . Let us first prove the uniqueness of ξϑ ∈ TZ. Assume ξ1 and ξ2 are two vector fields, defined and holomorphic near Z, such that ϑ + ξ1 − f ∗ ξ1 and ϑ + ξ2 − f ∗ ξ2 are holomorphic near Cf . Then, (ξ1 − ξ2 ) − f ∗ (ξ1 − ξ2 ) is holomorphic and vanishes along Cf . As a consequence, Df ◦ (ξ1 − ξ2 ) − (ξ1 − ξ2 ) ◦ f vanishes on Cf . Since Df ◦ (ξ1 − ξ2 ) vanishes on Cf , this forces ξ1 − ξ2 to vanish on Vf . In that case, f ∗ (ξ1 − ξ2 ) vanishes on Cf and so, ξ1 − ξ2 vanishes on Cf . This shows the uniqueness of ξϑ ∈ TZ. This also proves that if ϑ + ξ − f ∗ ξ is holomorphic and vanishes along Cf for some vector field ξ, defined and holomorphic near Z with ξ(z) = ξϑ (z) for z ∈ Z, then ϑ + ξ − f ∗ ξ is holomorphic and vanishes along Cf for any vector field ξ, defined and holomorphic near Z with ξ(z) = ξϑ (z) for z ∈ Z. Let us now prove the existence of ξϑ ∈ TZ. Note that Df ◦ ϑ is a map from TrCf to the tangent bundle TT. Note that it is not a vector field since for z ∈ T, the vector Df ◦ ϑ(z) belongs to Tf (z) T. However, since ϑ has at worst simple poles along Cf and since Df vanishes on Cf , the map Df ◦ ϑ extends holomorphically to T. Set  ξϑ f (c± ) := Df ◦ ϑ(c± ). Next, let ξ be any vector field, defined and holomorphic near Vf , coinciding with ξϑ on Vf . Then, ϑ − f ∗ ξ is holomorphic near Cf and we may set ξϑ (c± ) := (ϑ − f ∗ ξ)(c± ).  Recall that Y := {c+ } ∪ Vf ⊂ Z. It will be convenient to consider the linear map Ξf : Tf → TY defined by Ξf (ϑ) = ξϑ |Y . Lemma 17. The map Ξf : Tf → TY is an isomorphism. Proof. Since the dimensions of Tf and TY are both equal to three, it is enough to show that the map is injective. Assume ϑ ∈ Tf and ξϑ vanishes on {c+ } ∪ Vf . Let ξ be a vector field, defined and holomorphic near Z, which coincides with ξϑ on Z. We may assume that ξ identically vanishes near {c+ }∪Vf . Then, ϑ+f ∗ ξ −ξ = ϑ−ξ is holomorphic and vanishes on Cf . This shows that ϑ is holomorphic near Cf and b and vanishes at vanishes at c+ . As a consequence, ϑ is globally holomorphic on T, three points: c+ , +i∞ and −i∞. So, ϑ = 0.  5.2. From the cycle to the critical set. We may now transfer the local com- putations done near the cycle hxi to local computations done near the critical set Y. Lemma 18. For all ϑ ∈ Tf and all q ∈ Qf , hq, ϑihxi = ∇f q, Ξf (ϑ) Y . 14 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN Proof. Let ξ be a vector field, defined and holomorphic near Z, coinciding with ξϑ := Ξf (ϑ) on Z. Then, ϑ + ξ − f ∗ ξ is holomorphic near Cf . In addition, since ∇f q is holomorphic near c− , h∇f q, ξϑ iY = hq − f∗ q, ξiZ = hq, ξiCf − hf∗ q, ξiVf = hq, ξiCf − hq, f ∗ ξiCf = hq, −ϑiCf = hq, ϑihxi . In the second line, we used the fact that the only poles of q ⊗ f ∗ ξ in f −1 (Vf ) belong to Cf . For the last equality, we used the fact that q ⊗ ϑ is a globally meromorphic b whose poles are contained in Cf ∪ hxi, and that the sum of all residues 1-form on T, of a globally meromorphic 1-form on a compact Riemann surface is 0.  5.3. Completion of the proof. Assume by contradiction that dA|Tλ Λ and dB|Tλ Λ are not linearly independent. Then, there is a q ∈ Qf r{0} such that for all ϑ ∈ Tf , 0 = hq, ϑihxi = ∇f q, Ξf (ϑ) Y . According to Lemma 17, the map Ξf : Tf → TY is an isomorphism. In particular, it is surjective. It follows that for all ξ ∈ TY , h∇f q, ξiY = 0. As a consequence, ∇f q is holomorphic near Y and thus, has at most three simple b has at least four poles at c− , +i∞ and −i∞. A non zero quadratic differential on T poles, counting multiplicities. Thus, ∇f q = 0. According to Proposition 14, the map ∇f : Qf → Q1 (T; Y ) is injective. It follows that q = 0. Contradiction. This completes the proof of Theorem 1. Appendix A. Quadratic differentials A.1. Meromorphic quadratic differentials. A quadratic differential on a Rie- mann surface U is a section of the square of the cotangent bundle T∗ U ⊗ T∗ U . We shall usually think of a quadratic differential q as a field of quadratic forms. In particular, if ϑ is a vector field on U and φ is a function on U , then q(ϑ) is a function on U and q(φϑ) = φ2 q(ϑ). If ζ : U → C is a coordinate, we shall use the notation (dζ)2 = dζ ⊗ dζ - not be confused with 1-form d(ζ 2 ). Then, a quadratic differential q on U is of the form q = φ (dζ)2 for some function φ. We say that q is meromorphic on U if φ is meromorphic on U . In that case, the order of q at a point x ∈ U is ordx q := ordx φ, i.e., 0 if φ is holomorphic and does not vanish at x, k ≥ 1 if φ has a zero of multiplicity k at x, and −k ≤ −1 if φ has a pole of multiplicity k at x. A.2. Pullback. The derivative Df : TU → TV of a holomorphic map f : U → V naturally induces a pullback map f ∗ from quadratic differentials on V to quadratic differentials on U : f ∗ q := q ◦ Df. Lemma 19. If f : (U, x) → (V, y) is holomorphic at x, and q is meromorphic at y = f (x), then 2 + ordx (f ∗ q) = degx f · (2 + ordy q). TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 15 Proof. Choose local coordinates z : (U, x) → (C, 0) and w : (V, y) → (C, 0) such that w ◦ f = z k , with k := degx f . If q = φ (dw)2 , then f ∗ q = φ ◦ f · (kz k−1 dz)2 . Thus, 2 + ordx (f ∗ q) = 2 + ordx (φ ◦ f ) + (2k − 2) = 2k + k · ordy φ = k · (2 + ordy q).  A.3. Pushforward for finite degree covering maps. Assume f : U → V is a finite degree covering map. If q is a quadratic differential on U , we define a quadratic differential f∗ q on V by X f∗ q := g ∗ q. g inverse branch of f If q is holomorphic on U , then f∗ q is holomorphic on V . Lemma 20. Assume U := U b r{x} and V := Vb r{y} are punctured disks, f : U → V is a covering map ramifying at x with local degree degx f and q is meromorphic at x. Then, f∗ q has at worst a pole at y and 2 + ordx q 2 + ordy (f∗ q) ≥ . degx f Proof. The group of deck transformations of f : U → V is a cyclic group of order degx f . Note that X f ∗ (f∗ q) = h∗ q, h deck transformation of f ∗ and ordx h q = ordx q for all deck transformations h, so that ordx f ∗ (f∗ q) ≥ ordx q. Then, 2 + ordx f ∗ (f∗ q) 2 + ordx q 2 + ordy (f∗ q) = ≥ .  degx f degx f A.4. Integrable quadratic differentials. If q is a quadratic differential on U , we denote by |q| the positive (1, 1)-form on U defined by its action on pairs of vector fields (ϑ1 , ϑ2 ) as follows: 1 1 |q|(ϑ1 , ϑ2 ) := q(ϑ1 − iϑ2 ) − q(ϑ1 + iϑ2 ) . 4 4 If ζ : U → C is a coordinate and q = φ (dζ)2 , then i |q| = |φ| · dζ ∧ dζ̄. 2 We shall say that q is integrable on U if Z kqkL1 (U ) := |q| < ∞. U Note that q is integrable in a neighborhood of a pole if and only if the pole is simple. If f : U → V is an isomorphism and q is an integrable quadratic differential on V , then f ∗ q is integrable on U and kf ∗ qkL1 (U ) = kqkL1 (V ) . 16 K. BANERJEE, X. BUFF, J. CANELA, AND A. EPSTEIN A.5. Pushforward for infinite degree covering maps. Assume f : U → V is an infinite degree covering map. If q is an integrable quadratic differential on U , we may still define X f∗ q := g ∗ q. g inverse branch of f Indeed, the series converges in L1loc since if V 0 ⊂ V is a topological disk, so that the inverse branches g : V → U of f are defined on V 0 , and if U 0 := f −1 (V 0 ), then 0 X kg ∗ qkL1 (V 0 ) = kqkL1 (U 0 ) ≤ kqkL1 (U ) . The limit of a sequence of holomorphic functions converging in L1loc is itself holo- morphic. It follows that if q is holomorphic on U , then f∗ q is holomorphic on V. A.6. The Contraction Principle. Proposition 21. Let f : U → V be a covering map and let q be a holomorphic integrable quadratic differential on U . Then, kf∗ qkL1 (V ) ≤ kqkL1 (U ) and equality holds if and only if either q = 0, or the degree of f is finite and f ∗ (f∗ q) = deg(f )·q. Proof. The proof is an immediate application of the triangle inequality: for any topological disk V 0 ⊂ V , we have Z Z X Z X XZ Z |f∗ q| = g∗ q ≤ |g ∗ q| = |g ∗ q| = |q|, V0 V0 V0 V0 f −1 (V 0 ) where the sums range over the inverse branches g : V 0 → U of f . It follows that Z Z Z |f∗ q| ≤ |q| = |q| V f −1 (V ) U with equality if and only if for all inverse branches P g of f , we have g ∗ q = ψg f∗ q for some function ψg : V 0 → [0, 1] satisfying g ψg = 1. Setting φ g(y) := ψg (y), we see that q = φ f ∗ (f∗ q) for some function φ : U → [0, 1]. Since q and f ∗ (f∗ q) are holomorphic,Peither q = 0, or the function φ is constant, let us say equal to c ∈ [0, 1]. Since g ψg = 1, we have that deg(f ) · c = 1, which forces the degree of f to be finite with f ∗ (f∗ q) = deg(f ) · q.  A.7. Pairing quadratic differentials and vector fields. If q is a quadratic differential on U and ϑ is a vector field on U , we may consider the 1-form q ⊗ ϑ defined on U by its action on vector fields τ : 1  q ⊗ ϑ(τ ) = q(ϑ + τ ) − q(ϑ − τ ) . 4 Note that if q = φ (dζ)2 and ϑ = ψ d/dζ, then q ⊗ ϑ = φψ dζ. If x ∈ U , and if ϑ and q are meromorphic on U , we set hq, ϑix := residue(q ⊗ ϑ, x). If q has at worst a simple pole at x, then hq, ϑix only depends on θ := ϑ(0), and we use the notation hq, θix := hq, ϑix . TIPS OF TONGUES IN THE DOUBLE STANDARD FAMILY 17 Lemma 22. Let U := U b r{x} and V := Vb r{y} be punctured disks, let f : U → V be a covering map ramifying at x, let q be a meromorphic quadratic differential on b and let ϑ be a meromorphic vector field on U . Then U hf∗ q, ϑiy = hq, f ∗ ϑix . Proof. Let γ ⊂ V be a loop around y with basepoint a. Then Z XZ XZ Z (f∗ q) ⊗ ϑ = (g ∗ q) ⊗ ϑ = q ⊗ f ∗ϑ = q ⊗ f ∗ ϑ, γ g γr{a} g g(γr{a}) f −1 (γ) where the sum ranges over the inverse branches g of f defined on γr{a}.  References [A65] V. I. 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Email address: kbanerjee.maths@presiuniv.ac.in Presidency University, 86/1 College Street, Kolkata - 700073, West Bengal, India Email address: xavier.buff@math.univ-toulouse.fr Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France Email address: canela@uji.es Centre de Recerca Matemàtica; Barcelona Graduate School of Mathematics (BGS- Math), Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain Email address: a.l.epstein@warwick.ac.uk Mathematics Institute, University of Warwick, Coventry CV4 7AL - UK