Classification of Special Curves in the Space of Cubic Polynomials

Classification of Special Curves in the Space of Cubic Polynomials Abstract We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker and DeMarco for cubic polynomials. Let Perm(λ) be the algebraic curve consisting of those cubic polynomials that admit an orbit of period m and multiplier λ. We also prove that an irreducible component of Perm(λ) is special if and only if λ=0. 1 Introduction The space Polyd of complex polynomials of degree d≥2 modulo affine conjugacy forms a complex analytic space that admits a ramified parameterization by the affine space Aℂd−1. The study of the set of degree d polynomials with special dynamical features forms the core of the modern theory of holomorphic dynamics. We shall be concerned here with the distribution of the set of post-critically finite (PCF) polynomials for which all critical points have a finite orbit under iteration. This set is a countable union of points defined over a number field, see for example [29, Corollary 3]. It was proved in [32] that hyperbolic PCF quadratic polynomials equidistribute to the harmonic measure of the Mandelbrot set. This convergence was generalized in [4] in degree 2, and later in [21] in any degree where it was proved that under a mild assumption any sequence of Galois-invariant finite subsets of PCF polynomials converges in the sense of measures to the so-called bifurcation measure. This fact was further explored in [26]. The support of this measure has been characterized in several ways in a series of works [18–20,25], and it was shown by the second author [23] to have maximal Hausdorff dimension 2(d−1). In a beautiful recent article [3], Baker and DeMarco have proposed a way to describe the distribution of PCF polynomials from the point of view of the Zariski topology. They defined special algebraic subvarieties as those subvarieties Z⊂Polyd admitting a Zariski-dense subset formed by PCF polynomials, and asked about the classification of such varieties. More precisely, they offered a quite general conjecture [3, Conjecture 1.4] inspired by the André-Oort conjecture in arithmetic geometry stating that any polynomial (the conjecture is actually stated for any rational maps, and a stronger conjecture related to Pink and Zilber's conjectures can be found in [14].) lying in a special (proper) subvariety should admit a critical orbit relation (see also [43, Conjecture 6.56]). They gave a proof of a stronger version of this conjecture in the case the subvariety was isomorphic to an affine line. Our objective is to give the list of all special curves in the case d=3, thereby proving Baker–DeMarco's conjecture for cubic polynomials. The geometry of the space of cubic polynomials has been thoroughly explored in the seminal work [7] of Branner and Hubbard. Instead of using their parameterization, we shall follow [20] and consider the parameterization (c,a)↦Pc,a of the parameter space by the affine plane with   Pc,a(z):=13z3−c2z2+a3, z∈ℂ,(c,a)∈ℂ2. Observe that Pc,a then admits two critical points c0:=0 and c1:=c and that this map defines a finite branched cover of the moduli space Poly3 of cubic polynomials with marked critical points. Here is our main result. Theorem A. An irreducible curve C in the space Poly3 contains an infinite collection of post-critically finite polynomials if and only if one of the following holds. One of the two critical points is persistently pre-periodic on C, that is, there exist integers m>0 and k≥0 such that Pc,am+k(c0)=Pc,ak(c0) or Pc,am+k(c1)=Pc,ak(c1) for all (c,a)∈C. There is a persistent collision of the two critical orbits on C, that is, there exist (m,k)∈ℕ2∖{(1,1)} such that Pc,am(c1)=Pc,ak(c0) for all (c,a)∈C. The curve C is given by the equation {(c,a),12a3−c3−6c=0}, and coincides with the set of cubic polynomials having a non-trivial symmetry, that is, the set of parameters (c,a) for which Qc(z):=−z+c commutes with Pc,a.     □ Recall that for any integer m≥1 and any complex number λ∈ℂ the set Perm(λ) consisting of all polynomials Pc,a∈Poly3 that admit at least one periodic orbit of period m and multiplier λ is an algebraic curve (see Section 5 for a more precise description). The geometry of these curves has been explored by several authors, especially when λ=0. The irreducible components of Perm(0) have been proven to be smooth by Milnor [38], and the escape components of these curves have been described in terms of Puiseux series by Bonifant, Kiwi, and Milnor [5] (see also [31, Section 7]). On the other hand, DeMarco and Schiff [15] have given an algorithm to compute their Euler characteristic. In a recent preprint, Arfeux and Kiwi counted the number of irreducible components of Perm(0) when m is prime. From the point of view of pluripotential theory, the distribution of the sequence of curves (Perm(λ))m≥1 has been completely described by Bassanelli and Berteloot in [9] in the case |λ|≤1 (see also [24] for the case |λ|>1 and [8] for the case of quadratic rational maps). Inspired by a similar result from Baker and DeMarco, see [3, Theorem 1.1] we also give a characterization of those Perm(λ) that contain infinitely many PCF polynomials. This answers a conjecture of DeMarco in the case of cubic polynomials (see e.g., [43, Conjecture 6.59]). More precisely, we prove Theorem B. For any m≥1, the curve Perm(λ) contains infinitely many post-critically finite polynomials if and only if λ=0.     □ The general strategy of the proof of these two theorems was set up by Baker and DeMarco. They start with an irreducible algebraic curve C⊂Poly3 containing infinitely many PCF polynomials (in Theorem B the curve C is a component of some Perm(λ)). We observe however that they used at several key points their assumption that the curve C has a single branch at infinity. To remove this restriction we had to include two new ingredients: we construct a one parameter family of heights for which Thuillier–Yuan's equidistribution theorem [44, 46] applies; we investigate systematically the arithmetic properties of the coefficients of the expansion of the Böttcher coordinates and their dependence on the parameters c,a. We propose also a new way to build the symmetry by relying on a recent algebraization result of Xie [45] that gives a criterion for when a formal curve in the affine plane is a branch of an algebraic curve. A characteristic feature of our proofs is to look at the dynamics induced by cubic polynomials over various fields: over the complex numbers and over p-adic fields (see e.g., Section 4.1), over the field of Laurent series (see the proof of Proposition 3.6 and Section 5), and over a number field (see Section 3). We use at one point the universality theorem of McMullen [35] which is a purely Archimedean statement. Moreover, the work of Kiwi [31] on non-Archimedean cubic polynomials over a field of residual characteristic zero plays a key role in the proof of Theorem B. Let us describe in more detail how we proceed, and so pick an irreducible algebraic curve C⊂Poly3 containing infinitely many PCF polynomials. We may suppose that neither c0 nor c1 are persistently pre-periodic on C. By a theorem of McMullen [33, Lemma 2.1] this is equivalent to say that both critical points exhibit bifurcations at some (possibly different) points in C. There is a more quantitative way to describe the set of bifurcations using the Green function gc,a(z):=limn→∞13nlogmax{1,|Pn(z)|}. Indeed both functions g0(c,a):=gc,a(c0), g1(c,a):=gc,a(c1) are non-negative and pluri-subharmonic, and it is a fact [13, Section 5] that the support of the positive measure Δg0|C (resp. Δg1|C) is equal to the set of parameters where c0 (resp. c1) is unstable. The first step consists in proving that g0|C and g1|C are proportional, and this conclusion is obtained by applying an equidistribution result of points of small height due to Yuan [46] and Thuillier [44]. We first observe that C is necessarily defined over a number field K since it contains infinitely many PCF polynomials, so that we may introduce the functions g0,v,g1,v for all (not necessarily Archimedean) places v over K. These functions can now be used to build a one-parameter family of heights on C by setting   hs(p):=1deg(p)∑q,vmax{s0g0,v(q),s1g1,v(q)}, where s=(s0,s1)∈ℝ+2, and the sum ranges over all Galois conjugates q of p and over all places v over K. When s0 and s1 are positive integers, then we prove in Section 3 that the height hs is induced by a continuous semi-positive adelic metrization in the sense of Zhang on a suitable line bundle over C of positive degree, so that Thuillier–Yuan's theorem applies. This gives us sufficiently many restrictions on g0 and g1 which force their proportionality. The key arguments are Proposition 3.6 that is close in spirit to [3, Proposition 2.1 (3)], and the fact that the function max{g0,v,g1,v} is a proper continuous function on Poly3 for any place v. From the proportionality of g0 and g1 on a special curve, we are actually able to conclude the proof of Theorem B. This step is done in Section 5. We suppose by contradiction that our special curve C is an irreducible component of some Perm(λ) with λ≠0. Then each branch at infinity of C defines a cubic polynomial over the field of Laurent series ℂ((t)). We show that except when c0 or c1 is persistently preperiodic in C the multipliers of all periodic points are exploding on that branch by [31]. We then analyze the situation of a unicritical (that is, having a single critical point) polynomial in C and, computing the norm of the multiplier of its periodic points in a suitable field of residual characteristic 3, we are able to get the required contradiction. Let us come back to the proof of Theorem A. At this point, we have an irreducible algebraic curve C defined over a number field K and such that g0,v=g1,v at any place v over K. Recall that for any polynomial Pc,a there exists an analytic isomorphism near infinity conjugating the polynomial to the cubic monomial map. This isomorphism is referred to as the Böttcher coordinate φc,a of Pc,a. We prove that when c,a are defined over a number field then φc,a is a power series with coefficients in a number field whose domain of convergence is positive at any place, see Lemma 2.2 and Proposition 2.3. Building on an argument of Baker and DeMarco, we then show that outside a compact subset of the analytification of C (for any completion of K) the values of the Böttcher coordinates at c0 and c1 are proportional up to a root of unity (Theorem 4.1 (2)). The proof now takes a slight twist as we fix any polynomial P:=Pc,a that is not post-critically finite and for which (c,a) belongs to C(L) for some finite field extension L of K. We prove that any such polynomial admits a weak form of symmetry in the sense that there exists a curve ZP⊂ℙ1×ℙ1 that is stable by the map (P,P). To do so we apply [45, Theorem 1.5] as an alternative to the arguments of Baker and DeMarco in [3, Section 5.6]. In order to get a polynomial that commutes with P instead of a correspondence, we proceed as Baker and DeMarco and use Medvedev-Scanlon's result [39, Theorem 6.24] (see [40, Theorem 4.9] for another proof of this result). At this point, we have proved the following result that we feel is of independent interest. Theorem C. Pick any irreducible complex algebraic curve C⊂Poly3. Then the following assertions are equivalent: the curve C is special, for any critical point that is not persistenly pre-periodic on C, the set of PCF polynomials lying in C is equal to the set where this critical point is pre-periodic; the curve C is defined over a number field K and there exist integers (s0,s1)∈ℕ2∖{(0,0)} such that for any place v∈MK, we have   s0⋅g0,v=s1⋅g1,v, on the analytification of C over the completion of K w.r.t. the v-adic norm; the curve C is special, and for any sequence Xk⊂C of Galois-invariant finite sets of PCF polynomials with Xk≠Xl for l≠k, the probability measures μk equidistributed on Xk converge towards (a multiple of) the bifurcation measure Tbif∧[C] as k→∞; there exists a root of unity ζ, and integers q,m≥0 such that the polynomial Qc,a(z):=ζPc,am(z)+(1−ζ)c2 commutes with any iterate Pc,ak such that ζ3k=ζ, and Qc,a(Pc,aq(ci))=Pc,aq(cj) for some i,j∈{0,1} and all (c,a)∈C.     □ In (4) the current Tbif is defined as the ddc of the plurisubharmonic function g0+g1. Its support in ℂ2 is known to be equal to the set of unstable parameters, see for example, [20, Section 3]. Notice that for any curve C there exists a critical point which is not persistently pre-periodic on C since by [7] the set {g0=0}∩{g1=0} is compact in Aℂ2. In particular, the assertion (2) is consistent. To complete the proof of Theorem A, we analyze in more detail the possibilities for a cubic polynomial to satisfy the condition (5) in the previous theorem. Namely, we prove that the set of parameters admitting a non-trivial symmetry of degree 3m>1 is actually finite. Theorems A and C are proved in Section 7. We have deliberately chosen to write the entire article for cubic polynomials only. This simplifies the exposition, but many parts of the proof actually extend to a larger context. Let us briefly discuss the possible extensions and the limitations of our approach. All ingredients are present to prove Baker–DeMarco's conjecture for a curve in the space of polynomials of any degree d≥2. It is however not clear to the authors how to obtain the more precise classification of special curves in the same vein as in Theorem A. There are serious difficulties that lie beyond the methods presented here to handle higher dimensional special varieties V in Polyd. The main issue is the following. To apply Yuan's equidistribution theorem of points of small heights it is necessary to have a continuoussemi-positive adelic metric on an ample line bundle on a compactification of V, and we are at the moment very far from being able to check any of the three underlined conditions. Trying to understand special curves in the space of quadratic maps requires much more delicate estimates than in the case of polynomials. A first important step has been done by DeMarco, Wang and Ye in a recent article [16]. 2 The Böttcher Coordinate of a Polynomial In this section, K is any complete metrized field of characteristic zero containing a square-root ω of 13. It may or may not be endowed with a non-Archimedean norm. If X is an algebraic variety over K, then Xan denotes its analytification as a real-analytic or a complex variety if K is Archimedean, and as a Berkovich analytic space when K is non-Archimedean (see e.g. [2, Section 3.4–5]). 2.1 Basics As in the introduction, we denote by Poly3≃A2 the space of cubic polynomials defined by   Pc,a(z):=13z3−c2z2+a3. (1) It is a branched cover of the parameter space of cubic polynomials with marked critical points. The critical points of Pc,a are given by c0:=c and c1:=0. For a fixed (c,a)∈K2 the function 13log+|Pc,a(z)|−log+|z| is bounded on AK1,an so that the sequence 13nlog+|Pc,an(z)| converges uniformly to a continuous sub-harmonic function gc,a(z) that is called the Green function of Pc,a. We shall write g0(c,a):=gc,a(c0), g1(c,a):=gc,a(c1), and   G(c,a):=max{g0(c,a),g1(c,a)}. Proposition 2.1. The function G(c,a) extends continuously to the analytification AK2,an, and there exists a constant C=C(K)>0 such that   supAK2,an|G(c,a)−log+max{|a|,|c|}|≤C, and this constant vanishes when the residual characteristic of K is at least 5.     □ Proof. A proof of this fact is given in [7, Section 4] (see also [20, Section 6] for a more detailed proof) in the Archimedean case. When the normed field is non-Archimedean, it is proved in [21, Proposition 2.5] that the sequence hn:=max{13nlog+|Pc,an(c0)|,13nlog+|Pc,an(c1)|} converges uniformly on bounded sets in K2 to G(c,a). Since hn extends continuously to AK2,an, it follows that G too. The rest of the proposition also follows from op. cit.     ▪ 2.2 Expansion of the Böttcher coordinate For any cubic polynomial P∈K[z], we let the Böttcher coordinate of P be the only formal power series φ satisfying the equation   φ∘P(z)=φ(z)3 (2) which is of the form   φ(z)=ωz+α+∑k≥1akz−k, (3) with α,ak∈K for all k≥1. Recall that ω2=1/3. Lemma 2.2. Given any (c,a)∈K×K, the Böttcher coordinate φc,a(z) of the cubic polynomial Pc,a:=z33−c2z2+a3 exists, is unique, and satisfies   φc,a(z)=ω(z−c2)+∑k≥1ak(c,a)z−k, where   ak(c,a)∈ℤ[ω,12][c,a] with deg(ak)=k+1. (4) Moreover, the 2-adic (resp. 3-adic) norm of the coefficients of ak are bounded from above by 2k+1 (resp. 3k/2).     □ Proof. The defining equation (2) reads as follows:   (ω(z−c2)+∑k≥1ak(c,a)z−k)3 =ω(z33−c2z2+a3−c2)+∑k≥13kak(c,a)z3k(1−3c2z+3a3z3)k An immediate check shows that terms in z3 and z2 are identical on both sides of the equation. Identifying terms in z yields   3ω3(c2/4)+3ω2a1=0, so that a1=−ω4c2, whereas identifying constant terms, we get   3ω2a2+6ω2(−c/2)a1+ω3(−c3/8)=ω(a3−c/2) hence   a2=−5ω24c3+13ω(a3−c2). This shows (4) for k=1,2, since ω−1=3ω. We now proceed by induction. Suppose (4) has been proven for k. Identifying terms in z−(k−1) in the equation above, we get   3ω2ak+1−3cω2ak+3ω24c2ak−1  +ω∑i+j=kaiaj−ωc2∑i+j=k−1aiaj+∑i+j+l=k+1aiajal =∑l≥13lal[(1+3c2z+a3z3)−l]k+1−3l, where [(1+3c2z+a3z3)−l]j denotes the coefficient in z−j of the expansion of (1+3c2z+a3z3)−l in power of z−1. Observe that this coefficient belongs to ℤ[12][c,a], has 2-adic norm ≤2l, and is a polynomial in c,a of degree at most j. It follows that the polynomial   al(c,a)[(1+3c2z+a3z3)−l]k+1−3l is of degree at most k+1−3l+l+1=k+2−2l<k+1. The induction step is then easy to complete using again ω−1=3ω.     ▪ 2.3 Extending the Böttcher coordinate Recall that G(c,a)=max{g0(c,a),g1(c,a)}. Proposition 2.3. There exists a constant ρ=ρ(K)≥0 such that the Böttcher coordinate of Pc,a is converging in {z,log|z|>ρ+G(c,a)}. There exists another constant τ=τ(K)≥0 such that the map (c,a,z)↦φc,a(z) extends as an analytic map on the open set   {(c,a,z)∈AK2,an×AK1,an,gc,a(z)>G(c,a)+τ}, and φc,a defines an analytic isomorphism from Uc,a:={gc,a>G(c,a)+τ} to AK1,an∖D(0,eG(c,a)+τ)¯ satisfying the equation (2) on Uc,a. We have   gc,a(z)=log|φc,a(z)|K  on Uc,a. (5) Finally, τ=0 except if the residual characteristic of K is equal to 2 or 3.     □ We shall use the following lemma which follows easily from e.g. [21, Proposition 2.3]. Lemma 2.4. There exists a constant θ=θ(K)≥0  supAK1,an|gc,a(z)−log+|z||≤θ. Moreover, θ is equal to 0 except if the norm on K is Archimedean or the residual characteristic of K is equal to 2 or 3.     □ Proof of Proposition 2.3. Assume first that K is Archimedean, and set τ=0. In that case most of the statements are proved in [17] (see also [7, Section 1]). In particular, φc,a(z) is analytic in a neighborhood of ∞ and extends to Uc,a by invariance and defines an isomorphism between the claimed domains. It is moreover analytic in c,a,z. To estimate more precisely the radius of convergence of the power series (3), we rely on [7, Section 4] as formulated in [20, Section 6]. First choose C=CK>0 such that G(c,a)>log+max{|a|,|c|}−C. Then log|z|>C+G(c,a) implies |z−c2|>max{1,|a|,|c|}−|c2|≥12max{1,|a|,|c|} hence log|z−c2|>G(c,a)−log2, so that gc,a(z)>log|z−c2|−log4>G(c,a), and φ converges in {z,log|z|>G(c,a)+ρ} with ρ:=C as required. From now on, we assume that the norm on K is non-Archimedean. When the residual characteristic of K is different from 2 and 3, then (4) implies |ak|≤max{1,|c|,|a|}k+1 so that φ converges for |z|>max{1,|c|,|a|}, and log|φ(z)|=log|z|. Recall that we have G(c,a)=logmax{1,|c|,|a|} by Proposition 2.1 so that one can take ρ=0. Pick any z such that g(z)>G(c,a), and observe that |Pn(z)|→∞. Then we get   gc,a(z)=limn→∞13nlog|Pn(z)|=limn→∞13nlog|φ(Pn(z))|=log|φ(z)|=log|z|. (6) In particular the set {g>G(c,a)} is equal to AK1,an∖D(0,eG(c,a)¯), and φ is an analytic map from that open set onto itself. It is an isomorphism since log|φ(z)|=log|z| as soon as g(z)>G(c,a). The proposition is thus proved in this case with τ=0. In residual characteristic 2, |ak|≤(2max{1,|c|,|a|})k+1 whence φ converges for |z|>2max{1,|c|,|a|}, and as above log|φ|=log|z| in that range. Recall that G(c,a)−log+max{|c|,|a|}≥C=C(K), so that log|z|>G(c,a)+log2−CK implies |z|>2max{1,|c|,|a|}, which proves that the power series (3) converges for log|z|>G(c,a)+ρ with ρ=log2−C. Set τ:=ρ+θ where θ is the constant given by Lemma 2.4. Using log|φ(z)|=log|z| as above, we get that φc,a defines an analytic isomorphism from Uc,a:={gc,a>G(c,a)+τ} to AK1,an∖D(0,eG(c,a)+τ)¯. In residual characteristic 3, |ak|≤(31/2max{1,|c|,|a|})k+1 whence φ converges for |z|>31/2max{1,|c|,|a|}, and log|φ|=log|ωz| in that range. Recall that G(c,a)−log+max{|c|,|a|}≥C=C(K), so that log|z|>G(c,a)+log31/2−CK implies |z|>31/2max{1,|c|,|a|}, which proves that the power series (3) converges for log|z|>G(c,a)+ρ with ρ=log31/2−C. We conclude the proof putting τ:=ρ+θ as before.     ▪ Remark. It is possible to argue that τ=0 also in residual characteristic 2. Although we do not know the optimal constant τ in residual characteristic 3, the Böttcher coordinate is likely not to induce an isomorphism from {gc,a>G(c,a)} to AK1,an∖D(0,eG(c,a))¯.     □ 3 Curves in Poly3 In this section, we fix a number field K containing a square-root ω of 13 and take an irreducible curve C in Poly3 that is defined over K. Our aim is to build suitable height functions on C for which the distribution of points of small height can be described using Thuillier–Yuan's theorem. Our main statement is Theorem 3.12 below. Recall that given any finite set S of places of K containing all Archimedean places, OK,S denotes the ring of S-integers in K that is of elements of K of v-norm ≤1 for all v∉S. We also write Kv for the completion of K w.r.t. the v-adic norm. 3.1 Adelic series A formal power series ∑nanzn is said to be adelic on K if there exists a finite set S of places on K such that an∈OK,S for all n∈ℕ; and for each place v on K the series has a positive radius of convergence rv:=limsupn→∞|an|v−1/n>0. Observe that rv=1 for all but finitely many places. Lemma 3.1. Suppose α(t)=∑nantn is an adelic series with a0=0 and a1≠0. Then there exists an adelic series β such that β∘α(t)=t.     □ Proof. Suppose an∈OK,S for all n, and write β(t)=∑nbntn. The equation β∘α(t)=t amounts to b0=0, b1=a1−1, and the relations   bna1n+∑1≤k≤n−1bk[(∑j≤najtj)k]n=0, for any n≥2 where [⋅]n denotes the coefficient in tn of the power series inside the brackets. It follows that bn∈OK,S′ for all n where S′ is the union of S and all places v for which |a1|v>1. The convergence of the series follows from Cauchy–Kowalewskaia's method of majorant series or from the analytic implicit function theorem, see [12] and [41, p. 73].     ▪ Lemma 3.2. Pick k∈ℕ*, and suppose α(t)=∑n≥kantn is an adelic series with ak≠0. Then there exists an adelic series β such that β(t)k=α(t).     □ Proof. As in the previous proof, suppose an∈OK,S for all n, and write β(t)=b1t+∑n≥2bntn. We get b1k=a1, and for all n≥2  an=kb1k−1bn−k+Pn(b1,…,bn−k−1), where Pn is a polynomial with integral coefficients. This time all coefficients bn belong to a finite extension of K containing a fixed k-th root of a1, and S′ is the union of S and all places v such that |kb1k−1|v<1. The analyticity of the series is handled as in the previous proof.     ▪ Lemma 3.3. Pick k∈ℕ*, and suppose α(t)=∑n≥kantn is an adelic series with ak≠0. Then there exists an adelic series β such that α∘β(t)=tk.     □ Proof. The equation α∘β(t)=tk is equivalent to   (1+∑j≥2ajtj−1)k(1+∑l≥1αl(t+∑i≥2aiti)l)=1. Identifying terms of order tn, one obtains   kan+1+[(1+∑2≤j≤najtj−1)k(1+∑l≥1αl(t+∑1≤i≤naiti)l)]n=0 which shows that β is unique, has coefficients in OL,S′ where S′ contains S and all places at which |k|v<1. The fact that β is analytic at all places is a consequence of the inverse function theorem and the fact that the power series t↦(1+t)1/n:=1+1nt+(1/n)(1/n−1)2t2+O(t3) has a positive radius of convergence.     ▪ We shall also deal with adelic series at infinity which we define to be series of the form α(z)=∑0≤k≤Nbkzk+∑k≥1akzk with N∈ℕ, bk,ak∈OK,S and ∑k≥1aktk is an adelic series. Observe that this is equivalent to assuming that α(t−1)−1 is an adelic series. 3.2 Puiseux expansions We shall need the following facts on the Puiseux parameterizations of a curve defined over K. These are probably well-known but we include a proof for the convenience of the reader. Proposition 3.4. Suppose P∈K[x,y] is a polynomial such that P(0,0)=0 and P(0,y) is not identically zero. Denote by n:D^→D:={P=0} the normalization map, and pick any point c∈n−1(0)∈D^. Then one can find a finite extension L of K, a finite set of places S of L, a positive integer n>0, and an adelic series β(t)∈OL,S[[t]] such that there is an isomorphism of complete local rings OD^,c^≃L[[t]]; the formal map t↦(tn,β(t)) parameterizes the branch c in the sense that x(n(t))=tn, and y(n(t))=β(t).     □ A branch of D at the origin is by definition a point in n−1(0). Proof. We first reduce the situation to the case D is smooth at 0. To do so we blow-up the origin X1→A2 and let D1 be the strict transform of D. Since D^ is normal the map n lifts to a map n1:D^→D1, and we let p1 be the image of c in D1. In the coordinates (x,y)=(x′,x′y′) (or (x'y', y')) the point p1 has coordinates (0,y1) where y1 is the solution of a polynomial with values in K hence belongs to an algebraic extension of this field. We may thus choose charts (x,y)=(x′,(x′+c)y′) (or (x′(y′+c),x′)) with c∈K¯ such that c is now a branch of D1={P1=0} at the origin, and P1∈K¯[x′,y′]. We iterate this process of blowing-up to build a sequence of proper birational morphisms between smooth surfaces Xi+1→Xi, i=1,…,N until we arrive at the following situation for X:=XN: the strict transform C of D by π:X→A2 is smooth at a point p∈π−1(0) and intersects transversally the exceptional locus of π. The normalization map n:D^→D lifts to a map m:D^→C and the image of c by m is equal to p. Finally there exist coordinates z,w centered at p such that (x,y)=π(z,w)=(A(z,w),B(z,w)) with A,B∈K¯[z,w], the exceptional locus of π contains {z=0}, and C={R(z,w):=w−za(z)−wQ(z,w)=0} where a∈K¯[z], Q∈K¯[z,w] and Q(0,0)=0. Fix an algebraic extension L of K and S finitely many places of L such that A,B, R have their coefficients in OL,S. We now look for a power series γ(t)=∑k≥1γktk such that R(t,γ(t))=0. Its coefficients satisfy the relations   γk=[t2a(t)]k+[(∑j=1k−1γjtj)Q(t,∑j=1k−1γjtj)]k which implies that γ exists, is unique, and all its coefficients belongs to OL,S. It follows from the analytic implicit function theorem, that γ is also analytic as a power series in Lv[[t]] for any place v. Let us now consider the two power series (α(t),δ(t)):=π(t,γ(t)). They both belong to OL,S, are analytic at any place, and we have P(α(t),δ(t))=0. Since P(0,y) is not identically zero, we may write α(t)=tn(a+∑k≥1αktk) for some n>0 and a≠0. Replacing L by a suitable finite extension, and t by a′t for a suitable a′ we may suppose that a=1 and αk∈OL,S for all k. By Lemma 3.3, there exists an invertible power series a^(t)=t+∑k≥2aktk that is analytic at all places with coefficients ak∈OL,S and such that α°a^(t)=tn. Once this claim is proved one sets β(t):=δ°a^(t), so that π(a^(t),γ(a^(t)))=(tn,β(t)). Since m is injective and maps the smooth point c∈D^ to the smooth point p∈C, it induces an isomorphism of complete local rings OC,p^≃OD^,c^. Observe that the complete local ring OC,p^=L[[z,w]]/⟨R⟩ is isomorphic to L[[t]] by sending the class of a formal series Φ to Φ(t,γ(t))). Composing with the isomorphism of L[[t]] sending t to a^(t), we get an isomorphism OD^,c^≃L[[t]] such that (x(n(t)),y(n(t)))=π(n(t))=π(a^(t),γ(a^(t)))=(tn,β(t)) as required.     ▪ 3.3 Branches at infinity of a curve in Poly3 Consider an irreducible affine curve C⊂Poly3 defined over a number field K. We denote by Poly3¯≃ℙ2 the natural compactification of Poly3≃A2 using the affine coordinates (c,a). Let C¯ be the Zariski closure of the curve C in Poly3¯, and n:C^→C¯ be its normalization. A branch at infinity of C is a point in C^ lying over C¯∖C. Proposition 3.5. There exists a finite extension L of K such that the following holds. For any branch c of C at infinity there is an isomorphism of complete local rings OC^,c^≃L[[t]] such that c(n(t)),a(n(t)) are adelic series at infinity.     □ Proof. Pick a branch at infinity c of C. Let p* be the image of c in Poly3¯≃ℙ2. It is given in homogeneous coordinates by p*=[c*:a*:0] and since C is defined over K we may assume c*,a* are algebraic over K. To simplify the discussion we shall assume that c*=1 so that p*=[1:a*:0] (otherwise, p*=[0:1:0] and the arguments are completely analoguous). Let d be the degree of a defining equation P∈K[c,a] of C. Observe that Q(τ,α):=τdP(1τ,ατ−a*) is a polynomial vanishing at (0,0) such that Q(0,α) is not identically zero. Note that {Q=0} can be identified to an open Zariski subset of the completion of {P=0} in Poly3¯, and c with a branch of {Q=0} at the origin. Apply Proposition 3.4 to this branch c. We get a finite extension L, a finite set of places S of L containing all archimedean ones, a positive integer n, an isomorphism of complete local ring OC^,c≃L[[t]], and a power series β∈OL,S[[t]] that is analytic at all places such that α(n(t))=β(t) and τ(n(t))=tn. It follows that c(n(t))=t−n, and a(n(t))=t−nβ(t)−a*∈OL,S[[t]].     ▪ 3.4. Estimates for the Green functions on a curve in Poly3 In this section, we fix an irreducible curve C in Poly3 defined over a number field K and let L be a finite extension of K for which Proposition 3.5 applies. Fix a place v of L, and let g0,v(c,a) be the function g0,v evaluated at c,a in the completion Lv of L with respect to the v-adic norm. By [20] and [21, Proposition 2.4], the function g0,v is the uniform limit on compact sets of 13nlog+|Pc,an(c0)|v. It follows that its lift to the normalization of C is sub-harmonic (in the classical sense when v is Archimedean and in the sense of Thuillier [44] when v is non-Archimedean). To simplify notations, we also write g0,v(t):=g0,v(c(n(t)),a(n(t))) where the adelic series at infinity c(n(t)) and a(n(t)) are given as above. Proposition 3.6. For each branch c of C at infinity, one of the following two situations occur. For any place v of L, the function g0,v(t) extends as a locally bounded subharmonic function through c. There exist a finite set of place S of L, and two constants a(c)∈ℚ+* and b(c)∈OL,S such that g0,v(t)=a(c)log|t|v−1+log|b(c)|v+o(1) for any place v on L.     □ Remark. This key result is very similar to [3, Proposition 2.1]. Ghioca and Ye have proved that g0,v(t) actually extends to a continuous function at t=0 in case 1. We also refer to [14, Proposition 3.1] for a version of this result in the case of rational maps.     □ Notation. We endow the field L((t)) with the t-adic norm so that for any Laurent series Q=∑aktk we have |Q|t:=e−ordt(Q) with ordt(Q)=min{k,ak≠0}. The resulting valued field is complete and non-Archimedean. In order to avoid confusion, we denote by P(z)∈L((t))[z] the cubic polynomial induced by the family (Pc(n(t)),a(n(t)))t. Observe that the critical points of P are given by c0 and c1 which correspond to the adelic series at infinity 0 and c(n(t)), respectively.     □ Proof of Proposition 3.6. For each q∈ℕ*, we set eq:=|Pq(c0)|t, so that either the sequence {eq}q∈ℕ is bounded (that is, c0 belongs to the filled-in Julia set of P) or eq→∞ (exponentially fast). Suppose we are in the former case, and consider the sequence of subharmonic functions 13qlog+|Ptq(0)|v defined on a punctured disk Dv* centered at 0 in ALv1,an. Since 13qlog+|Ptq(0)|v=log+eq3qlog|t|v−1+O(1), the function   hq:=13qlog+|Ptq(0)|v−log+eq3qlog|t|v−1 is subharmonic on Dv* and locally bounded near 0. It thus extends as a subharmonic function to Dv by the next lemma. Lemma 3.7. Any subharmonic function on Dv* that is bounded from above in a neighborhood of the origin is the restriction of a subharmonic function on Dv.     □ Proof. In the Archimedean case, this follows from [28, Theorem 3.4.3]. Let us explain how to adapt these arguments to the non-Archimedean setting. Let h be a subharmonic function on Dv*, and suppose it is non-positive. For each integer n consider the function hn:=h+1nlog|t| with the convention hn(0)=−∞. Observe that hn is the pointwise decreasing limit of the sequence of subharmonic functions max{hn,−A} as A→∞, hence is subharmonic by [44, Proposition 3.1.9]. Letting n→∞, we get an increasing sequence of subharmonic functions that is locally bounded and converging pointwise on Dv* to h. The upper-semicontinuous regularization h* of limnhn is thus subharmonic on Dv* by [44, Proposition 3.1.9] and extends h as required.     ▪ Since 13qlog+|Ptq(0)|v converges uniformly on compact subsets in Dv* to g0,v, hq is uniformly bounded from above on its boundary, hence everywhere by the maximum principle. It follows from Hartog's theorem (see e.g., [27, Theorem 1.6.13] in the Archimedean case, and [22, Proposition 2.18] in the non-Archimedean case) that hq converges (in Lloc1 in the Archimedean case, and pointwise at any non-rigid point in the non-Archimedean case) to a subharmonic function, hence g0,v is subharmonic on Dv. But g0,v is non-negative so that (1) holds. Suppose that eq→∞. Recall that c(n(t)) and a(n(t)) are adelic series at infinity that belong to t−nOL,S[[t]] for a suitable integer n≥1. Write φt:=φPc(n(t)),a(n(t)). Lemma 3.8. There exists an integer q≥1 such that for any place v of L, there exists ϵ>0 such that Ptq(c0) belongs to the domain of convergence of φt for any |t|v<ϵ.     □ Proof. Indeed Ptq(c0) is an adelic series at infinity having a pole of order logeq. On the other hand, we have   G(t):=G(c(n(t)),a(n(t)))≤logmax{|c(n(t))|,|a(n(t))|}+C               ≤nlog|t|−1+O(1) by Proposition 2.1. By assumption we may take logeq to be as large as we want so that log|Ptq(c0)|v−G(t)→∞ for any fixed place v when |t|v→0. We conclude by Proposition 2.3.     ▪ Our objective is to estimate φt(Ptq(c0)). Recall from Lemma 2.2 that   φc,a(z)=ω(z−c2)+∑k≥1ak(c,a)z−k, with ak∈ℤ[ω,12][c,a] of degree ≤k+1. It follows that   ak:=ak(c(n(t)),a(n(t)))∈t−n(k+1)OL,S[[t]], so that one can define   φP(z):=φc(n(t)),a(n(t))(z)=ω(z−c(n(t))2)+∑k≥1akz−k as an element of the ring t−nzOL,S((t))[[(tnz)−1]]. On the other hand, Pc,aq(c0) is a polynomial in c,a of degree ≤3q with coefficients in ℤ[12,13] hence, if   c0:=c0(n(t)) and Pq(z):=Pc(n(t)),a(n(t))q(z), we have Pq(c0)∈t−3qnOL,S[[t]], so that   ak(Pq(c0))k∈t3qnk−n(k+1)OL,S[[t]]⊂tnkOL,S[[t]]. (7) It follows that Θ:=∑k≥1ak(Pq(c0))k converges as a formal power series and belongs to tnOL,S[[t]]. Observe that Lemma 3.8 shows that Θ is convergent at all places hence defines an adelic power series. Fix a place v of L and choose |t|v small enough. Then we get   φt(Ptq(c0))=ω(Ptq(0)−c(n(t))2)+Θ(t)=ω(Ptq(0)−c(n(t))2)+o(1). (8) By (8), for |t|v small enough, one obtains   g0,v(t)=13qlog|φt(Ptq(0))|v=13qlog|ω(Ptq(0)−12tn)|v+o(1)=13qlog|∑0≤k≤n0bk,0tk|v+o(1)==n03qlog|t|v−1+log|∑0≤k≤n0bk,0tn0−k|v+o(1) where bk,0∈OL,S, and bn0,0≠0. And the proof is complete with a(c):=n03q, and b(c)=bn0,0. Proposition 3.9. Fix any two positive integers s:=(s0,s1), and for any place v define   gs,v(c,a):=max{s0g0,v(c,a),s1g1,v(c,a)}. (9) Then there exists an integer q≥1 such that   gs,v(c,a)=13qmax{s0log+|Pc,aq(c0)|,s1log+|Pc,aq(c1)|} (10) for all but finitely many places.     □ Proof. During the proof S is a finite set of places on L that contains all Archimedean places and all places of residual characteristic 2 and 3. Pick any v∉S, and recall from [21, Proposition 2.5] that Gv(c,a)=log+max{|c|v,|a|v}. Suppose first that gs,v(c,a)=0. Then gc,a,v(c0)=gc,a,v(c1)=0 and Gv(c,a)=0 so that 13qlog+|Pc,aq(c0)|v=13qlog+|Pc,aq(c1)|v=0 for all q, and (9) holds in that case. Pick q large enough such that 3q>max{s1s0,s0s1}. Suppose now that 0<gs,v(c,a)=s0g0,v(c,a) so that s0g0,v(c,a)≥s1g1,v(c,a). Then   gc,a,v(Pc,aq(c0))=3qg0,v(c,a) ≥3qmin{s1s0,1}max{g0,v(c,a),g1,v(c,a)}>Gv(c,a). By (6), we get   gs,v(c,a)=s0g0,v(c,a)=s03qg0,v(Pc,aq(c0))=s03qlog+|Pc,aq(c0)|v. Now observe that either Pc,aq(c1) falls into the domain of definition of φc,a that is, log|Pc,aq(c1)|v>Gv(c,a) and g1,v(c,a)=13qlog+|Pc,aq(c1)|v, so that   gs,v(c,a)=max{s0g0,v(c,a),s1g1,v(c,a)}=13qmax{s0log+|Pc,aq(c0)|v,s1log+|Pc,aq(c1)|v}, as required. Or we have   s13qlog+|Pc,aq(c1)|v≤s13qlog+max{|a|v,|c|v}≤s0g0,v(c,a), and again (10) holds. We complete the proof by arguing in the same way when gs,v(c,a)=s1g1,v(c,a).     ▪ 3.5 Adelic semi-positive metrics on curves in Poly3 We fix a number field L and a finite set S of places of this field that contains all Archimedean places and all places of residual characteristics 2 and 3. We also assume that Propositions 3.5, 3.6, and 3.9 are all valid for these choices. Fix any pair of positive integers s0,s1∈ℕ*. For each place v, introduce the function   gs,v(c,a):=max{s0⋅g0,v(c,a),s1⋅g1,v(c,a)}, as in the previous section. Pick a branch at infinity c and choose parameterizations such that Proposition 3.6 is valid for g0,v and g1,v. Observe that   Gv(t)=max{g0,v(t),g1,v(t)}→∞ as t→0 by Proposition 2.1 so that either g0,v or g1,v tends to infinity near t=0. Since s0 and s1 are both positive, we obtain the existence of a(c)∈ℚ+* and b(c)∈OL,S such that   gs,v(t)=a(c)log|t|v−1+log|b(c)|v+o(1). (11) We replace the integers s0,s1 by suitable multiples such that the constants a(c) become integral (for all branch c), and we introduce the divisor D:=∑a(c)[c] on C^ where the sum is taken over all branches at infinity of C. Pick a place v, an open subset U⊂C^an,v and a section σ of the line bundle OC^(D) over U. By definition σ is a meromorphic function on U whose divisor of poles and zeroes satisfies div(σ)+D≥0. We set |σ|s,v:=|σ|ve−gs,v. Recall the notion of semi-positive metrics in the sense of Zhang from [11, Section 1.2.8 & 1.3.7]. We are now in position to prove Lemma 3.10. The metrization |⋅|s,v on the line bundle OC^(D) is continuous and semi-positive for any place v. The collection of metrizations {|⋅|s,v}v is adelic.     □ Proof. For any place v, and for any local section σ of the line bundle OC^(D) the function |σ|s,v is continuous by (11), therefore the metrization |⋅|s,v is continuous. Since gs,v is subharmonic on Cv,an, for any local section σ the function −log|σ|s,v is subharmonic on Cv,an. As it extends continuously to C^an,v, Lemma 3.7 implies that −log|σ|s,v is subharmonic on C^an,v. When v is Archimedean, the metrization is thus semi-positive by definition. When v is non-Archimedean the metrization is semi-positive by the next lemma. Finally the collection of metrizations {|⋅|s,v}v is adelic thanks to Proposition 3.9 and [21, Section 2.3]. Lemma 3.11. Suppose L→X is an ample line bundle on a smooth projective curve over a metrized field. Let |⋅| be any continuous metrization on L that is subharmonic in the sense that for any local section σ the function −log|σ| is subharmonic. Then the metrization |⋅| is semi-positive in the sense of Zhang.     □ Remark. This result holds true in arbitrary dimension over any field of the form L((t)) endowed with the t-adic norm where L has characteristic zero by [6].     □ Proof. One needs to show that |⋅| is the uniform limit of a sequence of semi-positive model metrics. By [44, Théorème 3.4.15], there exists an inductive set I and an increasing family of semi-positive model metrizations |⋅|i that are pointwise converging to |⋅|. Since |⋅| is continuous and X is compact, Dini's theorem applies and shows that the convergence is uniform. Observe that the notion of semi-positivity used in op. cit. for model metrics coincides with the one in [47] as explained in [44, Section 4.3.2].     ▪ We have thus obtained Theorem 3.12. Pick any positive integers s0,s1>0. Then there exists a positive integer n≥1, and a non-zero effective and integral divisor D on C^ such that the collection of subharmonic functions   gs,v(c,a):=max{ns0⋅g0,v(c,a),ns1⋅g1,v(c,a)}, (c,a)∈Cv,an induces a semi-positive adelic metrization on the line bundle OC^(D).     □ Remark. Since the metrization |⋅|s,v is continuous its curvature form (see [11]) does not charge any point, and is given by the pull-back by n of the positive measure Δgs,v restricted to the set of regular points on C, see [44, Section 3.4.3]. To simplify notations we shall simply write this curvature form as Δgs,v.     □ Remark. The line bundle OC^(D) is defined over the same number field as C.     □ Remark. It is likely that gs,v defines a semi-positive adelic metrization on an ample line bundle over a suitable compactification of Poly3, but this seems quite delicate to prove for arbitrary s=(s0,s1)∈(ℕ*)2.     □ 4. Green Functions on Special Curves This section is devoted to the proof of Theorem 4.1 below. If K is a number field, and v a place of K, recall the definition of τv=τ(Kv) from Proposition 2.3, and that τv=0 if the residual characteristic of K is larger than 5. Theorem 4.1. Let C be an irreducible curve in the space Poly3 of complex cubic polynomials parameterized as in (1). Suppose that C contains infinitely many post-critically finite parameters and that neither c0 nor c1 is persistently pre-periodic. Then the following holds. The curve C is defined over a number field K and there exist positive integers s0,s1 such that for any place v of K  s0g0,v(c,a)=s1g1,v(c,a) for all (c,a)∈Cv,an. For any branch c of C at infinity, there exists an integer q≥1 and a root of unity ζ such that for any place v of K, one has   (φc,a(Pc,aq(c0)))s0=ζ⋅(φc,a(Pc,aq(c1)))s1 (12) on the connected component of {g0,v>τv/s0}={g1,v>τv/s1} in Cv,an whose closure in C^ contains c.     □ A remark is in order about the second assertion of the theorem. Remark. We shall prove that for any parameter on the connected component {g0,v>τv/s0}={g1,v>τv/s1} in Cv,an whose closure in C^ contains c, the two points Pc,aq(c0) and Pc,aq(c1) belong to the domain of definition of the Böttcher coordinate φc,a for q large so that (12) is consistent. We shall prove that (12) holds as an equality of adelic series at infinity. 4.1. Green functions are proportional The set of post-critically finite polynomials is a countable union of varieties   Vn,m:={Pc,an0+m0(c0)=Pc,an0(c0)}∩{Pc,an1+m1(c1)=Pc,an1(c1)} with n=(n0,n1)∈ℕ2 and m=(m0,m1)∈(ℕ*)2, and each Vn,m is cut out by two polynomial equations with coefficients in ℤ[12,13]. Since Vn,m(ℂ) are all contained in a fixed compact set by [7] (see also [20, Proposition 6.2]), it is a finite set, hence all its solutions are defined over a number field. It follows that C is an irreducible curve containing infinitely many algebraic points (cn,an). Let Q∈ℂ[c,a] be a defining equation for C with at least one coefficient equal to 1 and pick σ an element of the Galois group of ℂ over the algebraic closure of ℚ. Then Q∘σ vanishes also on {(cn,an)} hence everywhere on C, and therefore Q∘σ=λQ for some λ∈ℂ*. Since one coefficient of Q is 1, we get λ=1 and Q∈K[c,a] for a number field K. Recall that we denote by n:C^→C¯ the normalization of the completion C¯ of C in Poly3¯≃ℙ2. Pick any pair of positive integers s=(s0,s1) and scale them such that Theorem 3.12 applies with n=1. This gives us a non-zero effective divisor Ds supported on C^∖n−1(C). Replacing s by a suitable multiple, we may suppose that it is very ample and pick a rational function ϕ on C^ whose divisor of poles and zeroes is greater or equal to −Ds. Observe in particular that ϕ is a regular function on n−1(C) that vanishes at finitely many points. Consider the height hs induced by the semi-positive adelic metrics given by gs,v, see Theorem 3.12. If (c,a) is a point in n−1(C) that is defined over a finite extension K, denote by O(c,a) its orbit under the action of the absolute Galois group of K, and by deg(c,a) the cardinality of this orbit. Fix a rational function ϕ as above that is not vanishing at (c,a) (this exists since −Ds is very ample). Let MK be the set of places of K. By [11, Section 3.1.3], since ϕ(c,a)≠0 we have   hs(c,a)=1deg(c,a)∑O(c,a)∑v∈MK−log|ϕ|s,v(c′,a′)=1deg(c,a)∑O(c,a)∑v∈MK(gs,v−log|ϕ|v)(c′,a′)=1deg(c,a)∑O(c,a)∑v∈MKgs,v(c′,a′)≥0 where the last equality follows from the product formula. We now estimate the total height of the curve C^ using [11, (1.2.6) & (1.3.10)]. Choose any two meromorphic functions ϕ0,ϕ1 such that div(ϕ0)+Ds and div(ϕ1)+Ds are both effective with disjoint support included in n−1(C). Let σ0 and σ1 be the associated sections of OC^(Ds). Let ∑ni[ci,ai] be the divisor of zeroes of σ0, and ∑n′j[c′j,a′j] be the divisor of zeroes of σ1. Then   hs(C^)=∑v∈MK(div^(σ0)⋅div^(σ1)|C^)v=∑inihs(ci,ai)−∑v∈MK∫C^log|σ0|s,vΔgs,v=∑v∈MK∫C^gs,vΔgs,v≥0, where the third equality follows from Poincaré–Lelong formula and writing log|σ0|s,v=log|ϕ|v−gs,v with ϕ∈K(C) defining the section σ0. The formula for the height of a closed point implies that for all post-critically finite polynomials Pcn,an we have hs(cn,an)=0. Since PCF polynomials are Zariski dense in C, the essential minimum of hs is non-positive. By the arithmetic Hilbert–Samuel theorem (see [44, Théorème 4.3.6], [1, Proposition 3.3.3], or [47, Theorem 5.2]), we get hs(C^)=0 hence we may apply Thuillier–Yuan's theorem (see [44, 46] and [10, Théorème 4.2]). It follows that the sequence of probability measures μn,v that are equidistributed on O(cn,an) in C^v,an converges to a probability measure μ∞,v that is proportional to Δgs,v. We may thus write μ∞,v=w(s)Δgs,v where w(s)∈ℝ+* is equal to the inverse of the mass of Δgs,v, that is, to deg(Ds)−1. Applying the arguments of the previous paragraphs to all choices of positive integers (s0,s1) (suitably scaled so that Theorem 3.12 applies), we conclude from the equidistribution of PCF polynomials that the measure μ∞,v is independent of s. We now observe that gs,v is homogeneous in s (i.e., gτs,v=τgs,v for any τ∈ℝ+*), and continuous with respect to this parameter. It follows that w(s) is also continuous on (ℝ+*)2, and μ∞,v=w(s)Δgs,v for all s∈(ℝ+*)2. From now on we fix an Archimedean place v. We shall treat the non-Archimedean case latter. We work in n−1(Cv,an) which is the complement of finitely many points in the analytification of the smooth projective curve C^v,an. To simplify notation we write g0,v, g1,v instead of g0,v∘n,g1,v∘n. Recall that by [20, Theorem 2.5] (see also [33, Theorem 2.2] or [14, Theorem 1.1]) the equality g0,v=0 on n−1(Cv,an) implies that c0 is persistently pre-periodic. Since we assumed that both c0 and c1 are not persistently pre-periodic, the functions g0,v and g1,v are not identically zero on n−1(Cv,an). Recall also that g0,v is harmonic where it is positive and that the support of Δg0,v is exactly the boundary of {g0,v=0} (see e.g. [20, Proposition 6.7]). In particular Δg0,v is a non-zero positive measure, and its mass is finite by Proposition 3.6. Observe now that gs,v→g0,v uniformly on compact sets when s tends to (1,0), hence Δgs,v→Δg0,v and Δg0,v=t0μ∞,v for some positive t0. In the same way, we get Δgs,v→Δg1,v as s→(0,1) which implies that the three positive measures μ∞,v, Δg0,v and Δg1,v are proportional. We may thus find s0,s1>0 such that the function Hv:=s0g0,v−s1g1,v is harmonic on n−1(Cv,an). Recall from [34] that the bifurcation locus of the family Pc,a parameterized by (c,a)∈n−1(Cv,an) is defined as the set where either c0 or c1 is unstable (or active in the terminology of [20]). It follows from [20] that the bifurcation locus is equal to the union of the support of Δg0,v and Δg1,v, hence to the support of μ∞,v. Suppose now that Hv is not identically zero. Then this support is included in the locus {Hv=0} which is real-analytic. This is impossible by McMullen's universality theorem, since the Hausdorff dimension of the bifurcation locus of any one-dimensional analytic family is equal to 2, see [35, Corollary 1.6]. We have proved that s0g0,v=s1g1,v on n−1(Cv,an) hence on Cv,an for some positive real numbers s0,s1>0. Since g0,v and g1,v are proportional, and Gv=max{g0,v,g1,v} is proper on Cv,an, it follows that g0,v is unbounded near any branch at infinity. By Proposition 3.6, g0,v admits an expansion of the form g0,v(t)=a(c)log|t|−1+O(1) with a(c)∈ℚ+* on the branch c hence is locally superharmonic on that branch. It follows that Δg0,v is a signed measure in C^an,v whose negative part is a divisor D0 with positive rational coefficients at any point of C^∖n−1(C). The same being true for Δg1,v, we obtain the equality of divisors s0D0=s1D1. This implies that s0/s1 is rational, and we can assume s0 and s1 to be integers. This ends the proof of the first statement in the case the place is Archimedean. Assume now that v is non-Archimedean. One cannot copy the proof we gave in the Archimedean setting since we used the fact that c0 is not persistently pre-periodic iff Δg0,v=0, and McMullen's universality theorem, two facts that are valid only over ℂ. Instead we apply Proposition 3.6. For each s′=(s′0,s′1) the function gs′,v extends near any branch c at infinity as an upper-semicontinuous function gs′,v^ whose Laplacian puts some non-positive mass at c. When s′0,s′1≠0 then gs′,v defines a positive continuous metric on OC^(Ds′) hence Δgs′,v^{c}=−ordc(Ds′)<0. This mass is in particular independent of the place. We get that   −Δg0,v^{c}≥lims→(1,0)−Δgs,v^{c}=ordc(D0)>0. We infer that the mass of Δg0,v is equal to the degree of D0 hence is non-zero. We may now argue as in the Archimedean case, and prove that Δg0,v and Δg1,v are proportional. The coefficient of proportionality is the only t>0 such that D0=tD1 hence t=s0/s1. Then Hv:=s0g0,v−s1g1,v is harmonic on C and bounded near any branch at infinity by Proposition 3.6, hence defines a harmonic function on the compact curve C^an,v. It follows Hv is a constant (in the non-Archimedean case by [44, Proposition 2.3.2]) which is necessarily zero since it is zero at all post-critically finite parameters. We have completed the proof of Theorem 4.1 (1). We mention here the following result that follows from the previous argument. Corollary 4.2. Let C be an irreducible curve in Poly3 defined over a number field K that contains infinitely many post-critically finite parameters and that neither c0 nor c1 is persistently pre-periodic. Pick an Archimedean place v. Pick any sequence Xn⊂C(K¯) of Galois-invariant finite sets of postcritically finite parameters such that Xn≠Xm for m≠n. Let μn be the measure equidistributed on Xn⊂Cv,an. Then the sequence μn converges weakly to (a multiple of) Tbif∧[C] as n→∞.     □ Recall that Tbif is defined as the ddc of the plurisubharmonic function g0+g1, and [C] is the current of integration over the analytic curve Cv,an. Proof. Let s0,s1>0 be given by Theorem 4.1. As seen above, the sequence μn converges weakly towards ddcmax{s0⋅g0,s1⋅g1}=s0⋅ddcg0 on Cv,an. It thus only remains to prove that ddc(g0|Cv,an)=κ⋅Tbif∧[C] for some κ>0. Recall that Tbif=ddc(g0+g1). By Theorem 4.1, on C,   g0+g1=g0+s1s0⋅g0=(1+s0s1)g0. Let κ:=1+s0s1. We thus have ddc(g0|Cv,an)=κ−1⋅ddc((g0+g1)|Cv,an). Finally, since g0+g1 is continuous, we have Tbif∧[C]=ddc((g0+g1)|Cv,an), which ends the proof.     ▪ 4.2 Values of the Böttcher coordinates at critical points are proportional near infinity In this section, we prove Theorem 4.1 (2). Let us fix a branch at infinity c of an irreducible curve C containing infinitely many PCF polynomials, and an isomorphism of complete local rings OC^,c≃L[[t]], such that c(n(t))=t−n, and a(n(t))∈OL,S((t)) is an adelic series. Write Pt=Pc(n(t)),a(n(t)), and φt=φPt. By Lemma 3.8 there exists an integer q≥1 large enough such that Ptq(c0) and Ptq(c1) both lie in the domain of convergence of the Böttcher coordinate φt for t small enough, and (8) holds, that is,   φt(Ptq(cε))=ω(Ptq(cε)−c(n(t))2)+Θ(t), where Θ is an adelic series vanishing at 0. We now fix a place v and compute using Proposition 2.3 for |t|v≪1. We get   |φt(P0q(t))|vs0|φt(P1q(t))|vs1=exp(s0⋅gc(n(t)),a(n(t))(P0q(t)))exp(s1⋅gc(n(t)),a(n(t))(P1q(t)))=exp(3qs0⋅g0,v(c(n(t)),a(n(t))))exp(3qs1⋅g1,v(c(n(t)),a(n(t)))=1, (⋆) where the last equality follows from Theorem 4.1 (1). Applying (⋆) in the case of an Archimedean place, we see that the complex analytic map   t↦(φt(P0q(t)))s0(φt(P1q(t)))s1 has a modulus constant equal to 1, hence is a constant, say ζ. Since both power series φt(P0q(t)) and φt(P1q(t)) have their coefficients in OL,S, we conclude that ζ∈OL,S. But |ζ|v=1 for all place v over L by (⋆) hence it is a root of unity. Note also that the equality φt(P0q(t))s0=ζφt(P1q(t))s1 holds as equality between adelic series, so that it is also true for analytic functions at any place. To conclude the proof of Theorem 4.1, pick a place v of L and consider the connected component U of {g0,v>τv/s0}={g1,v>τv/s1} in Cv,an whose closure in C^ contains c. We need to argue that Pc,aq(c0) and Pc,aq(c1) belong to the domain of convergence of the Böttcher coordinate φv,c,a for any c,a∈U. Recall that s0g0,v(c,a)=s1g1,v(c,a) for some positive integers s0,s1. It follows that   min{gc,a(Pc,aq(c0)),gc,a(Pc,aq(c1))}=3qmin{gc,a(c0),gc,a(c1)} ≥3qmin{s0s1,s1s0}max{gc,a(c0),gc,a(c1)} >G(c,a)+max{s0gc,a(c0),s1gc,a(c1)}>G(c,a)+τv for q large enough and we conclude by Proposition 2.3. The proof of Theorem 4.1 is now complete. 5 Special Curves Having a Periodic Orbit with a Constant Multiplier In this section, we prove Theorem B. Pick an integer m≥1, a complex number λ∈ℂ, and consider the set of polynomials Pc,a that admit a periodic orbits of period m and multiplier λ. It follows from [42, p. 225] that this set is an algebraic curve in Poly3 (see also [36, Appendix D], [9, Theorem 2.1] or [21, Section 6.2]). Let us be more precise: Theorem 5.1 (Silverman). For any integer m≥1, there exists a polynomial pm∈ℚ[c,a,λ] with the following properties. For any λ∈ℂ∖{1}, pm(c,a,λ)=0 if and only if Pc,a has a cycle of exact period m and multiplier λ. When λ=1, then pm(c,a,1)=0 if and only if there exists an integer k dividing m such that Pc,a has a cycle of exact period k whose multiplier is a primitive m/k-th root of unity.     □ We now come to the proof of Theorem B. One implication is easy. For any integer m≥1, the curve Perm(0) is contained in the union of the two curves {(c,a)∈ℂ2; Pc,am(c0)=c0} and {(c,a)∈ℂ2;Pc,am(c1)=c1}. According to lemma 5.2 below, it contains infinitely many post-critically finite parameters. Lemma 5.2. Pick n≥0, k>0 and i∈{0,1}. Any irreducible component C of the set {(c,a),Pc,an+k(ci)=Pc,an(ci)} contains infinitely many post-critically finite parameters.     □ Proof. We argue over the complex numbers, and use the terminology and results from [20]. In particular, a critical point ci, i=0,1 is said to be active at a parameter (c,a) if the family of analytic functions Pc,an(ci) is normal in a neighborhood of (c,a). Suppose that C is an irreducible component of the set   {(c,a),Pc,an+k(ci)=Pc,an(ci)}, where n≥0, k>0 and i∈{0,1}. To fix notation we suppose i=0. Observe that gc,a(c0)=0 on C, and since G(c,a)=max{gc,a(c0),gc,a(c1)} is a proper function on Poly3 (see Proposition 2.1) it follows that gc,a(c1) is also proper on C. In particular, c1 has an unbounded orbit when c,a∈C is close enough to infinity in Poly3. It follows from for example, [20, Theorem 2.5] (which builds on [33, Theorem 2.2]) that c1 is active at at least one point (c0,a0) on C. The arguments of [20, Lemma 2.3] based on Montel's theorem show that (c0,a0) is accumulated by parameters for which c1 is pre-periodic to a repelling cycle, hence by post-critically finite polynomials. In particular, it contains infinitely many post-critically finite parameters.     ▪ For the converse implication, we proceed by contradiction and suppose that we can find a complex number λ≠0, an integer m≥1, and an irreducible component C of Perm(λ) containing infinitely many post-critically finite polynomials. Observe that, whenever 0<|λ|≤1, any parameter (c,a)∈C⊂Perm(λ) has a non-repelling cycle which is not super-attracting. In particular, at least one of its critical points has an infinite forward orbit (see e.g., [37]). It follows that Perm(λ) contains no post-critically finite parameter when 0<|λ|≤1. This argument is however not sufficient to conclude in general. But we shall see that a combination of this argument applied at a place of residual characteristic 3 together with the study of the explosion of multipliers on a branch at infinity of C gives a contradiction. Proposition 5.3. Suppose C is an irreducible component of Perm(λ) with λ∈ℂ* and m≥1 containing infinitely many post-critically finite polynomials. Then one of the two critical points is persistently preperiodic on C and λ is equal to the multiplier of a repelling periodic orbit of a post-critically finite quadratic polynomial.     □ We may thus assume that the curve C is included in {Pc,an(c0)=Pc,ak(c0)} (or {Pc,an(c1)=Pc,ak(c1)}) for some integers n>k≥0. Observe that the equation Pc,an(c0)=Pc,ak(c0) (resp. Pc,an(c1)=Pc,ak(c1)) is equivalent to the vanishing of a polynomial of the form 31−3na3n+l.o.t (resp. 31−3n(a3−c36)3n+l.o.t). It follows that the closure of C in Poly3¯ intersects the line at infinity in a set included in {[1:0:0],[ζ:1:0]} with ζ3=6 (see also [9, Theorem 4.2]). Consider the curve of unicritical polynomials c0=c1, which is defined by the equation c=0. It intersects the line at infinity at [0:1:0], so that Bezout' theorem implies the existence of a parameter (c,a)∈C which is unicritical. We conjugate Pc,a by a suitable affine map to a polynomial Q(z)=z3+t. This unicritical polynomial has a preperiodic critical orbit. Proposition 5.4 below implies |λ|v<1 at any place v of residual characteristic 3. By the previous proposition, Q also has a periodic orbit whose multiplier is equal to the multiplier of a repelling orbit of a quadratic polynomial having a preperiodic critical point. Proposition 5.4 now gives |λ|v=1 for this place, hence a contradiction. The proof of Theorem B is complete. Proposition 5.4. Suppose Q(z)=zd+t is a post-critically finite unicritical polynomial of degree d≥2, and let λ≠0 be the multiplier of some periodic orbit of P. Then λ belongs to some number field K, and given any non-Archimedean place v of K we have: |λ|v<1 if the residual characteristic of Kv divides d; |λ|v=1 if the residual characteristic of Kv is prime to d.     □ Proof. Since Q is post-critically finite, t satisfies a polynomial equation with integral coefficients hence belongs to a number field. Its periodic points are solutions of a polynomial of the form Qn(z)−z so that the periodic points of Q and their multipliers also belong to a number field. We may thus fix a number field containing t, λ, and fix a place v of K of residual characteristic p≥2. Observe that the completion of the algebraic closure of the completion of K with respect to the norm induced by v is a complete algebraically closed normed field isometric to the p-adic field ℂp. We consider the action of Q on the Berkovich analytification of the affine plane over that field. To simplify notation we denote by |⋅| the norm on ℂp. Suppose that |t|>1. Then we have |Q(0)|=|t|>1, and thus |Qn(0)|=|Qn−1(0)|d=|t|dn→∞ by an immediate induction. This would imply the critical point to have an infinite orbit contradicting our assumption that Q is post-critically finite. We thus have |t|≤1. This implies that any point having a bounded orbit lies in the closed unit ball {z,|z|≤1}. Indeed the same induction as before yields |Q(z)|=|z|d and |Qn(z)|=|Qn−1(z)|d=|z|dn→∞ for any |z|>1. Pick any periodic point w of period k with multiplier λ=(Qk)′(w)≠0. Observe that Q′(z)=dzd−1. Suppose first that p divides d so that |d|<1. Since |Qj(w)|≤1 for all j≥0 by what precedes, we have   |λ|=∏j=0k−1|Q′(Qj(w))|≤|d|k<1. Suppose now that p is prime to d, hence |d|=1. Observe that one has Q(B(z))=B(Q(z)) for any |z|≤1 where B(z)={w,|z−w|<1} is the open ball of center z and radius 1. Since the critical point 0 has a finite orbit, two situations may arise. Either B(0) is strictly preperiodic, and thus cannot contain any periodic orbit. Or B(0) is periodic, and is contained in the basin of attraction of some attracting periodic orbit. Since 0 has a finite orbit, it has to be periodic. In both cases this implies the orbit of w to be included in the annulus {|z|=1}. We thus have   |λ|=∏i=0k−1|Q′(Qi(w))|=∏i=0k−1|d(Qi(w))d−1|=1 , which concludes the proof.     ▪ Proof of Proposition 5.3. Since C contains infinitely many post-critically finite polynomials we may assume it is defined over a number field K. Let C^ be the normalization of the completion of C in Poly3¯. Pick any branch c of C at infinity (i.e., a point in C^ which projects to the line at infinity in Poly3¯). By Proposition 3.5 we may choose an isomorphism of complete local rings OC^,c^≃L[[t]] such that c(n(t)),a(n(t)) are adelic series, that is, formal Laurent series with coefficients in OL,S((t)) that are analytic at all places. In the remainder of the proof, we fix an Archimedean place, and embed L into the field of complex numbers (endowed with its standard norm). We may suppose c(n(t)),a(n(t)) are holomorphic in 0<|t|<ϵ for some ϵ, and meromorphic at 0. We get a one-parameter family of cubic polynomials Pt:=Pc(n(t)),a(n(t)) parameterized by the punctured disk Dϵ*={0<|t|<ϵ}. Consider the subvariety Z:={(z,t),Ptm(z)=z}⊂ℂ×Dϵ*. The projection map Z→Dϵ* is a finite cover which is unramified if ϵ is chosen small enough. By reducing ϵ if necessary, and replacing t by tN, we may thus assume that Z→Dϵ* is a trivial cover. In other words, there exists a meromorphic function t↦p(t) such that Ptm(p(t))=p(t) and (Ptm)′(p(t))=λ. As in Section 3, we denote by P(z)∈ℂ((t))[z] the cubic polynomial induced by the family Pt. It induces a continuous map on the analytification Aℂ((t))1,an, for which the point p∈A1(ℂ((t))) corresponding to p(t) is periodic of period m with multiplier (Pm)′(p)=λ. Observe that P has two critical points c0 and c1 corresponding to the meromorphic functions 0 and c(n(t)), respectively. Lemma 5.5. If c0 is not pre-periodic for P, then |Pq(c0)|t tends to infinity when q→∞.     □ Proof. Observe that our assumption is equivalent to the fact that c0 is not persistently pre-periodic on C. We claim that g0(t):=gPt(c0) tends to infinity when t→0. Suppose first that c1 is persistently pre-periodic on C. Then the function g1 is identically zero on C, so that G|C=max{g0,g1}|C=g0. Since G is proper by Proposition 2.1, and (c(n(t)),a(n(t))) tends to infinity in Poly3¯ when t→0, we conclude that g0(t)→∞. When c1 is not persistently pre-periodic on C, the two functions g0(t) and g1(t):=gPt(c1) are proportional on c by Theorem 4.1 (1). As before max{g0,g1}→∞ as t→0 so that again g0(t)→∞. By Proposition 3.6, we can find a>0 such that g0(Pt)=alog|t|−1+O(1). And [20, Lemma 6.4] implies the existence of a constant C>0 such that gPt(z)≤logmax{|z|,|c(n(t))|,|a(n(t))|}+C for all t. Since gPt∘Pt=3gPt, we conclude that for all q≥1 (observe that the statement of the lemma is incorrectly stated in [20], and the constant C is actually independent on P.)   logmax{|Ptq(c0)|,|c(n(t))|,|a(n(t))|}≥3qgt(0)−C=3qalog|t|−1+O(1). This implies |Ptq(c0)|t≥3qa|t|t→∞ when q→∞ as required.     ▪ We continue the proof of Proposition 5.3. Suppose neither c0 nor c1 is persistently pre-periodic so that the previous lemma applies to both critical points. Translating its conclusion over the non-Archimedean field ℂ((t)), we get that Pq(c0) and Pq(c1) both tend to infinity when q→∞. We may thus apply [31, Theorem 1.1 (ii)], and [31, Corollary 1.4] (which is directly inspired from a result of Bezivin). We conclude that all periodic cycles of P are repelling so that |(Pm)′(p)|t>1. This contradicts |λ|t=1. Suppose next that c0 is persistently pre-periodic or periodic (which implies c1 not to be persistently pre-periodic). Then c0 is pre-periodic whereas c1 escapes to infinity by the previous lemma. Observe that if c0 is eventually mapped to a point in the Julia set of P, then [31, Theorem 1.1 (iii) (a)] combined with [31, Corollary 1.4] implies that all cycles of P are repelling which gives a contradiction. We can thus apply [31, Theorem 1.1 (iii) (b)) to P, and the preperiodic critical point c0(=0) is contained in a closed ball B={z∈ℂ((t)),|z|t≤r} for some positive r>0 that is periodic of exact period n. Since B is fixed by the polynomial Pn(z)=∑j≥2bjzj with coefficients bj∈ℂ((t)), the radius r satisfies an equation of the form |bj|rj=r for some j hence r=|t|tl for some l∈ℚ. To simplify the discussion to follow we do a suitable base change t→tN, and we conjugate P by the automorphism z↦t−lz so that B becomes the closed unit ball. Observe that 0 remains a critical point of P after this conjugacy. Recall that the closed unit ball B defines the Gauss point xg∈Aℂ((t))1,an for which we have   |Q(xg)|:=supz∈B|Q(z)|t=max|qi| for all Q=∑qizi∈ℂ((t))[z]. Since B is fixed by Pn, it follows that xg is also fixed by Pn. This is equivalent to say that Pn can be written as Pn(z)=∑i=13naizi where max|ai|=1. For any z∈ℂ((t)) of norm 1, denote by z˜ the unique complex number such that |z−z˜|t<1. Lemma 5.6. We have a1=0, |a0|≤|a2|=1, and |ai|<1 for all i≥3; and the complex quadratic polynomial P˜(z):=a2˜z2+a0˜ has a preperiodic critical orbit.     □ Lemma 5.7. The orbit of the periodic point p intersects the ball B.     □ Replacing p by its image by a suitable iterate of P we may suppose that it belongs to B, that is, |p|t≤1. In fact we have |Pi(p)|t=1 for all i≥0. Indeed if it were not the case, then the open unit ball would be periodic. Since it contains a critical point, it would be contained in the basin of attraction of an attracting periodic orbit which yields a contradiction. Observe also that the period of p is necessarily a multiple of n, say nk with k≥1. To render the computation of the multiplier of p easier, we conjugate Pn by z↦a2z. Since |a2|=1, we still have |p|t=1, and the equality a2=1 is now satisfied. By Lemma 5.6, we get supB|Q|<1 with Q:=Pn−P˜, so that   (Pnk)′(p)=∏i=0k−1(Pn)′(Pni(p))=(P˜k)′(p˜). But the multiplier of p is equal to λ∈ℂ. Hence it is equal to the multiplier of a repelling periodic orbit of some quadratic polynomial (namely P˜) having a preperiodic critical orbit, as was to be shown.     ▪ Proof of Lemma 5.6. The point 0 is critical for P hence a1=0. Since the Gauss point is fixed by Pn, we have maxi≥2|ai|=1. Let d≥2 be the maximum over all integers i such that |ai|=1. The number of critical points of Pn lying in the closed unit ball (counted with multiplicity) is precisely equal to d−1. Since the exact period of xg is n, and the other point escapes to infinity, the ball B contains a unique critical point of Pn namely 0. It follows that d=2, and |a2|=1>maxi≥3|ai|. Finally 0 is preperiodic by Pn, hence the complex quadratic polynomial Pn˜ has a preperiodic critical orbit.     ▪ Proof of Lemma 5.7. Since the multiplier of p is λ∈ℂ, its t-adic norm is 1, hence a small ball U centered at p of positive radius is included in the filled-in Julia set of P. By [31, Corollary 4.8], U is eventually mapped into B, hence the claim.     ▪ 6. A Polynomial on a Special Curve Admits a Symmetry We fix K a number field, and s0,s1 two positive integers such that s0 and s1 are coprime. We shall say that a cubic polynomial P:=Pc,a with c,a in a finite extension L of K satisfies the condition (P) if the following holds: ( gQc,a,v=gPc,a,v) For any place v of L, we have s0gP,v(c0)=s1gP,v(c1). ( P2) Given any place v of L, if min{gP,v(Pn(c0)),gP,v(Pn(c1))}>Gv(P)+τv for some integer n≥1, then   φP,v(Pn(c0))s0φP,v(Pn(c1))s1 is a root of unity lying in K. Recall the definition of the constant τv:=τ(Lv) from Proposition 2.3. Observe that if the condition in (P2) never occurs, then Gv(c,a)=0 for all places v of L, hence P is post-critically finite (see e.g., [21, Theorem 3.2]). We prove here the following Theorem 6.1. Suppose P=Pc,a is a cubic polynomial defined over a number field L satisfying the assumptions (P) which is not post-critically finite and such that min{gP,v(Pq(c0)),gP,v(Pq(c1))}>Gv(P)+τv for some integer q and some place v of L. Then there exists a root of unity ζ∈K, an integer q′≤C(K,q), and an integer m≥0 such that the polynomial Q(z):=ζPm(z)+(1−ζ)c2 commutes with all iterates Pk such that ζ3k=ζ, and either Q(Pq′(c0))=Pq′(c1), or Q(Pq′(c1))=Pq′(c0).     □ Remark. We shall prove along the way that there exists an integer k≥1 with ζ3k=ζ so that the commutativity statement is non-empty.     □ 6.1. Algebraization of adelic branches at infinity The material of this section is taken from [45]. Let K be a number field. For any place v on K, denote by Kv the completion of K w.r.t. the v-adic norm. We cover the line at infinity H∞ of the compactification of the affine space AK2=SpecK[x,y] by ℙK2 by charts Uα=SpecK[xα,yα] centered at α∈H∞(K) such that α={(xα,yα)=(0,0)}, H∞∩Uα={xα=0}, and xα=1/x, yα=y/x+c for some c∈K (or xα=1/y, yα=x/y). Fix S a finite set of places of K. By definition, an adelic branch s at infinity defined over the ring OK,S is a formal branch based at a point α∈H∞(K) given in coordinates xα,yα as above by a formal Puiseux series   yα=∑j≥1ajxαj/m∈OK,S[[xα1/m]] such that ∑j≥1ajxj is an adelic series. Observe that for any place v∈S, then the radius of convergence is a least 1. In the sequel, we set rs,α,v to be the minimum between 1 and the radius of convergence over Kv of this Puiseux series. Any adelic branch s based at α at infinity thus defines an analytic curve in an (unbounded) open subset of Av2,an:   Zv(s):={(xα,yα)∈Uα(Kv);yαm=∑j≥1ajxαj,0<|xα|v<rs,α,v}. Theorem 6.2 (Xie). Suppose s1,…,sl are adelic branches at infinity, and let {Bv}v∈MK be a set of positive real numbers such that Bv=1 for all but finitely many places. Assume that there exists a sequence of distinct points pn=(xn,yn)∈A2(K) such that for all n and for each place v∈MK then either we have max{|xn|v,|yn|v}≤Bv or pn∈∪i=1lZv(si). Then there exists an algebraic curve Z defined over K such that any branch of Z at infinity is contained in the set {s1,…,sl} and pn belongs to Z(K) for all n large enough.     □ 6.2 Construction of an invariant correspondence Our aim is to prove the following statement. Theorem 6.3. Suppose P=Pc,a is a cubic polynomial satisfying the assumptions (P). Then there exists a (possibly reducible) algebraic curve ZP⊂A1×A1 such that: ϕ(ZP)=ZP with ϕ(x,y):=(P(x),P(y)); for all n large enough, we have (Pn(c0),Pn(c1))∈ZP; any branch at infinity of ZP is given by an equation φP(x)s0=ζ⋅φP(y)s1 for some root of unity ζ∈K.     □ Proof. The proof is a direct application of Xie's theorem. Recall that the set UK of roots of unity that is contained in the number field K is finite. Recall that for each place v over L, we let gP,v:=limn13nlog+|Pn|v be the Green function of P, and write Gv(P)=max{gP,v(c0),gP,v(c1)}. Lemma 6.4. For any ζ∈UK, there exists an adelic branch cζ based at a point q∈H∞(L) such that for any place v the analytic curve Zv(cζ) is defined by the equation {φP,v(x)s0=ζ⋅φP,v(y)s1} in the range min{|x|v,|y|v}>exp(Gv(P)+τv).     □ Define (xn,yn):=(Pn(c0),Pn(c1))∈A2(L), and consider the family of all adelic curves cζ given by Lemma 6.4 for all ζ∈UK. We shall first check that all hypotheses of Xie's theorem are satisfied. To do so pick any integer n and any place v on L. Suppose first that gP,v(c0)=0. Since gP,v(Pn(c0))=3ngP,v(c0)=0, we get |xn|v≤eCv=:Bv by Lemma 2.4. The same upper bound applies to |yn|v since gP,v(c1)=0 by (P1) so that max{|xn|v,|yn|v}≤Bv in this case. Observe that Bv=1 for all but finitely many places v of L by Lemma 2.4. Suppose now that gP,v(c0)>0 so that gP,v(c1)>0 by (P1). Fix N large enough such that gP,v(PN(c0))>Gv(P)+τv and gP,v(PN(c1))>Gv(P)+τv. Then PN(c0) and PN(c1) lie in the domain of definition of the Böttcher coordinate by Proposition 2.3. Since   gP,v(Pn(c0))=3n−NgP,v(PN(c0))≥gP,v(PN(c0))>Gv(P), we may also evaluate φP at xn for all n≥N. The same holds for yn and we get that   φP(xn)s0φP(yn)s1 is a root of unity ζ∈K by (P2) hence (xn,yn) belongs to Zv(cζ) for all n≥N. Xie's theorem thus applies to the sequence {(xn,yn)}n≥N, and we get a (possibly reducible) curve Z1⊂A1×A1 that contains infinitely many points (xn,yn) and such that each of its branch at infinity is equal to cζ for some ζ∈UK. Recall that ϕ(x,y)=(P(x),P(y)), and pick any integer n≥1. Let Z be an irreducible component of Z1. Then ϕn(Z) is an irreducible curve defined over L whose branches at infinity are the images under ϕn of the branches at infinity of Z. Fix ζ∈UK and pick (x,y)∈Zv(cζ). Then (x′,y′)=(P(x),P(y)) satisfies   φP(x′)s0φP(y′)s1=φP(x)3s0φP(y)3s1=ζ3, hence ϕ(cζ)=cζ3. We conclude that any branch at infinity of ϕn(Z) is of the form cζ for some ζ∈UK. Since two irreducible curves having a branch at infinity in common are equal, we see that Z is pre-periodic for the morphism ϕ so that ϕl+k(Z)=ϕk(Z) for some l,k>0. Setting ZP:=∪j=kl+k−1ϕj(Z), we obtain a (possibly reducible) curve defined over L such that ϕ(ZP)=ZP and (xn,yn)∈ZP for all n≥k. This concludes the proof of the theorem, since Z1 has only finitely many irreducible components.     ▪ Proof of Lemma 6.4. Recall from Lemma 2.2 that   φP(z)=ω(z−c2)+∑k≥1akzk, is an adelic series at infinity in the sense of Section 3.1 , and therefore   φP−1(z)=1ωz+c2+∑k≥1bkzk, by Lemma 3.1. We may assume that ak,bk∈OK,S. Recall from Proposition 2.3 that φP,v induces an analytic isomorphism between {z,gP,v(z)>Gv(P)+τv} and {z′,|z′|v>exp(Gv(P)+τv)}. By Lemma 3.1 the formal map φP−1 defines an adelic series at infinity in the terminology of Section 3.1. For each place v, this series coincides with the inverse map of φP on the complement of the closed disk of radius exp(Gv(P)+τv) hence its domain of convergence is exactly {z′,|z′|v>exp(Gv(P)+τv)}. It follows that   Zv:={(x,y),φP(x)s0=ζφP(y)s1} defines an analytic curve in the domain min{gP,v(x),gP,v(y)}>Gv(P)+τv, whose image under the isomorphism (x′,y′):=(φP,v(x),φP,v(y)) is given by   Z′v:={(x′,y′),(x′)s0=ζ(y′)s1} where min{|x′|v,|y′|v}>exp(Gv(P)+τv). Pick any ξ∈ℚ¯ such that ξs1ζ=1. Let cζ be the adelic branch at infinity defined by the formal Laurent series (φP−1(t−s1),φP−1(ξt−s0)). For all places v, the analytic curve Zv(cζ) is included in Zv. Since s0 and s1 are coprime, for any pair (x′,y′) with (x′)s0=ζ(y′)s1 and min{|x′|v,|y′|v}>exp(Gv(P)+τv), there exists 0<|t|v<exp(−Gv(P)+τvmin{s0,s1}) such that x′=t−s1 and y′=ξt−s0. This proves that Zv(cζ)=Zv for all place as required.     ▪ 6.3 Invariant correspondences are graphs Let Z0,…,Zp−1 be the irreducible components of ZP such that ϕ(Zi)=Zi+1 (the index computed modulo p). Since we assumed P not to be post-critically finite, it is non-special in the sense of [40]. We may thus apply Theorem 4.9 of op. cit. (or [39, Theorem 6.24]) to the component Z0 of ZP that is ϕp-invariant. It implies that after exchanging x and y if necessary, Z0 is the graph of a polynomial map, that is, Z0={(Q(t),t)} for some Q∈L[t] such that Q∘Pp=Pp∘Q. Observe that by [30] the two polynomials P and Q share a common iterate when deg(Q)≥2 since we assumed P not to be post-critically finite. We now work at an Archimedean place. Recall that the branch at infinity of Z0 is of the form φP(x)s0=ζφP(y)s1 for some ζ∈UK. Since s0 and s1 are coprime, it follows that s0=1 and s1=deg(Q), and therefore s1 is a power of 3, say s1=3m. We get   φP(Q(t))=ζφP(t)3m=ζφP(Pm(t)). (13) for all t of large enough norm. By Lemma 2.2, we get that φP(t)=ω(t−c2)+o(1) so that   ω(Q(t)−c2)=ωζ(Pm(t)−c2)+o(1) (14) which implies Q(t):=ζPm(t)+(1−ζ)c2 since a polynomial which tends to 0 at infinity is identically zero. At this point, recall our assumption that min{gP,v(Pq(c0)),gP,v(Pq(c1))}>Gv(P)+τv for some integer q and some place v of L. Then by (P2) φP(Pq(c0))s0=ξφP(Pq(c1))s1 for some root of unity ξ∈K which implies φP(Pq+n(c0))=ξ3nφP(Pq+n(c1))3m. Since for some n large enough the point (Pq+n(c0),Pq+n(c1)) belongs to Z0, we get ξ3n=ζ. Now observe that the least integer n such that ξ3n=ζ is less that the cardinality of UK. We get the existence of q′≤C(K,q) such that φP(Pq′(c0))=ζφP(Pq′(c1))3m. Since φP is injective, the equation (13) shows that Pq′(c0)=Q(Pq′(c1)). Observe that ζ3p=ζ. Indeed, since Z0 is ϕp-invariant and since ϕ(cζ)=cζ3, we get cζ3p=cζ, hence ζ3p=ζ. We now pick any integer k≥1 such that ζ3k=ζ. Then for all t large enough, we have   φP(Q∘Pk(t))=ζφP(Pk(t))3m=ζφP3m+k(t), whereas,   φP(Pk∘Q(t))=φP(Q(t))3k=ζ3kφP3k+m(t). Since φP is injective on a neighborhood of ∞, and since ζ3k=ζ by assumption, we conclude that Q∘Pk=Pk∘Q. This concludes the proof of Theorem 6.3. 7 Classification of Special Curves In this section, we prove Theorems C and A. Before starting the proofs, let us introduce some notation. Pick q,m≥0 and ζ a root of unity. We let Z(q,m,ζ) be the algebraic set of those (c,a)∈A2 such that the polynomial Qc,a:=ζPc,am+(1−ζ)c2 commutes with all iterates Pc,ak of Pc,a such that ζ3k=ζ, and either Qc,a(Pc,aq(c0))=Pc,aq(c1), or Qc,a(Pc,aq(c1))=Pc,aq(c0). Observe that, when m≥1, Q has degree 3m and when (c,a) belongs to a fixed normed field K then the Green function gQ:=limn13nmlog+|Qn| is equal to gc,a. Indeed since Q and Pk commute they have the same filled-in Julia set, KQ coincides with the filled-in Julia set KP of P. And gP (resp. gQ) is the unique continuous sub-harmonic function g on AK1,an that is zero on KP, harmonic outside, with a logarithmic growth at infinity gP(z)=log|z|+O(1) (resp. gQ(z)=log|z|+O(1)). As KP=KQ, this gives gc,a=gQ. 7.1. Proof of Theorem C The implication (1)⇒(3) is exactly point 1. of Theorem 4.1. The implication (3)⇒(4) follows from Corollary 4.2 when s0 and s1 are both non-zero. Indeed, by assumption the critical point c0 is pre-periodic for Pc,a if and only if c1 is. Moreover, we observe that c0 is active at at least one parameter of C, since the function G is proper, hence that the set for which it is pre-periodic is infinite, by e.g. [20, Lemma 2.3). When s1=0, then g0,v≡0 on C at all places. By [20, Theorem 2.5] there exist n>0 and k≥0 such that C is an irreducible component of {(c,a)∈A2;Pc,an+k(c0)=Pc,ak(c0)}. By Theorem 3.12 (applied to arbitrary weights) the family of functions {g1,v}v∈MK induces a semi-positive adelic metric on some ample line bundle on the normalization of the completion of C so that Thuillier–Yuan's theorem applies. This gives (4) by observing that g0+g1=g1. The case s0=0 is treated similarly. The implication (4)⇒(1) is obvious. To prove (2)⇒(1), we observe that if c0 is not persistently pre-periodic on C then it is active at at least one parameter by [20, Theorem 2.5] and that the set of parameters for which it is pre-periodic is infinite by e.g. [20, Lemma 2.3]. We now prove (3)⇒(2). We suppose c0 is not persistently pre-periodic on C. Pick some parameter (c,a)∈C and suppose c0 is pre-periodic. We need to show that Pc,a is post-critically finite. Since c0 is not persistently pre-periodic on C we have s1≠0 (again by [20, Theorem 2.5] applied at any Archimedean place). In the case s0=0 then c1 is persistently pre-periodic and Pc,a is clearly post-critically finite. We may thus assume that s0 and s1 are both non-zero and the functions g0,v,g1,v vanish on the same set in Cv,an for any place v of K. Observe that c0 being pre-periodic implies (c,a) to be defined over a number field. It follows that for all the Galois-conjugates (c′,a′) of (c,a) (over the defining field K of the curve C) we have Gv(c′,a′):=max{g0,v(c′,a′),g1,v(c′,a′)}=0. It follows from [29] or [21, Theorem 3.2] that Pc,a is post-critically finite. Let us now prove (3)⇒(5). We suppose C is special. If either c0 or c1 is persistently pre-periodic in C, the assertion (5) holds true with ζ=1 and i=j by [20, Theorem 2.5]. Assume from now on that we are not in this case. Replacing K by a finite extension we may assume that all roots of unity ζ appearing in Theorem 6.1 2. belong to K, since there are at most the number of branches at infinity of C of such roots of unity. Let B be the set of all (c,a)∈C(L) where L is a finite extension of K such that Pc,a is not post-critically finite. Given a place v of K we also define the subset Bv of B of parameters c,a such that g0,v(c,a)>0. This set is infinite since post-critically finite polynomials form a bounded set in Cv,an. Lemma 7.1. For any (c,a)∈B, the polynomial Pc,a satisfies (P1) and (P2).     □ Pick q large enough such that 3q>max{s0/s1,s1/s0}, and fix a place v of residual characteristic ≥5. Now choose any (c,a)∈Bv. Then g1,v(c,a) is also positive and min{gc,a,v(Pq(c0)),gc,a,v(Pq(c1))}>Gv(c,a) so that Theorem 6.1 applies by the previous lemma. We get a positive integer q′ (bounded from above by a constant C depending only on K and q), a root of unity ζ∈K and an integer m≥0 such that (c,a)∈Z(q′,m,ζ). Since gQ=gP, and Q(Pq′(c0))=Pq′(c1) we have   3mgP,v(Pq′(c0))=gP,v(Q(Pq′(c0)))=gP,v(Pq′(c1)) so that 3m=s0s1. We conclude that the algebraic set consisting of the union of the curves Z(q′,m,ζ) with 3m=s0s1, q′≤C and ζ ranging over all roots of unity lying in K contains Bv. It follows that C is an irreducible component of one of these curves. To end the proof of the theorem, we are left with proving (5)⇒(3). Suppose that C is an irreducible component of Z(m,q,ζ) for some m≥0 and q≥0 and some root of unity ζ. Observe that Z(q,m,ζ) hence C are defined over a number field say K. When m≥1, then for all place v of that number field we have gQc,a,v=gPc,a,v for all (c,a)∈C(L) for some finite extension L of K. In particular Qc,a(Pc,aq(ci))=Pc,aq(cj) implies 3mgi,v(c,a)=gj,v(c,a) which proves (3) (with s0=0 or s1=0 if i=j). When m=0 and ζ≠1 and C≠{c=0}, then Qc,a(c0)=(1−ζ)c/2≠0 hence i≠j. It follows that gPc,a,v∘Qc,a=gPc,a,v hence g0,v=g1,v. When C={c=0}, then c0=c1 so that again g0,v=g1,v. Finally when m=0 and ζ=1, then i≠j and Pc,aq(c0)=Pc,aq(c1) hence g0,v=g1,v at all places. Proof of Lemma 7.1. Pick (c,a)∈B. By the point 1. of Theorem 4.1, Pc,a satisfies (P1). To check (P2), we need to introduce a few sets. Fix any place v of K, and for any integer n≥0, define the open subset of Cv,an  Ωn,v:={(c′,a′),min{gc′,a′,v(Pc′,a′n(c0)),gc′,a′,v(Pc′,a′n(c1))}>Gv(c′,a′)+τv}. On Ωn,v one can define the analytic map   Mn(c′,a′):=φc′,a′(Pc′,a′n(c0))s0φc′,a′(Pc′,a′n(c1))s1. Observe that Ωn+1,v⊂Ωn,v, and Mk+l(c′,a′)=Mk(c′,a′)3l on Ωk,v for all integers k,l≥0. We also define the increasing sequence of open sets   Un,v:={(c′,a′),Gv(c′,a′)>τv3n−1}⊂Cv,an. Since Gv is subharmonic and proper on Cv,an, the set Un,v contains no bounded component by the maximum principle. Lemma 7.2. Suppose 3r≥max{s0/s1,s1/s0}. Then we have Ωn,v⊂Un,v and Un,v⊂Ωn+r,v.     □ By the point 2. of Theorem 4.1, one can find two integers q≥1 and N≥1 such that Mq is well-defined and constant equal to a root of unity lying in K in each component of UN,v. Let V be the connected component of Ωn,v containing (c,a). This open set might or might not be bounded. By the previous lemma, if n≥max{r+N,q}, then Un−r,v⊂Ωn,v so that Mn is well-defined on Un−r,v. Since all components of Un−r,v are unbounded, and Mn=Mq3n−q in UN,r, we conclude that Mn is locally constant in Un−r,v (hence on V) equal to a root of unity lying in K. When n≤n0=max{r+N,q}, then (Mn)3n0−n=Mn0 which we know is constant in V equal to a root of unity lying in K. We conclude that Mn is constant on V equal to a root of unity lying in a fixed extension K′ of K that only depends on the constants r,N and q. Since these constants are in turn independent of the place v, we conclude the proof of the lemma replacing K by K′.     ▪ Proof of Lemma 7.2. Pick (c′,a′)∈Ωn,v. We may suppose that Gv(c′,a′)=gc′,a′,v(c0) so that Gv(c′,a′)=gc′,a′,v(c0)=13ngc′,a′,v(Pc′,a′n(c0))>13n(Gv(c′,a′)+τv) which implies (c′,a′)∈Un,v. Conversely suppose (c′,a′)∈Un,v. As before we may suppose that Gv(c′,a′)=gc′,a′,v(c0) so that   gc′,a′,v(Pn+r(c0))≥gc′,a′,v(Pn(c0))=3ngc′,a′,v(c0) =3nGv(c′,a′)=Gv(c′,a′)+(3n−1)Gv(c′,a′)>Gv(c′,a′)+τv. Similarly we have   gc′,a′,v(Pn+r(c1))=3n+rs0s1gc′,a′,v(c1)≥3ngc′,a′,v(c0)>Gv(c′,a′)+τv hence (c′,a′)∈Ωn+r,v.     ▪ 7.2. Proof of Theorem A According to the implication (1)⇒(5) of Theorem C, any irreducible algebraic curve C of Poly3 containing infinitely many post-critically finite polynomials is a component of some Z(q,m,ζ) so that Theorem A reduces to the following. Proposition 7.3. The set Z(q,m,1) is equal to the union {Pc,am+q(c1)=Pc,aq(c0)}∪{Pc,am+q(c0)=Pc,aq(c1)}, hence contains an algebraic curve. Moreover, one has Z(1,0,1)=Z(0,0,1). The set Z(q,m,−1) is infinite if and only if m=0, and we have Z(q,0,−1)={12a3−c3−6c=0} for any q≥0. if ζ2≠1, the set Z(q,m,ζ) is finite.     □ We shall rely on the following observation. Denote by Crit(P) the set of critical points of the polynomial P. Lemma 7.4. Pick any (c,a)∈Z(q,m,ζ), and suppose that   Qc,a=ζPc,am+(1−ζ)c2 is a polynomial that commutes with Pc,ak and ζ is a (3k−1)-root of unity. Then we have   Qc,a(Crit(Pc,ak+m))=Qc,a(Crit(Pc,am))∪Crit(Pc,ak).     □ Proof. Write P=Pc,a and Q=Qc,a. Differentiate the equality Pk∘Q=Q∘Pk. Since Q′=ζ⋅(Pm)′, we get   Crit(Q∘Pk)=P−k(Crit(Q))∪Crit(Pk) =P−k(Crit(Pm))∪Crit(Pk)=Crit(Pk+m), and therefore   Crit(Pk+m)=Crit(Pk∘Q)=Crit(Pm)∪Q−1(Crit(Pk)), and we conclude taking the image of both sides by Q.     ▪ We now come to the proof of the Proposition. Proof of Proposition 7.3. We may and shall assume that all objects are defined over the field of complex numbers: 1. Suppose Z(q,0,ζ) contains an irreducible curve C. We shall prove that either ζ=±1, or C={c=0}. Observe that for any (c,a)∈Z(q,0,ζ), the polynomial Qc,a is an affine map which commutes with Pc,ak, hence gc,a(Qc,a(z))=gc,a(z) for all z∈ℂ. Without loss of generality, we may suppose that Qc,a(Pc,aq(c0))=Pc,aq(c1), hence G(c,a)=g0(c,a)=g1(c,a). Suppose that Z(q,0,ζ) contains an irreducible curve C. If g0 vanishes identically on C then g1 also, and this implies both critical points to be persistently pre-periodic so that all polynomials in C are post-critically finite. This cannot happen, so that we can find an open set U in C such that G(c,a)>0 for all (c,a)∈U. Pick any parameter (c,a) in U. We have Qc,a(Crit(Pc,ak))=Crit(Pc,ak) by Lemma 7.4, so that Qc,a(c0),Qc,a(c1)∈Crit(Pc,ak). Since   Crit(Pc,ak)=∪0≤j≤k−1Pc,a−j(Crit(Pc,a)) we get gc,a(α)=3−jgc,a(c0)<gc,a(c0)=G(c,a) for any α lying in Crit(Pc,ak) but not in Crit(Pc,a). However, gc,a(Qc,a(c0))=gc,a(Qc,a(c0))=G(c,a), therefore, we have Qc,a(c0),Qc,a(c1)∈Crit(Pc,a)={c0,c1}. This implies either (1−ζ)c=0, or (1+ζ)c=0, hence ζ=±1 or C={c=0} as required. 2. Suppose now that C is an irreducible curve included in Z(q,m,ζ) with m>0. We claim that either ζ=1, or C={c=0} as above. We proceed similarly as in the previous case. We suppose that Z(q,m,ζ) is infinite. For any (c,a)∈Z(q,m,ζ), the polynomial Qc,a commutes with Pc,ak for some k, and has degree 3m>1. In particular we have equality of Green functions gQc,a=gc,a. Without loss of generality we may (and shall) assume Qc,a(Pc,aq(c0))=Pc,aq(c1), which implies g0(c,a)=3mg1(c,a). Assume now by contradiction that ζ≠1. Proceeding as in the previous case, we can find an open set U⊂C such that G(c,a)=g0(c,a)>0 for all (c,a)∈U. Pick now (c,a)∈U.     □ Lemma 7.5. For any α∈Pc,a−m{c0}, we have Qc,a(α)∈{c0,Qc,a(c1)}.     □ Observe that   Qc,a(α)=ζPc,am(α)+(1−ζ)c2=ζc0+(1−ζ)c2=(1−ζ)c2. The equality Qc,a(c1)=Qc,a(α) therefore gives Qc,a(c1)=ζPc,am(c1)+(1−ζ)c2=(1−ζ)c2, and we find Pc,am(c1)=0=c0 so that C is a component of Z(1,m,1). The equality c0=Qc,a(α), implies (1−ζ)c2=0 so that either ζ=1, or C equals {c=0}. 3. We have Z(q,0,−1)=Z(0,0,−1) for all q≥0. Fix q≥0, and pick any (c,a)∈Z(q,0,−1). Observe that (−1)3=−1 hence Qc,a(z)=−z+c commutes with Pc,a by definition. A direct computation shows that this happens if and only if (c,a) belongs to the curve D1:={12a3−c3−6c=0}. One can also check that Qc,a(c0)=c1 for any parameter on D1, and this implies (Qc,a∘Pc,aq)(c0)=Pc,aq(Qc,a(c0))=Pc,aq(c1) for any q≥0. This implies the claim. 4. The irreducible curve D0={c=0} is included in Z(q,m,ζ) if and only if m=0 and ζ=1. Observe that any polynomial P:=P0,a in D0 is unicritical with a single critical point at 0, so that D0 is included in Z(q,0,1) for all q≥0. Observe also that g0=g1>0 on a non-empty open subset of D0. Suppose that D0 is included in Z(q,m,ζ) for some positive integer m>0. Then the Green function of Q:=Q0,a is equal to gP and the equation Q(Pq(c0))=Pq(c1) implies gP(c0)=3mgP(c1)>gP(c1) at least when P is close enough to infinity. This is absurd. Suppose now that D0 is included in Z(q,0,ζ) with ζ≠1 so that Q(z)=ζz. One checks by induction that for any integer k≥1 one has   Pk(z)=qkz3k+ska3z3k−3+l.o.t with qk,sk∈ℚ+*. Choose k minimal such that ζ3k=ζ. We get   Q−1∘Pk∘Q(z)=qkz3k+ska3ζ3k−4z3k−3+l.o.t≠Pk which yields a contradiction, and concludes the proof of our claim. 5. We may now prove the proposition. The first statement follows from the definition of Z(q,m,1), since in that case we have Q=Pm which always commutes with P. Moreover, the curve Z(1,0,1) is given by the equation   0=Pc,a(c0)−Pc,a(c1)=a3−(a3−c36)=c36, whence Z(1,0,1)={c=0}=Z(0,0,1). For the second statement, suppose first that Z(q,m,−1) is infinite. By the second step, we have m=0, or D0={c=0} is included in Z(q,m,−1). The fourth step rules out the latter possibility so that m=0. Conversely if m=0 we may apply the third step to conclude that Z(q,0,−1) is a curve equal to D1={12a3−c3−6c=0}. For the third statement, pick ζ≠±1 and suppose by contradiction that Z(q,m,ζ) is infinite. The first and second step imply that Z(q,m,ζ) contains D0 which is impossible by Step 4. This concludes the proof of the proposition.     ▪ Proof of Lemma 7.5. Take α∈Pc,a−m{c0}, and observe that α∈Crit(Pc,ak+m). According to Lemma 7.4, we have   Qc,a(α)∈Crit(Pc,ak)∪Qc,a(Crit(Pc,am)) and gc,a(Qc,a(α))=3mgc,a(α)=3m⋅3−mgc,a(c0)=gc,a(c0)=G(c,a)>0. Pick any point z∈Crit(Pc,ak)∪Qc,a(Crit(Pc,am)), and suppose it is equal to neither c0 nor Qc,a(c1). Then we are in one of the following (excluding) cases: z is a preimage of c0 under Pc,aj for some 1≤j≤k−1, and gc,a(z)<gP(c0); z is a preimage of c1 under Pc,aj for some 0≤j≤k−1, in which case gc,a(z)≤gc,a(c1)<gc,a(c0); z∈Qc,a(Crit(Pc,am)), so that gc,a(z)=3mgc,a(w) for some point w∈Crit(Pc,am)=∪0≤j≤m−1Pc,a−j(Crit(Pc,a)). In the last case two sub-cases arise. When w is a preimage of c0, we find   gc,a(z)=3mgc,a(w)≥3m13m−1gc,a(c0)>gc,a(c0). Otherwise w is a preimage of c1 distinct from c1 since z≠Qc,a(c1). And we find gc,a(z)=3mgc,a(w)≤3m−1gc,a(c1)<gc,a(c0). Since gc,a(Qc,a(α))=gc,a(c0) we conclude that z≠Qc,a(α) as required. Funding This work was supported by the ERC-starting grant project “Nonarcomp” [307856 to CF]., and was partially supported by ANR project “Lambda” [ANR-13-BS01-0002 to TG] Acknowledgements We thank Xavier Buff and Laura De Marco for discussions at a preliminary stage of this project, and the referee for his/her careful reading of this article and his/her constructive remarks. While finishing the writing of this article we have learned that Dragos Ghioca and Hexi Ye have independently obtained a proof of Theorem A. Their approach differs from ours in the sense that they directly prove the continuity of the metrizations induced by the functions g0,v and g1,v. We get around this problem by considering metrizations induced by max{s0g0,v,s1g1,v} for positive s0,s1 instead. We warmly thank D. Ghioca and H. Ye for sharing with us their preprint. 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Classification of Special Curves in the Space of Cubic Polynomials

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Abstract

Abstract We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker and DeMarco for cubic polynomials. Let Perm(λ) be the algebraic curve consisting of those cubic polynomials that admit an orbit of period m and multiplier λ. We also prove that an irreducible component of Perm(λ) is special if and only if λ=0. 1 Introduction The space Polyd of complex polynomials of degree d≥2 modulo affine conjugacy forms a complex analytic space that admits a ramified parameterization by the affine space Aℂd−1. The study of the set of degree d polynomials with special dynamical features forms the core of the modern theory of holomorphic dynamics. We shall be concerned here with the distribution of the set of post-critically finite (PCF) polynomials for which all critical points have a finite orbit under iteration. This set is a countable union of points defined over a number field, see for example [29, Corollary 3]. It was proved in [32] that hyperbolic PCF quadratic polynomials equidistribute to the harmonic measure of the Mandelbrot set. This convergence was generalized in [4] in degree 2, and later in [21] in any degree where it was proved that under a mild assumption any sequence of Galois-invariant finite subsets of PCF polynomials converges in the sense of measures to the so-called bifurcation measure. This fact was further explored in [26]. The support of this measure has been characterized in several ways in a series of works [18–20,25], and it was shown by the second author [23] to have maximal Hausdorff dimension 2(d−1). In a beautiful recent article [3], Baker and DeMarco have proposed a way to describe the distribution of PCF polynomials from the point of view of the Zariski topology. They defined special algebraic subvarieties as those subvarieties Z⊂Polyd admitting a Zariski-dense subset formed by PCF polynomials, and asked about the classification of such varieties. More precisely, they offered a quite general conjecture [3, Conjecture 1.4] inspired by the André-Oort conjecture in arithmetic geometry stating that any polynomial (the conjecture is actually stated for any rational maps, and a stronger conjecture related to Pink and Zilber's conjectures can be found in [14].) lying in a special (proper) subvariety should admit a critical orbit relation (see also [43, Conjecture 6.56]). They gave a proof of a stronger version of this conjecture in the case the subvariety was isomorphic to an affine line. Our objective is to give the list of all special curves in the case d=3, thereby proving Baker–DeMarco's conjecture for cubic polynomials. The geometry of the space of cubic polynomials has been thoroughly explored in the seminal work [7] of Branner and Hubbard. Instead of using their parameterization, we shall follow [20] and consider the parameterization (c,a)↦Pc,a of the parameter space by the affine plane with   Pc,a(z):=13z3−c2z2+a3, z∈ℂ,(c,a)∈ℂ2. Observe that Pc,a then admits two critical points c0:=0 and c1:=c and that this map defines a finite branched cover of the moduli space Poly3 of cubic polynomials with marked critical points. Here is our main result. Theorem A. An irreducible curve C in the space Poly3 contains an infinite collection of post-critically finite polynomials if and only if one of the following holds. One of the two critical points is persistently pre-periodic on C, that is, there exist integers m>0 and k≥0 such that Pc,am+k(c0)=Pc,ak(c0) or Pc,am+k(c1)=Pc,ak(c1) for all (c,a)∈C. There is a persistent collision of the two critical orbits on C, that is, there exist (m,k)∈ℕ2∖{(1,1)} such that Pc,am(c1)=Pc,ak(c0) for all (c,a)∈C. The curve C is given by the equation {(c,a),12a3−c3−6c=0}, and coincides with the set of cubic polynomials having a non-trivial symmetry, that is, the set of parameters (c,a) for which Qc(z):=−z+c commutes with Pc,a.     □ Recall that for any integer m≥1 and any complex number λ∈ℂ the set Perm(λ) consisting of all polynomials Pc,a∈Poly3 that admit at least one periodic orbit of period m and multiplier λ is an algebraic curve (see Section 5 for a more precise description). The geometry of these curves has been explored by several authors, especially when λ=0. The irreducible components of Perm(0) have been proven to be smooth by Milnor [38], and the escape components of these curves have been described in terms of Puiseux series by Bonifant, Kiwi, and Milnor [5] (see also [31, Section 7]). On the other hand, DeMarco and Schiff [15] have given an algorithm to compute their Euler characteristic. In a recent preprint, Arfeux and Kiwi counted the number of irreducible components of Perm(0) when m is prime. From the point of view of pluripotential theory, the distribution of the sequence of curves (Perm(λ))m≥1 has been completely described by Bassanelli and Berteloot in [9] in the case |λ|≤1 (see also [24] for the case |λ|>1 and [8] for the case of quadratic rational maps). Inspired by a similar result from Baker and DeMarco, see [3, Theorem 1.1] we also give a characterization of those Perm(λ) that contain infinitely many PCF polynomials. This answers a conjecture of DeMarco in the case of cubic polynomials (see e.g., [43, Conjecture 6.59]). More precisely, we prove Theorem B. For any m≥1, the curve Perm(λ) contains infinitely many post-critically finite polynomials if and only if λ=0.     □ The general strategy of the proof of these two theorems was set up by Baker and DeMarco. They start with an irreducible algebraic curve C⊂Poly3 containing infinitely many PCF polynomials (in Theorem B the curve C is a component of some Perm(λ)). We observe however that they used at several key points their assumption that the curve C has a single branch at infinity. To remove this restriction we had to include two new ingredients: we construct a one parameter family of heights for which Thuillier–Yuan's equidistribution theorem [44, 46] applies; we investigate systematically the arithmetic properties of the coefficients of the expansion of the Böttcher coordinates and their dependence on the parameters c,a. We propose also a new way to build the symmetry by relying on a recent algebraization result of Xie [45] that gives a criterion for when a formal curve in the affine plane is a branch of an algebraic curve. A characteristic feature of our proofs is to look at the dynamics induced by cubic polynomials over various fields: over the complex numbers and over p-adic fields (see e.g., Section 4.1), over the field of Laurent series (see the proof of Proposition 3.6 and Section 5), and over a number field (see Section 3). We use at one point the universality theorem of McMullen [35] which is a purely Archimedean statement. Moreover, the work of Kiwi [31] on non-Archimedean cubic polynomials over a field of residual characteristic zero plays a key role in the proof of Theorem B. Let us describe in more detail how we proceed, and so pick an irreducible algebraic curve C⊂Poly3 containing infinitely many PCF polynomials. We may suppose that neither c0 nor c1 are persistently pre-periodic on C. By a theorem of McMullen [33, Lemma 2.1] this is equivalent to say that both critical points exhibit bifurcations at some (possibly different) points in C. There is a more quantitative way to describe the set of bifurcations using the Green function gc,a(z):=limn→∞13nlogmax{1,|Pn(z)|}. Indeed both functions g0(c,a):=gc,a(c0), g1(c,a):=gc,a(c1) are non-negative and pluri-subharmonic, and it is a fact [13, Section 5] that the support of the positive measure Δg0|C (resp. Δg1|C) is equal to the set of parameters where c0 (resp. c1) is unstable. The first step consists in proving that g0|C and g1|C are proportional, and this conclusion is obtained by applying an equidistribution result of points of small height due to Yuan [46] and Thuillier [44]. We first observe that C is necessarily defined over a number field K since it contains infinitely many PCF polynomials, so that we may introduce the functions g0,v,g1,v for all (not necessarily Archimedean) places v over K. These functions can now be used to build a one-parameter family of heights on C by setting   hs(p):=1deg(p)∑q,vmax{s0g0,v(q),s1g1,v(q)}, where s=(s0,s1)∈ℝ+2, and the sum ranges over all Galois conjugates q of p and over all places v over K. When s0 and s1 are positive integers, then we prove in Section 3 that the height hs is induced by a continuous semi-positive adelic metrization in the sense of Zhang on a suitable line bundle over C of positive degree, so that Thuillier–Yuan's theorem applies. This gives us sufficiently many restrictions on g0 and g1 which force their proportionality. The key arguments are Proposition 3.6 that is close in spirit to [3, Proposition 2.1 (3)], and the fact that the function max{g0,v,g1,v} is a proper continuous function on Poly3 for any place v. From the proportionality of g0 and g1 on a special curve, we are actually able to conclude the proof of Theorem B. This step is done in Section 5. We suppose by contradiction that our special curve C is an irreducible component of some Perm(λ) with λ≠0. Then each branch at infinity of C defines a cubic polynomial over the field of Laurent series ℂ((t)). We show that except when c0 or c1 is persistently preperiodic in C the multipliers of all periodic points are exploding on that branch by [31]. We then analyze the situation of a unicritical (that is, having a single critical point) polynomial in C and, computing the norm of the multiplier of its periodic points in a suitable field of residual characteristic 3, we are able to get the required contradiction. Let us come back to the proof of Theorem A. At this point, we have an irreducible algebraic curve C defined over a number field K and such that g0,v=g1,v at any place v over K. Recall that for any polynomial Pc,a there exists an analytic isomorphism near infinity conjugating the polynomial to the cubic monomial map. This isomorphism is referred to as the Böttcher coordinate φc,a of Pc,a. We prove that when c,a are defined over a number field then φc,a is a power series with coefficients in a number field whose domain of convergence is positive at any place, see Lemma 2.2 and Proposition 2.3. Building on an argument of Baker and DeMarco, we then show that outside a compact subset of the analytification of C (for any completion of K) the values of the Böttcher coordinates at c0 and c1 are proportional up to a root of unity (Theorem 4.1 (2)). The proof now takes a slight twist as we fix any polynomial P:=Pc,a that is not post-critically finite and for which (c,a) belongs to C(L) for some finite field extension L of K. We prove that any such polynomial admits a weak form of symmetry in the sense that there exists a curve ZP⊂ℙ1×ℙ1 that is stable by the map (P,P). To do so we apply [45, Theorem 1.5] as an alternative to the arguments of Baker and DeMarco in [3, Section 5.6]. In order to get a polynomial that commutes with P instead of a correspondence, we proceed as Baker and DeMarco and use Medvedev-Scanlon's result [39, Theorem 6.24] (see [40, Theorem 4.9] for another proof of this result). At this point, we have proved the following result that we feel is of independent interest. Theorem C. Pick any irreducible complex algebraic curve C⊂Poly3. Then the following assertions are equivalent: the curve C is special, for any critical point that is not persistenly pre-periodic on C, the set of PCF polynomials lying in C is equal to the set where this critical point is pre-periodic; the curve C is defined over a number field K and there exist integers (s0,s1)∈ℕ2∖{(0,0)} such that for any place v∈MK, we have   s0⋅g0,v=s1⋅g1,v, on the analytification of C over the completion of K w.r.t. the v-adic norm; the curve C is special, and for any sequence Xk⊂C of Galois-invariant finite sets of PCF polynomials with Xk≠Xl for l≠k, the probability measures μk equidistributed on Xk converge towards (a multiple of) the bifurcation measure Tbif∧[C] as k→∞; there exists a root of unity ζ, and integers q,m≥0 such that the polynomial Qc,a(z):=ζPc,am(z)+(1−ζ)c2 commutes with any iterate Pc,ak such that ζ3k=ζ, and Qc,a(Pc,aq(ci))=Pc,aq(cj) for some i,j∈{0,1} and all (c,a)∈C.     □ In (4) the current Tbif is defined as the ddc of the plurisubharmonic function g0+g1. Its support in ℂ2 is known to be equal to the set of unstable parameters, see for example, [20, Section 3]. Notice that for any curve C there exists a critical point which is not persistently pre-periodic on C since by [7] the set {g0=0}∩{g1=0} is compact in Aℂ2. In particular, the assertion (2) is consistent. To complete the proof of Theorem A, we analyze in more detail the possibilities for a cubic polynomial to satisfy the condition (5) in the previous theorem. Namely, we prove that the set of parameters admitting a non-trivial symmetry of degree 3m>1 is actually finite. Theorems A and C are proved in Section 7. We have deliberately chosen to write the entire article for cubic polynomials only. This simplifies the exposition, but many parts of the proof actually extend to a larger context. Let us briefly discuss the possible extensions and the limitations of our approach. All ingredients are present to prove Baker–DeMarco's conjecture for a curve in the space of polynomials of any degree d≥2. It is however not clear to the authors how to obtain the more precise classification of special curves in the same vein as in Theorem A. There are serious difficulties that lie beyond the methods presented here to handle higher dimensional special varieties V in Polyd. The main issue is the following. To apply Yuan's equidistribution theorem of points of small heights it is necessary to have a continuoussemi-positive adelic metric on an ample line bundle on a compactification of V, and we are at the moment very far from being able to check any of the three underlined conditions. Trying to understand special curves in the space of quadratic maps requires much more delicate estimates than in the case of polynomials. A first important step has been done by DeMarco, Wang and Ye in a recent article [16]. 2 The Böttcher Coordinate of a Polynomial In this section, K is any complete metrized field of characteristic zero containing a square-root ω of 13. It may or may not be endowed with a non-Archimedean norm. If X is an algebraic variety over K, then Xan denotes its analytification as a real-analytic or a complex variety if K is Archimedean, and as a Berkovich analytic space when K is non-Archimedean (see e.g. [2, Section 3.4–5]). 2.1 Basics As in the introduction, we denote by Poly3≃A2 the space of cubic polynomials defined by   Pc,a(z):=13z3−c2z2+a3. (1) It is a branched cover of the parameter space of cubic polynomials with marked critical points. The critical points of Pc,a are given by c0:=c and c1:=0. For a fixed (c,a)∈K2 the function 13log+|Pc,a(z)|−log+|z| is bounded on AK1,an so that the sequence 13nlog+|Pc,an(z)| converges uniformly to a continuous sub-harmonic function gc,a(z) that is called the Green function of Pc,a. We shall write g0(c,a):=gc,a(c0), g1(c,a):=gc,a(c1), and   G(c,a):=max{g0(c,a),g1(c,a)}. Proposition 2.1. The function G(c,a) extends continuously to the analytification AK2,an, and there exists a constant C=C(K)>0 such that   supAK2,an|G(c,a)−log+max{|a|,|c|}|≤C, and this constant vanishes when the residual characteristic of K is at least 5.     □ Proof. A proof of this fact is given in [7, Section 4] (see also [20, Section 6] for a more detailed proof) in the Archimedean case. When the normed field is non-Archimedean, it is proved in [21, Proposition 2.5] that the sequence hn:=max{13nlog+|Pc,an(c0)|,13nlog+|Pc,an(c1)|} converges uniformly on bounded sets in K2 to G(c,a). Since hn extends continuously to AK2,an, it follows that G too. The rest of the proposition also follows from op. cit.     ▪ 2.2 Expansion of the Böttcher coordinate For any cubic polynomial P∈K[z], we let the Böttcher coordinate of P be the only formal power series φ satisfying the equation   φ∘P(z)=φ(z)3 (2) which is of the form   φ(z)=ωz+α+∑k≥1akz−k, (3) with α,ak∈K for all k≥1. Recall that ω2=1/3. Lemma 2.2. Given any (c,a)∈K×K, the Böttcher coordinate φc,a(z) of the cubic polynomial Pc,a:=z33−c2z2+a3 exists, is unique, and satisfies   φc,a(z)=ω(z−c2)+∑k≥1ak(c,a)z−k, where   ak(c,a)∈ℤ[ω,12][c,a] with deg(ak)=k+1. (4) Moreover, the 2-adic (resp. 3-adic) norm of the coefficients of ak are bounded from above by 2k+1 (resp. 3k/2).     □ Proof. The defining equation (2) reads as follows:   (ω(z−c2)+∑k≥1ak(c,a)z−k)3 =ω(z33−c2z2+a3−c2)+∑k≥13kak(c,a)z3k(1−3c2z+3a3z3)k An immediate check shows that terms in z3 and z2 are identical on both sides of the equation. Identifying terms in z yields   3ω3(c2/4)+3ω2a1=0, so that a1=−ω4c2, whereas identifying constant terms, we get   3ω2a2+6ω2(−c/2)a1+ω3(−c3/8)=ω(a3−c/2) hence   a2=−5ω24c3+13ω(a3−c2). This shows (4) for k=1,2, since ω−1=3ω. We now proceed by induction. Suppose (4) has been proven for k. Identifying terms in z−(k−1) in the equation above, we get   3ω2ak+1−3cω2ak+3ω24c2ak−1  +ω∑i+j=kaiaj−ωc2∑i+j=k−1aiaj+∑i+j+l=k+1aiajal =∑l≥13lal[(1+3c2z+a3z3)−l]k+1−3l, where [(1+3c2z+a3z3)−l]j denotes the coefficient in z−j of the expansion of (1+3c2z+a3z3)−l in power of z−1. Observe that this coefficient belongs to ℤ[12][c,a], has 2-adic norm ≤2l, and is a polynomial in c,a of degree at most j. It follows that the polynomial   al(c,a)[(1+3c2z+a3z3)−l]k+1−3l is of degree at most k+1−3l+l+1=k+2−2l<k+1. The induction step is then easy to complete using again ω−1=3ω.     ▪ 2.3 Extending the Böttcher coordinate Recall that G(c,a)=max{g0(c,a),g1(c,a)}. Proposition 2.3. There exists a constant ρ=ρ(K)≥0 such that the Böttcher coordinate of Pc,a is converging in {z,log|z|>ρ+G(c,a)}. There exists another constant τ=τ(K)≥0 such that the map (c,a,z)↦φc,a(z) extends as an analytic map on the open set   {(c,a,z)∈AK2,an×AK1,an,gc,a(z)>G(c,a)+τ}, and φc,a defines an analytic isomorphism from Uc,a:={gc,a>G(c,a)+τ} to AK1,an∖D(0,eG(c,a)+τ)¯ satisfying the equation (2) on Uc,a. We have   gc,a(z)=log|φc,a(z)|K  on Uc,a. (5) Finally, τ=0 except if the residual characteristic of K is equal to 2 or 3.     □ We shall use the following lemma which follows easily from e.g. [21, Proposition 2.3]. Lemma 2.4. There exists a constant θ=θ(K)≥0  supAK1,an|gc,a(z)−log+|z||≤θ. Moreover, θ is equal to 0 except if the norm on K is Archimedean or the residual characteristic of K is equal to 2 or 3.     □ Proof of Proposition 2.3. Assume first that K is Archimedean, and set τ=0. In that case most of the statements are proved in [17] (see also [7, Section 1]). In particular, φc,a(z) is analytic in a neighborhood of ∞ and extends to Uc,a by invariance and defines an isomorphism between the claimed domains. It is moreover analytic in c,a,z. To estimate more precisely the radius of convergence of the power series (3), we rely on [7, Section 4] as formulated in [20, Section 6]. First choose C=CK>0 such that G(c,a)>log+max{|a|,|c|}−C. Then log|z|>C+G(c,a) implies |z−c2|>max{1,|a|,|c|}−|c2|≥12max{1,|a|,|c|} hence log|z−c2|>G(c,a)−log2, so that gc,a(z)>log|z−c2|−log4>G(c,a), and φ converges in {z,log|z|>G(c,a)+ρ} with ρ:=C as required. From now on, we assume that the norm on K is non-Archimedean. When the residual characteristic of K is different from 2 and 3, then (4) implies |ak|≤max{1,|c|,|a|}k+1 so that φ converges for |z|>max{1,|c|,|a|}, and log|φ(z)|=log|z|. Recall that we have G(c,a)=logmax{1,|c|,|a|} by Proposition 2.1 so that one can take ρ=0. Pick any z such that g(z)>G(c,a), and observe that |Pn(z)|→∞. Then we get   gc,a(z)=limn→∞13nlog|Pn(z)|=limn→∞13nlog|φ(Pn(z))|=log|φ(z)|=log|z|. (6) In particular the set {g>G(c,a)} is equal to AK1,an∖D(0,eG(c,a)¯), and φ is an analytic map from that open set onto itself. It is an isomorphism since log|φ(z)|=log|z| as soon as g(z)>G(c,a). The proposition is thus proved in this case with τ=0. In residual characteristic 2, |ak|≤(2max{1,|c|,|a|})k+1 whence φ converges for |z|>2max{1,|c|,|a|}, and as above log|φ|=log|z| in that range. Recall that G(c,a)−log+max{|c|,|a|}≥C=C(K), so that log|z|>G(c,a)+log2−CK implies |z|>2max{1,|c|,|a|}, which proves that the power series (3) converges for log|z|>G(c,a)+ρ with ρ=log2−C. Set τ:=ρ+θ where θ is the constant given by Lemma 2.4. Using log|φ(z)|=log|z| as above, we get that φc,a defines an analytic isomorphism from Uc,a:={gc,a>G(c,a)+τ} to AK1,an∖D(0,eG(c,a)+τ)¯. In residual characteristic 3, |ak|≤(31/2max{1,|c|,|a|})k+1 whence φ converges for |z|>31/2max{1,|c|,|a|}, and log|φ|=log|ωz| in that range. Recall that G(c,a)−log+max{|c|,|a|}≥C=C(K), so that log|z|>G(c,a)+log31/2−CK implies |z|>31/2max{1,|c|,|a|}, which proves that the power series (3) converges for log|z|>G(c,a)+ρ with ρ=log31/2−C. We conclude the proof putting τ:=ρ+θ as before.     ▪ Remark. It is possible to argue that τ=0 also in residual characteristic 2. Although we do not know the optimal constant τ in residual characteristic 3, the Böttcher coordinate is likely not to induce an isomorphism from {gc,a>G(c,a)} to AK1,an∖D(0,eG(c,a))¯.     □ 3 Curves in Poly3 In this section, we fix a number field K containing a square-root ω of 13 and take an irreducible curve C in Poly3 that is defined over K. Our aim is to build suitable height functions on C for which the distribution of points of small height can be described using Thuillier–Yuan's theorem. Our main statement is Theorem 3.12 below. Recall that given any finite set S of places of K containing all Archimedean places, OK,S denotes the ring of S-integers in K that is of elements of K of v-norm ≤1 for all v∉S. We also write Kv for the completion of K w.r.t. the v-adic norm. 3.1 Adelic series A formal power series ∑nanzn is said to be adelic on K if there exists a finite set S of places on K such that an∈OK,S for all n∈ℕ; and for each place v on K the series has a positive radius of convergence rv:=limsupn→∞|an|v−1/n>0. Observe that rv=1 for all but finitely many places. Lemma 3.1. Suppose α(t)=∑nantn is an adelic series with a0=0 and a1≠0. Then there exists an adelic series β such that β∘α(t)=t.     □ Proof. Suppose an∈OK,S for all n, and write β(t)=∑nbntn. The equation β∘α(t)=t amounts to b0=0, b1=a1−1, and the relations   bna1n+∑1≤k≤n−1bk[(∑j≤najtj)k]n=0, for any n≥2 where [⋅]n denotes the coefficient in tn of the power series inside the brackets. It follows that bn∈OK,S′ for all n where S′ is the union of S and all places v for which |a1|v>1. The convergence of the series follows from Cauchy–Kowalewskaia's method of majorant series or from the analytic implicit function theorem, see [12] and [41, p. 73].     ▪ Lemma 3.2. Pick k∈ℕ*, and suppose α(t)=∑n≥kantn is an adelic series with ak≠0. Then there exists an adelic series β such that β(t)k=α(t).     □ Proof. As in the previous proof, suppose an∈OK,S for all n, and write β(t)=b1t+∑n≥2bntn. We get b1k=a1, and for all n≥2  an=kb1k−1bn−k+Pn(b1,…,bn−k−1), where Pn is a polynomial with integral coefficients. This time all coefficients bn belong to a finite extension of K containing a fixed k-th root of a1, and S′ is the union of S and all places v such that |kb1k−1|v<1. The analyticity of the series is handled as in the previous proof.     ▪ Lemma 3.3. Pick k∈ℕ*, and suppose α(t)=∑n≥kantn is an adelic series with ak≠0. Then there exists an adelic series β such that α∘β(t)=tk.     □ Proof. The equation α∘β(t)=tk is equivalent to   (1+∑j≥2ajtj−1)k(1+∑l≥1αl(t+∑i≥2aiti)l)=1. Identifying terms of order tn, one obtains   kan+1+[(1+∑2≤j≤najtj−1)k(1+∑l≥1αl(t+∑1≤i≤naiti)l)]n=0 which shows that β is unique, has coefficients in OL,S′ where S′ contains S and all places at which |k|v<1. The fact that β is analytic at all places is a consequence of the inverse function theorem and the fact that the power series t↦(1+t)1/n:=1+1nt+(1/n)(1/n−1)2t2+O(t3) has a positive radius of convergence.     ▪ We shall also deal with adelic series at infinity which we define to be series of the form α(z)=∑0≤k≤Nbkzk+∑k≥1akzk with N∈ℕ, bk,ak∈OK,S and ∑k≥1aktk is an adelic series. Observe that this is equivalent to assuming that α(t−1)−1 is an adelic series. 3.2 Puiseux expansions We shall need the following facts on the Puiseux parameterizations of a curve defined over K. These are probably well-known but we include a proof for the convenience of the reader. Proposition 3.4. Suppose P∈K[x,y] is a polynomial such that P(0,0)=0 and P(0,y) is not identically zero. Denote by n:D^→D:={P=0} the normalization map, and pick any point c∈n−1(0)∈D^. Then one can find a finite extension L of K, a finite set of places S of L, a positive integer n>0, and an adelic series β(t)∈OL,S[[t]] such that there is an isomorphism of complete local rings OD^,c^≃L[[t]]; the formal map t↦(tn,β(t)) parameterizes the branch c in the sense that x(n(t))=tn, and y(n(t))=β(t).     □ A branch of D at the origin is by definition a point in n−1(0). Proof. We first reduce the situation to the case D is smooth at 0. To do so we blow-up the origin X1→A2 and let D1 be the strict transform of D. Since D^ is normal the map n lifts to a map n1:D^→D1, and we let p1 be the image of c in D1. In the coordinates (x,y)=(x′,x′y′) (or (x'y', y')) the point p1 has coordinates (0,y1) where y1 is the solution of a polynomial with values in K hence belongs to an algebraic extension of this field. We may thus choose charts (x,y)=(x′,(x′+c)y′) (or (x′(y′+c),x′)) with c∈K¯ such that c is now a branch of D1={P1=0} at the origin, and P1∈K¯[x′,y′]. We iterate this process of blowing-up to build a sequence of proper birational morphisms between smooth surfaces Xi+1→Xi, i=1,…,N until we arrive at the following situation for X:=XN: the strict transform C of D by π:X→A2 is smooth at a point p∈π−1(0) and intersects transversally the exceptional locus of π. The normalization map n:D^→D lifts to a map m:D^→C and the image of c by m is equal to p. Finally there exist coordinates z,w centered at p such that (x,y)=π(z,w)=(A(z,w),B(z,w)) with A,B∈K¯[z,w], the exceptional locus of π contains {z=0}, and C={R(z,w):=w−za(z)−wQ(z,w)=0} where a∈K¯[z], Q∈K¯[z,w] and Q(0,0)=0. Fix an algebraic extension L of K and S finitely many places of L such that A,B, R have their coefficients in OL,S. We now look for a power series γ(t)=∑k≥1γktk such that R(t,γ(t))=0. Its coefficients satisfy the relations   γk=[t2a(t)]k+[(∑j=1k−1γjtj)Q(t,∑j=1k−1γjtj)]k which implies that γ exists, is unique, and all its coefficients belongs to OL,S. It follows from the analytic implicit function theorem, that γ is also analytic as a power series in Lv[[t]] for any place v. Let us now consider the two power series (α(t),δ(t)):=π(t,γ(t)). They both belong to OL,S, are analytic at any place, and we have P(α(t),δ(t))=0. Since P(0,y) is not identically zero, we may write α(t)=tn(a+∑k≥1αktk) for some n>0 and a≠0. Replacing L by a suitable finite extension, and t by a′t for a suitable a′ we may suppose that a=1 and αk∈OL,S for all k. By Lemma 3.3, there exists an invertible power series a^(t)=t+∑k≥2aktk that is analytic at all places with coefficients ak∈OL,S and such that α°a^(t)=tn. Once this claim is proved one sets β(t):=δ°a^(t), so that π(a^(t),γ(a^(t)))=(tn,β(t)). Since m is injective and maps the smooth point c∈D^ to the smooth point p∈C, it induces an isomorphism of complete local rings OC,p^≃OD^,c^. Observe that the complete local ring OC,p^=L[[z,w]]/⟨R⟩ is isomorphic to L[[t]] by sending the class of a formal series Φ to Φ(t,γ(t))). Composing with the isomorphism of L[[t]] sending t to a^(t), we get an isomorphism OD^,c^≃L[[t]] such that (x(n(t)),y(n(t)))=π(n(t))=π(a^(t),γ(a^(t)))=(tn,β(t)) as required.     ▪ 3.3 Branches at infinity of a curve in Poly3 Consider an irreducible affine curve C⊂Poly3 defined over a number field K. We denote by Poly3¯≃ℙ2 the natural compactification of Poly3≃A2 using the affine coordinates (c,a). Let C¯ be the Zariski closure of the curve C in Poly3¯, and n:C^→C¯ be its normalization. A branch at infinity of C is a point in C^ lying over C¯∖C. Proposition 3.5. There exists a finite extension L of K such that the following holds. For any branch c of C at infinity there is an isomorphism of complete local rings OC^,c^≃L[[t]] such that c(n(t)),a(n(t)) are adelic series at infinity.     □ Proof. Pick a branch at infinity c of C. Let p* be the image of c in Poly3¯≃ℙ2. It is given in homogeneous coordinates by p*=[c*:a*:0] and since C is defined over K we may assume c*,a* are algebraic over K. To simplify the discussion we shall assume that c*=1 so that p*=[1:a*:0] (otherwise, p*=[0:1:0] and the arguments are completely analoguous). Let d be the degree of a defining equation P∈K[c,a] of C. Observe that Q(τ,α):=τdP(1τ,ατ−a*) is a polynomial vanishing at (0,0) such that Q(0,α) is not identically zero. Note that {Q=0} can be identified to an open Zariski subset of the completion of {P=0} in Poly3¯, and c with a branch of {Q=0} at the origin. Apply Proposition 3.4 to this branch c. We get a finite extension L, a finite set of places S of L containing all archimedean ones, a positive integer n, an isomorphism of complete local ring OC^,c≃L[[t]], and a power series β∈OL,S[[t]] that is analytic at all places such that α(n(t))=β(t) and τ(n(t))=tn. It follows that c(n(t))=t−n, and a(n(t))=t−nβ(t)−a*∈OL,S[[t]].     ▪ 3.4. Estimates for the Green functions on a curve in Poly3 In this section, we fix an irreducible curve C in Poly3 defined over a number field K and let L be a finite extension of K for which Proposition 3.5 applies. Fix a place v of L, and let g0,v(c,a) be the function g0,v evaluated at c,a in the completion Lv of L with respect to the v-adic norm. By [20] and [21, Proposition 2.4], the function g0,v is the uniform limit on compact sets of 13nlog+|Pc,an(c0)|v. It follows that its lift to the normalization of C is sub-harmonic (in the classical sense when v is Archimedean and in the sense of Thuillier [44] when v is non-Archimedean). To simplify notations, we also write g0,v(t):=g0,v(c(n(t)),a(n(t))) where the adelic series at infinity c(n(t)) and a(n(t)) are given as above. Proposition 3.6. For each branch c of C at infinity, one of the following two situations occur. For any place v of L, the function g0,v(t) extends as a locally bounded subharmonic function through c. There exist a finite set of place S of L, and two constants a(c)∈ℚ+* and b(c)∈OL,S such that g0,v(t)=a(c)log|t|v−1+log|b(c)|v+o(1) for any place v on L.     □ Remark. This key result is very similar to [3, Proposition 2.1]. Ghioca and Ye have proved that g0,v(t) actually extends to a continuous function at t=0 in case 1. We also refer to [14, Proposition 3.1] for a version of this result in the case of rational maps.     □ Notation. We endow the field L((t)) with the t-adic norm so that for any Laurent series Q=∑aktk we have |Q|t:=e−ordt(Q) with ordt(Q)=min{k,ak≠0}. The resulting valued field is complete and non-Archimedean. In order to avoid confusion, we denote by P(z)∈L((t))[z] the cubic polynomial induced by the family (Pc(n(t)),a(n(t)))t. Observe that the critical points of P are given by c0 and c1 which correspond to the adelic series at infinity 0 and c(n(t)), respectively.     □ Proof of Proposition 3.6. For each q∈ℕ*, we set eq:=|Pq(c0)|t, so that either the sequence {eq}q∈ℕ is bounded (that is, c0 belongs to the filled-in Julia set of P) or eq→∞ (exponentially fast). Suppose we are in the former case, and consider the sequence of subharmonic functions 13qlog+|Ptq(0)|v defined on a punctured disk Dv* centered at 0 in ALv1,an. Since 13qlog+|Ptq(0)|v=log+eq3qlog|t|v−1+O(1), the function   hq:=13qlog+|Ptq(0)|v−log+eq3qlog|t|v−1 is subharmonic on Dv* and locally bounded near 0. It thus extends as a subharmonic function to Dv by the next lemma. Lemma 3.7. Any subharmonic function on Dv* that is bounded from above in a neighborhood of the origin is the restriction of a subharmonic function on Dv.     □ Proof. In the Archimedean case, this follows from [28, Theorem 3.4.3]. Let us explain how to adapt these arguments to the non-Archimedean setting. Let h be a subharmonic function on Dv*, and suppose it is non-positive. For each integer n consider the function hn:=h+1nlog|t| with the convention hn(0)=−∞. Observe that hn is the pointwise decreasing limit of the sequence of subharmonic functions max{hn,−A} as A→∞, hence is subharmonic by [44, Proposition 3.1.9]. Letting n→∞, we get an increasing sequence of subharmonic functions that is locally bounded and converging pointwise on Dv* to h. The upper-semicontinuous regularization h* of limnhn is thus subharmonic on Dv* by [44, Proposition 3.1.9] and extends h as required.     ▪ Since 13qlog+|Ptq(0)|v converges uniformly on compact subsets in Dv* to g0,v, hq is uniformly bounded from above on its boundary, hence everywhere by the maximum principle. It follows from Hartog's theorem (see e.g., [27, Theorem 1.6.13] in the Archimedean case, and [22, Proposition 2.18] in the non-Archimedean case) that hq converges (in Lloc1 in the Archimedean case, and pointwise at any non-rigid point in the non-Archimedean case) to a subharmonic function, hence g0,v is subharmonic on Dv. But g0,v is non-negative so that (1) holds. Suppose that eq→∞. Recall that c(n(t)) and a(n(t)) are adelic series at infinity that belong to t−nOL,S[[t]] for a suitable integer n≥1. Write φt:=φPc(n(t)),a(n(t)). Lemma 3.8. There exists an integer q≥1 such that for any place v of L, there exists ϵ>0 such that Ptq(c0) belongs to the domain of convergence of φt for any |t|v<ϵ.     □ Proof. Indeed Ptq(c0) is an adelic series at infinity having a pole of order logeq. On the other hand, we have   G(t):=G(c(n(t)),a(n(t)))≤logmax{|c(n(t))|,|a(n(t))|}+C               ≤nlog|t|−1+O(1) by Proposition 2.1. By assumption we may take logeq to be as large as we want so that log|Ptq(c0)|v−G(t)→∞ for any fixed place v when |t|v→0. We conclude by Proposition 2.3.     ▪ Our objective is to estimate φt(Ptq(c0)). Recall from Lemma 2.2 that   φc,a(z)=ω(z−c2)+∑k≥1ak(c,a)z−k, with ak∈ℤ[ω,12][c,a] of degree ≤k+1. It follows that   ak:=ak(c(n(t)),a(n(t)))∈t−n(k+1)OL,S[[t]], so that one can define   φP(z):=φc(n(t)),a(n(t))(z)=ω(z−c(n(t))2)+∑k≥1akz−k as an element of the ring t−nzOL,S((t))[[(tnz)−1]]. On the other hand, Pc,aq(c0) is a polynomial in c,a of degree ≤3q with coefficients in ℤ[12,13] hence, if   c0:=c0(n(t)) and Pq(z):=Pc(n(t)),a(n(t))q(z), we have Pq(c0)∈t−3qnOL,S[[t]], so that   ak(Pq(c0))k∈t3qnk−n(k+1)OL,S[[t]]⊂tnkOL,S[[t]]. (7) It follows that Θ:=∑k≥1ak(Pq(c0))k converges as a formal power series and belongs to tnOL,S[[t]]. Observe that Lemma 3.8 shows that Θ is convergent at all places hence defines an adelic power series. Fix a place v of L and choose |t|v small enough. Then we get   φt(Ptq(c0))=ω(Ptq(0)−c(n(t))2)+Θ(t)=ω(Ptq(0)−c(n(t))2)+o(1). (8) By (8), for |t|v small enough, one obtains   g0,v(t)=13qlog|φt(Ptq(0))|v=13qlog|ω(Ptq(0)−12tn)|v+o(1)=13qlog|∑0≤k≤n0bk,0tk|v+o(1)==n03qlog|t|v−1+log|∑0≤k≤n0bk,0tn0−k|v+o(1) where bk,0∈OL,S, and bn0,0≠0. And the proof is complete with a(c):=n03q, and b(c)=bn0,0. Proposition 3.9. Fix any two positive integers s:=(s0,s1), and for any place v define   gs,v(c,a):=max{s0g0,v(c,a),s1g1,v(c,a)}. (9) Then there exists an integer q≥1 such that   gs,v(c,a)=13qmax{s0log+|Pc,aq(c0)|,s1log+|Pc,aq(c1)|} (10) for all but finitely many places.     □ Proof. During the proof S is a finite set of places on L that contains all Archimedean places and all places of residual characteristic 2 and 3. Pick any v∉S, and recall from [21, Proposition 2.5] that Gv(c,a)=log+max{|c|v,|a|v}. Suppose first that gs,v(c,a)=0. Then gc,a,v(c0)=gc,a,v(c1)=0 and Gv(c,a)=0 so that 13qlog+|Pc,aq(c0)|v=13qlog+|Pc,aq(c1)|v=0 for all q, and (9) holds in that case. Pick q large enough such that 3q>max{s1s0,s0s1}. Suppose now that 0<gs,v(c,a)=s0g0,v(c,a) so that s0g0,v(c,a)≥s1g1,v(c,a). Then   gc,a,v(Pc,aq(c0))=3qg0,v(c,a) ≥3qmin{s1s0,1}max{g0,v(c,a),g1,v(c,a)}>Gv(c,a). By (6), we get   gs,v(c,a)=s0g0,v(c,a)=s03qg0,v(Pc,aq(c0))=s03qlog+|Pc,aq(c0)|v. Now observe that either Pc,aq(c1) falls into the domain of definition of φc,a that is, log|Pc,aq(c1)|v>Gv(c,a) and g1,v(c,a)=13qlog+|Pc,aq(c1)|v, so that   gs,v(c,a)=max{s0g0,v(c,a),s1g1,v(c,a)}=13qmax{s0log+|Pc,aq(c0)|v,s1log+|Pc,aq(c1)|v}, as required. Or we have   s13qlog+|Pc,aq(c1)|v≤s13qlog+max{|a|v,|c|v}≤s0g0,v(c,a), and again (10) holds. We complete the proof by arguing in the same way when gs,v(c,a)=s1g1,v(c,a).     ▪ 3.5 Adelic semi-positive metrics on curves in Poly3 We fix a number field L and a finite set S of places of this field that contains all Archimedean places and all places of residual characteristics 2 and 3. We also assume that Propositions 3.5, 3.6, and 3.9 are all valid for these choices. Fix any pair of positive integers s0,s1∈ℕ*. For each place v, introduce the function   gs,v(c,a):=max{s0⋅g0,v(c,a),s1⋅g1,v(c,a)}, as in the previous section. Pick a branch at infinity c and choose parameterizations such that Proposition 3.6 is valid for g0,v and g1,v. Observe that   Gv(t)=max{g0,v(t),g1,v(t)}→∞ as t→0 by Proposition 2.1 so that either g0,v or g1,v tends to infinity near t=0. Since s0 and s1 are both positive, we obtain the existence of a(c)∈ℚ+* and b(c)∈OL,S such that   gs,v(t)=a(c)log|t|v−1+log|b(c)|v+o(1). (11) We replace the integers s0,s1 by suitable multiples such that the constants a(c) become integral (for all branch c), and we introduce the divisor D:=∑a(c)[c] on C^ where the sum is taken over all branches at infinity of C. Pick a place v, an open subset U⊂C^an,v and a section σ of the line bundle OC^(D) over U. By definition σ is a meromorphic function on U whose divisor of poles and zeroes satisfies div(σ)+D≥0. We set |σ|s,v:=|σ|ve−gs,v. Recall the notion of semi-positive metrics in the sense of Zhang from [11, Section 1.2.8 & 1.3.7]. We are now in position to prove Lemma 3.10. The metrization |⋅|s,v on the line bundle OC^(D) is continuous and semi-positive for any place v. The collection of metrizations {|⋅|s,v}v is adelic.     □ Proof. For any place v, and for any local section σ of the line bundle OC^(D) the function |σ|s,v is continuous by (11), therefore the metrization |⋅|s,v is continuous. Since gs,v is subharmonic on Cv,an, for any local section σ the function −log|σ|s,v is subharmonic on Cv,an. As it extends continuously to C^an,v, Lemma 3.7 implies that −log|σ|s,v is subharmonic on C^an,v. When v is Archimedean, the metrization is thus semi-positive by definition. When v is non-Archimedean the metrization is semi-positive by the next lemma. Finally the collection of metrizations {|⋅|s,v}v is adelic thanks to Proposition 3.9 and [21, Section 2.3]. Lemma 3.11. Suppose L→X is an ample line bundle on a smooth projective curve over a metrized field. Let |⋅| be any continuous metrization on L that is subharmonic in the sense that for any local section σ the function −log|σ| is subharmonic. Then the metrization |⋅| is semi-positive in the sense of Zhang.     □ Remark. This result holds true in arbitrary dimension over any field of the form L((t)) endowed with the t-adic norm where L has characteristic zero by [6].     □ Proof. One needs to show that |⋅| is the uniform limit of a sequence of semi-positive model metrics. By [44, Théorème 3.4.15], there exists an inductive set I and an increasing family of semi-positive model metrizations |⋅|i that are pointwise converging to |⋅|. Since |⋅| is continuous and X is compact, Dini's theorem applies and shows that the convergence is uniform. Observe that the notion of semi-positivity used in op. cit. for model metrics coincides with the one in [47] as explained in [44, Section 4.3.2].     ▪ We have thus obtained Theorem 3.12. Pick any positive integers s0,s1>0. Then there exists a positive integer n≥1, and a non-zero effective and integral divisor D on C^ such that the collection of subharmonic functions   gs,v(c,a):=max{ns0⋅g0,v(c,a),ns1⋅g1,v(c,a)}, (c,a)∈Cv,an induces a semi-positive adelic metrization on the line bundle OC^(D).     □ Remark. Since the metrization |⋅|s,v is continuous its curvature form (see [11]) does not charge any point, and is given by the pull-back by n of the positive measure Δgs,v restricted to the set of regular points on C, see [44, Section 3.4.3]. To simplify notations we shall simply write this curvature form as Δgs,v.     □ Remark. The line bundle OC^(D) is defined over the same number field as C.     □ Remark. It is likely that gs,v defines a semi-positive adelic metrization on an ample line bundle over a suitable compactification of Poly3, but this seems quite delicate to prove for arbitrary s=(s0,s1)∈(ℕ*)2.     □ 4. Green Functions on Special Curves This section is devoted to the proof of Theorem 4.1 below. If K is a number field, and v a place of K, recall the definition of τv=τ(Kv) from Proposition 2.3, and that τv=0 if the residual characteristic of K is larger than 5. Theorem 4.1. Let C be an irreducible curve in the space Poly3 of complex cubic polynomials parameterized as in (1). Suppose that C contains infinitely many post-critically finite parameters and that neither c0 nor c1 is persistently pre-periodic. Then the following holds. The curve C is defined over a number field K and there exist positive integers s0,s1 such that for any place v of K  s0g0,v(c,a)=s1g1,v(c,a) for all (c,a)∈Cv,an. For any branch c of C at infinity, there exists an integer q≥1 and a root of unity ζ such that for any place v of K, one has   (φc,a(Pc,aq(c0)))s0=ζ⋅(φc,a(Pc,aq(c1)))s1 (12) on the connected component of {g0,v>τv/s0}={g1,v>τv/s1} in Cv,an whose closure in C^ contains c.     □ A remark is in order about the second assertion of the theorem. Remark. We shall prove that for any parameter on the connected component {g0,v>τv/s0}={g1,v>τv/s1} in Cv,an whose closure in C^ contains c, the two points Pc,aq(c0) and Pc,aq(c1) belong to the domain of definition of the Böttcher coordinate φc,a for q large so that (12) is consistent. We shall prove that (12) holds as an equality of adelic series at infinity. 4.1. Green functions are proportional The set of post-critically finite polynomials is a countable union of varieties   Vn,m:={Pc,an0+m0(c0)=Pc,an0(c0)}∩{Pc,an1+m1(c1)=Pc,an1(c1)} with n=(n0,n1)∈ℕ2 and m=(m0,m1)∈(ℕ*)2, and each Vn,m is cut out by two polynomial equations with coefficients in ℤ[12,13]. Since Vn,m(ℂ) are all contained in a fixed compact set by [7] (see also [20, Proposition 6.2]), it is a finite set, hence all its solutions are defined over a number field. It follows that C is an irreducible curve containing infinitely many algebraic points (cn,an). Let Q∈ℂ[c,a] be a defining equation for C with at least one coefficient equal to 1 and pick σ an element of the Galois group of ℂ over the algebraic closure of ℚ. Then Q∘σ vanishes also on {(cn,an)} hence everywhere on C, and therefore Q∘σ=λQ for some λ∈ℂ*. Since one coefficient of Q is 1, we get λ=1 and Q∈K[c,a] for a number field K. Recall that we denote by n:C^→C¯ the normalization of the completion C¯ of C in Poly3¯≃ℙ2. Pick any pair of positive integers s=(s0,s1) and scale them such that Theorem 3.12 applies with n=1. This gives us a non-zero effective divisor Ds supported on C^∖n−1(C). Replacing s by a suitable multiple, we may suppose that it is very ample and pick a rational function ϕ on C^ whose divisor of poles and zeroes is greater or equal to −Ds. Observe in particular that ϕ is a regular function on n−1(C) that vanishes at finitely many points. Consider the height hs induced by the semi-positive adelic metrics given by gs,v, see Theorem 3.12. If (c,a) is a point in n−1(C) that is defined over a finite extension K, denote by O(c,a) its orbit under the action of the absolute Galois group of K, and by deg(c,a) the cardinality of this orbit. Fix a rational function ϕ as above that is not vanishing at (c,a) (this exists since −Ds is very ample). Let MK be the set of places of K. By [11, Section 3.1.3], since ϕ(c,a)≠0 we have   hs(c,a)=1deg(c,a)∑O(c,a)∑v∈MK−log|ϕ|s,v(c′,a′)=1deg(c,a)∑O(c,a)∑v∈MK(gs,v−log|ϕ|v)(c′,a′)=1deg(c,a)∑O(c,a)∑v∈MKgs,v(c′,a′)≥0 where the last equality follows from the product formula. We now estimate the total height of the curve C^ using [11, (1.2.6) & (1.3.10)]. Choose any two meromorphic functions ϕ0,ϕ1 such that div(ϕ0)+Ds and div(ϕ1)+Ds are both effective with disjoint support included in n−1(C). Let σ0 and σ1 be the associated sections of OC^(Ds). Let ∑ni[ci,ai] be the divisor of zeroes of σ0, and ∑n′j[c′j,a′j] be the divisor of zeroes of σ1. Then   hs(C^)=∑v∈MK(div^(σ0)⋅div^(σ1)|C^)v=∑inihs(ci,ai)−∑v∈MK∫C^log|σ0|s,vΔgs,v=∑v∈MK∫C^gs,vΔgs,v≥0, where the third equality follows from Poincaré–Lelong formula and writing log|σ0|s,v=log|ϕ|v−gs,v with ϕ∈K(C) defining the section σ0. The formula for the height of a closed point implies that for all post-critically finite polynomials Pcn,an we have hs(cn,an)=0. Since PCF polynomials are Zariski dense in C, the essential minimum of hs is non-positive. By the arithmetic Hilbert–Samuel theorem (see [44, Théorème 4.3.6], [1, Proposition 3.3.3], or [47, Theorem 5.2]), we get hs(C^)=0 hence we may apply Thuillier–Yuan's theorem (see [44, 46] and [10, Théorème 4.2]). It follows that the sequence of probability measures μn,v that are equidistributed on O(cn,an) in C^v,an converges to a probability measure μ∞,v that is proportional to Δgs,v. We may thus write μ∞,v=w(s)Δgs,v where w(s)∈ℝ+* is equal to the inverse of the mass of Δgs,v, that is, to deg(Ds)−1. Applying the arguments of the previous paragraphs to all choices of positive integers (s0,s1) (suitably scaled so that Theorem 3.12 applies), we conclude from the equidistribution of PCF polynomials that the measure μ∞,v is independent of s. We now observe that gs,v is homogeneous in s (i.e., gτs,v=τgs,v for any τ∈ℝ+*), and continuous with respect to this parameter. It follows that w(s) is also continuous on (ℝ+*)2, and μ∞,v=w(s)Δgs,v for all s∈(ℝ+*)2. From now on we fix an Archimedean place v. We shall treat the non-Archimedean case latter. We work in n−1(Cv,an) which is the complement of finitely many points in the analytification of the smooth projective curve C^v,an. To simplify notation we write g0,v, g1,v instead of g0,v∘n,g1,v∘n. Recall that by [20, Theorem 2.5] (see also [33, Theorem 2.2] or [14, Theorem 1.1]) the equality g0,v=0 on n−1(Cv,an) implies that c0 is persistently pre-periodic. Since we assumed that both c0 and c1 are not persistently pre-periodic, the functions g0,v and g1,v are not identically zero on n−1(Cv,an). Recall also that g0,v is harmonic where it is positive and that the support of Δg0,v is exactly the boundary of {g0,v=0} (see e.g. [20, Proposition 6.7]). In particular Δg0,v is a non-zero positive measure, and its mass is finite by Proposition 3.6. Observe now that gs,v→g0,v uniformly on compact sets when s tends to (1,0), hence Δgs,v→Δg0,v and Δg0,v=t0μ∞,v for some positive t0. In the same way, we get Δgs,v→Δg1,v as s→(0,1) which implies that the three positive measures μ∞,v, Δg0,v and Δg1,v are proportional. We may thus find s0,s1>0 such that the function Hv:=s0g0,v−s1g1,v is harmonic on n−1(Cv,an). Recall from [34] that the bifurcation locus of the family Pc,a parameterized by (c,a)∈n−1(Cv,an) is defined as the set where either c0 or c1 is unstable (or active in the terminology of [20]). It follows from [20] that the bifurcation locus is equal to the union of the support of Δg0,v and Δg1,v, hence to the support of μ∞,v. Suppose now that Hv is not identically zero. Then this support is included in the locus {Hv=0} which is real-analytic. This is impossible by McMullen's universality theorem, since the Hausdorff dimension of the bifurcation locus of any one-dimensional analytic family is equal to 2, see [35, Corollary 1.6]. We have proved that s0g0,v=s1g1,v on n−1(Cv,an) hence on Cv,an for some positive real numbers s0,s1>0. Since g0,v and g1,v are proportional, and Gv=max{g0,v,g1,v} is proper on Cv,an, it follows that g0,v is unbounded near any branch at infinity. By Proposition 3.6, g0,v admits an expansion of the form g0,v(t)=a(c)log|t|−1+O(1) with a(c)∈ℚ+* on the branch c hence is locally superharmonic on that branch. It follows that Δg0,v is a signed measure in C^an,v whose negative part is a divisor D0 with positive rational coefficients at any point of C^∖n−1(C). The same being true for Δg1,v, we obtain the equality of divisors s0D0=s1D1. This implies that s0/s1 is rational, and we can assume s0 and s1 to be integers. This ends the proof of the first statement in the case the place is Archimedean. Assume now that v is non-Archimedean. One cannot copy the proof we gave in the Archimedean setting since we used the fact that c0 is not persistently pre-periodic iff Δg0,v=0, and McMullen's universality theorem, two facts that are valid only over ℂ. Instead we apply Proposition 3.6. For each s′=(s′0,s′1) the function gs′,v extends near any branch c at infinity as an upper-semicontinuous function gs′,v^ whose Laplacian puts some non-positive mass at c. When s′0,s′1≠0 then gs′,v defines a positive continuous metric on OC^(Ds′) hence Δgs′,v^{c}=−ordc(Ds′)<0. This mass is in particular independent of the place. We get that   −Δg0,v^{c}≥lims→(1,0)−Δgs,v^{c}=ordc(D0)>0. We infer that the mass of Δg0,v is equal to the degree of D0 hence is non-zero. We may now argue as in the Archimedean case, and prove that Δg0,v and Δg1,v are proportional. The coefficient of proportionality is the only t>0 such that D0=tD1 hence t=s0/s1. Then Hv:=s0g0,v−s1g1,v is harmonic on C and bounded near any branch at infinity by Proposition 3.6, hence defines a harmonic function on the compact curve C^an,v. It follows Hv is a constant (in the non-Archimedean case by [44, Proposition 2.3.2]) which is necessarily zero since it is zero at all post-critically finite parameters. We have completed the proof of Theorem 4.1 (1). We mention here the following result that follows from the previous argument. Corollary 4.2. Let C be an irreducible curve in Poly3 defined over a number field K that contains infinitely many post-critically finite parameters and that neither c0 nor c1 is persistently pre-periodic. Pick an Archimedean place v. Pick any sequence Xn⊂C(K¯) of Galois-invariant finite sets of postcritically finite parameters such that Xn≠Xm for m≠n. Let μn be the measure equidistributed on Xn⊂Cv,an. Then the sequence μn converges weakly to (a multiple of) Tbif∧[C] as n→∞.     □ Recall that Tbif is defined as the ddc of the plurisubharmonic function g0+g1, and [C] is the current of integration over the analytic curve Cv,an. Proof. Let s0,s1>0 be given by Theorem 4.1. As seen above, the sequence μn converges weakly towards ddcmax{s0⋅g0,s1⋅g1}=s0⋅ddcg0 on Cv,an. It thus only remains to prove that ddc(g0|Cv,an)=κ⋅Tbif∧[C] for some κ>0. Recall that Tbif=ddc(g0+g1). By Theorem 4.1, on C,   g0+g1=g0+s1s0⋅g0=(1+s0s1)g0. Let κ:=1+s0s1. We thus have ddc(g0|Cv,an)=κ−1⋅ddc((g0+g1)|Cv,an). Finally, since g0+g1 is continuous, we have Tbif∧[C]=ddc((g0+g1)|Cv,an), which ends the proof.     ▪ 4.2 Values of the Böttcher coordinates at critical points are proportional near infinity In this section, we prove Theorem 4.1 (2). Let us fix a branch at infinity c of an irreducible curve C containing infinitely many PCF polynomials, and an isomorphism of complete local rings OC^,c≃L[[t]], such that c(n(t))=t−n, and a(n(t))∈OL,S((t)) is an adelic series. Write Pt=Pc(n(t)),a(n(t)), and φt=φPt. By Lemma 3.8 there exists an integer q≥1 large enough such that Ptq(c0) and Ptq(c1) both lie in the domain of convergence of the Böttcher coordinate φt for t small enough, and (8) holds, that is,   φt(Ptq(cε))=ω(Ptq(cε)−c(n(t))2)+Θ(t), where Θ is an adelic series vanishing at 0. We now fix a place v and compute using Proposition 2.3 for |t|v≪1. We get   |φt(P0q(t))|vs0|φt(P1q(t))|vs1=exp(s0⋅gc(n(t)),a(n(t))(P0q(t)))exp(s1⋅gc(n(t)),a(n(t))(P1q(t)))=exp(3qs0⋅g0,v(c(n(t)),a(n(t))))exp(3qs1⋅g1,v(c(n(t)),a(n(t)))=1, (⋆) where the last equality follows from Theorem 4.1 (1). Applying (⋆) in the case of an Archimedean place, we see that the complex analytic map   t↦(φt(P0q(t)))s0(φt(P1q(t)))s1 has a modulus constant equal to 1, hence is a constant, say ζ. Since both power series φt(P0q(t)) and φt(P1q(t)) have their coefficients in OL,S, we conclude that ζ∈OL,S. But |ζ|v=1 for all place v over L by (⋆) hence it is a root of unity. Note also that the equality φt(P0q(t))s0=ζφt(P1q(t))s1 holds as equality between adelic series, so that it is also true for analytic functions at any place. To conclude the proof of Theorem 4.1, pick a place v of L and consider the connected component U of {g0,v>τv/s0}={g1,v>τv/s1} in Cv,an whose closure in C^ contains c. We need to argue that Pc,aq(c0) and Pc,aq(c1) belong to the domain of convergence of the Böttcher coordinate φv,c,a for any c,a∈U. Recall that s0g0,v(c,a)=s1g1,v(c,a) for some positive integers s0,s1. It follows that   min{gc,a(Pc,aq(c0)),gc,a(Pc,aq(c1))}=3qmin{gc,a(c0),gc,a(c1)} ≥3qmin{s0s1,s1s0}max{gc,a(c0),gc,a(c1)} >G(c,a)+max{s0gc,a(c0),s1gc,a(c1)}>G(c,a)+τv for q large enough and we conclude by Proposition 2.3. The proof of Theorem 4.1 is now complete. 5 Special Curves Having a Periodic Orbit with a Constant Multiplier In this section, we prove Theorem B. Pick an integer m≥1, a complex number λ∈ℂ, and consider the set of polynomials Pc,a that admit a periodic orbits of period m and multiplier λ. It follows from [42, p. 225] that this set is an algebraic curve in Poly3 (see also [36, Appendix D], [9, Theorem 2.1] or [21, Section 6.2]). Let us be more precise: Theorem 5.1 (Silverman). For any integer m≥1, there exists a polynomial pm∈ℚ[c,a,λ] with the following properties. For any λ∈ℂ∖{1}, pm(c,a,λ)=0 if and only if Pc,a has a cycle of exact period m and multiplier λ. When λ=1, then pm(c,a,1)=0 if and only if there exists an integer k dividing m such that Pc,a has a cycle of exact period k whose multiplier is a primitive m/k-th root of unity.     □ We now come to the proof of Theorem B. One implication is easy. For any integer m≥1, the curve Perm(0) is contained in the union of the two curves {(c,a)∈ℂ2; Pc,am(c0)=c0} and {(c,a)∈ℂ2;Pc,am(c1)=c1}. According to lemma 5.2 below, it contains infinitely many post-critically finite parameters. Lemma 5.2. Pick n≥0, k>0 and i∈{0,1}. Any irreducible component C of the set {(c,a),Pc,an+k(ci)=Pc,an(ci)} contains infinitely many post-critically finite parameters.     □ Proof. We argue over the complex numbers, and use the terminology and results from [20]. In particular, a critical point ci, i=0,1 is said to be active at a parameter (c,a) if the family of analytic functions Pc,an(ci) is normal in a neighborhood of (c,a). Suppose that C is an irreducible component of the set   {(c,a),Pc,an+k(ci)=Pc,an(ci)}, where n≥0, k>0 and i∈{0,1}. To fix notation we suppose i=0. Observe that gc,a(c0)=0 on C, and since G(c,a)=max{gc,a(c0),gc,a(c1)} is a proper function on Poly3 (see Proposition 2.1) it follows that gc,a(c1) is also proper on C. In particular, c1 has an unbounded orbit when c,a∈C is close enough to infinity in Poly3. It follows from for example, [20, Theorem 2.5] (which builds on [33, Theorem 2.2]) that c1 is active at at least one point (c0,a0) on C. The arguments of [20, Lemma 2.3] based on Montel's theorem show that (c0,a0) is accumulated by parameters for which c1 is pre-periodic to a repelling cycle, hence by post-critically finite polynomials. In particular, it contains infinitely many post-critically finite parameters.     ▪ For the converse implication, we proceed by contradiction and suppose that we can find a complex number λ≠0, an integer m≥1, and an irreducible component C of Perm(λ) containing infinitely many post-critically finite polynomials. Observe that, whenever 0<|λ|≤1, any parameter (c,a)∈C⊂Perm(λ) has a non-repelling cycle which is not super-attracting. In particular, at least one of its critical points has an infinite forward orbit (see e.g., [37]). It follows that Perm(λ) contains no post-critically finite parameter when 0<|λ|≤1. This argument is however not sufficient to conclude in general. But we shall see that a combination of this argument applied at a place of residual characteristic 3 together with the study of the explosion of multipliers on a branch at infinity of C gives a contradiction. Proposition 5.3. Suppose C is an irreducible component of Perm(λ) with λ∈ℂ* and m≥1 containing infinitely many post-critically finite polynomials. Then one of the two critical points is persistently preperiodic on C and λ is equal to the multiplier of a repelling periodic orbit of a post-critically finite quadratic polynomial.     □ We may thus assume that the curve C is included in {Pc,an(c0)=Pc,ak(c0)} (or {Pc,an(c1)=Pc,ak(c1)}) for some integers n>k≥0. Observe that the equation Pc,an(c0)=Pc,ak(c0) (resp. Pc,an(c1)=Pc,ak(c1)) is equivalent to the vanishing of a polynomial of the form 31−3na3n+l.o.t (resp. 31−3n(a3−c36)3n+l.o.t). It follows that the closure of C in Poly3¯ intersects the line at infinity in a set included in {[1:0:0],[ζ:1:0]} with ζ3=6 (see also [9, Theorem 4.2]). Consider the curve of unicritical polynomials c0=c1, which is defined by the equation c=0. It intersects the line at infinity at [0:1:0], so that Bezout' theorem implies the existence of a parameter (c,a)∈C which is unicritical. We conjugate Pc,a by a suitable affine map to a polynomial Q(z)=z3+t. This unicritical polynomial has a preperiodic critical orbit. Proposition 5.4 below implies |λ|v<1 at any place v of residual characteristic 3. By the previous proposition, Q also has a periodic orbit whose multiplier is equal to the multiplier of a repelling orbit of a quadratic polynomial having a preperiodic critical point. Proposition 5.4 now gives |λ|v=1 for this place, hence a contradiction. The proof of Theorem B is complete. Proposition 5.4. Suppose Q(z)=zd+t is a post-critically finite unicritical polynomial of degree d≥2, and let λ≠0 be the multiplier of some periodic orbit of P. Then λ belongs to some number field K, and given any non-Archimedean place v of K we have: |λ|v<1 if the residual characteristic of Kv divides d; |λ|v=1 if the residual characteristic of Kv is prime to d.     □ Proof. Since Q is post-critically finite, t satisfies a polynomial equation with integral coefficients hence belongs to a number field. Its periodic points are solutions of a polynomial of the form Qn(z)−z so that the periodic points of Q and their multipliers also belong to a number field. We may thus fix a number field containing t, λ, and fix a place v of K of residual characteristic p≥2. Observe that the completion of the algebraic closure of the completion of K with respect to the norm induced by v is a complete algebraically closed normed field isometric to the p-adic field ℂp. We consider the action of Q on the Berkovich analytification of the affine plane over that field. To simplify notation we denote by |⋅| the norm on ℂp. Suppose that |t|>1. Then we have |Q(0)|=|t|>1, and thus |Qn(0)|=|Qn−1(0)|d=|t|dn→∞ by an immediate induction. This would imply the critical point to have an infinite orbit contradicting our assumption that Q is post-critically finite. We thus have |t|≤1. This implies that any point having a bounded orbit lies in the closed unit ball {z,|z|≤1}. Indeed the same induction as before yields |Q(z)|=|z|d and |Qn(z)|=|Qn−1(z)|d=|z|dn→∞ for any |z|>1. Pick any periodic point w of period k with multiplier λ=(Qk)′(w)≠0. Observe that Q′(z)=dzd−1. Suppose first that p divides d so that |d|<1. Since |Qj(w)|≤1 for all j≥0 by what precedes, we have   |λ|=∏j=0k−1|Q′(Qj(w))|≤|d|k<1. Suppose now that p is prime to d, hence |d|=1. Observe that one has Q(B(z))=B(Q(z)) for any |z|≤1 where B(z)={w,|z−w|<1} is the open ball of center z and radius 1. Since the critical point 0 has a finite orbit, two situations may arise. Either B(0) is strictly preperiodic, and thus cannot contain any periodic orbit. Or B(0) is periodic, and is contained in the basin of attraction of some attracting periodic orbit. Since 0 has a finite orbit, it has to be periodic. In both cases this implies the orbit of w to be included in the annulus {|z|=1}. We thus have   |λ|=∏i=0k−1|Q′(Qi(w))|=∏i=0k−1|d(Qi(w))d−1|=1 , which concludes the proof.     ▪ Proof of Proposition 5.3. Since C contains infinitely many post-critically finite polynomials we may assume it is defined over a number field K. Let C^ be the normalization of the completion of C in Poly3¯. Pick any branch c of C at infinity (i.e., a point in C^ which projects to the line at infinity in Poly3¯). By Proposition 3.5 we may choose an isomorphism of complete local rings OC^,c^≃L[[t]] such that c(n(t)),a(n(t)) are adelic series, that is, formal Laurent series with coefficients in OL,S((t)) that are analytic at all places. In the remainder of the proof, we fix an Archimedean place, and embed L into the field of complex numbers (endowed with its standard norm). We may suppose c(n(t)),a(n(t)) are holomorphic in 0<|t|<ϵ for some ϵ, and meromorphic at 0. We get a one-parameter family of cubic polynomials Pt:=Pc(n(t)),a(n(t)) parameterized by the punctured disk Dϵ*={0<|t|<ϵ}. Consider the subvariety Z:={(z,t),Ptm(z)=z}⊂ℂ×Dϵ*. The projection map Z→Dϵ* is a finite cover which is unramified if ϵ is chosen small enough. By reducing ϵ if necessary, and replacing t by tN, we may thus assume that Z→Dϵ* is a trivial cover. In other words, there exists a meromorphic function t↦p(t) such that Ptm(p(t))=p(t) and (Ptm)′(p(t))=λ. As in Section 3, we denote by P(z)∈ℂ((t))[z] the cubic polynomial induced by the family Pt. It induces a continuous map on the analytification Aℂ((t))1,an, for which the point p∈A1(ℂ((t))) corresponding to p(t) is periodic of period m with multiplier (Pm)′(p)=λ. Observe that P has two critical points c0 and c1 corresponding to the meromorphic functions 0 and c(n(t)), respectively. Lemma 5.5. If c0 is not pre-periodic for P, then |Pq(c0)|t tends to infinity when q→∞.     □ Proof. Observe that our assumption is equivalent to the fact that c0 is not persistently pre-periodic on C. We claim that g0(t):=gPt(c0) tends to infinity when t→0. Suppose first that c1 is persistently pre-periodic on C. Then the function g1 is identically zero on C, so that G|C=max{g0,g1}|C=g0. Since G is proper by Proposition 2.1, and (c(n(t)),a(n(t))) tends to infinity in Poly3¯ when t→0, we conclude that g0(t)→∞. When c1 is not persistently pre-periodic on C, the two functions g0(t) and g1(t):=gPt(c1) are proportional on c by Theorem 4.1 (1). As before max{g0,g1}→∞ as t→0 so that again g0(t)→∞. By Proposition 3.6, we can find a>0 such that g0(Pt)=alog|t|−1+O(1). And [20, Lemma 6.4] implies the existence of a constant C>0 such that gPt(z)≤logmax{|z|,|c(n(t))|,|a(n(t))|}+C for all t. Since gPt∘Pt=3gPt, we conclude that for all q≥1 (observe that the statement of the lemma is incorrectly stated in [20], and the constant C is actually independent on P.)   logmax{|Ptq(c0)|,|c(n(t))|,|a(n(t))|}≥3qgt(0)−C=3qalog|t|−1+O(1). This implies |Ptq(c0)|t≥3qa|t|t→∞ when q→∞ as required.     ▪ We continue the proof of Proposition 5.3. Suppose neither c0 nor c1 is persistently pre-periodic so that the previous lemma applies to both critical points. Translating its conclusion over the non-Archimedean field ℂ((t)), we get that Pq(c0) and Pq(c1) both tend to infinity when q→∞. We may thus apply [31, Theorem 1.1 (ii)], and [31, Corollary 1.4] (which is directly inspired from a result of Bezivin). We conclude that all periodic cycles of P are repelling so that |(Pm)′(p)|t>1. This contradicts |λ|t=1. Suppose next that c0 is persistently pre-periodic or periodic (which implies c1 not to be persistently pre-periodic). Then c0 is pre-periodic whereas c1 escapes to infinity by the previous lemma. Observe that if c0 is eventually mapped to a point in the Julia set of P, then [31, Theorem 1.1 (iii) (a)] combined with [31, Corollary 1.4] implies that all cycles of P are repelling which gives a contradiction. We can thus apply [31, Theorem 1.1 (iii) (b)) to P, and the preperiodic critical point c0(=0) is contained in a closed ball B={z∈ℂ((t)),|z|t≤r} for some positive r>0 that is periodic of exact period n. Since B is fixed by the polynomial Pn(z)=∑j≥2bjzj with coefficients bj∈ℂ((t)), the radius r satisfies an equation of the form |bj|rj=r for some j hence r=|t|tl for some l∈ℚ. To simplify the discussion to follow we do a suitable base change t→tN, and we conjugate P by the automorphism z↦t−lz so that B becomes the closed unit ball. Observe that 0 remains a critical point of P after this conjugacy. Recall that the closed unit ball B defines the Gauss point xg∈Aℂ((t))1,an for which we have   |Q(xg)|:=supz∈B|Q(z)|t=max|qi| for all Q=∑qizi∈ℂ((t))[z]. Since B is fixed by Pn, it follows that xg is also fixed by Pn. This is equivalent to say that Pn can be written as Pn(z)=∑i=13naizi where max|ai|=1. For any z∈ℂ((t)) of norm 1, denote by z˜ the unique complex number such that |z−z˜|t<1. Lemma 5.6. We have a1=0, |a0|≤|a2|=1, and |ai|<1 for all i≥3; and the complex quadratic polynomial P˜(z):=a2˜z2+a0˜ has a preperiodic critical orbit.     □ Lemma 5.7. The orbit of the periodic point p intersects the ball B.     □ Replacing p by its image by a suitable iterate of P we may suppose that it belongs to B, that is, |p|t≤1. In fact we have |Pi(p)|t=1 for all i≥0. Indeed if it were not the case, then the open unit ball would be periodic. Since it contains a critical point, it would be contained in the basin of attraction of an attracting periodic orbit which yields a contradiction. Observe also that the period of p is necessarily a multiple of n, say nk with k≥1. To render the computation of the multiplier of p easier, we conjugate Pn by z↦a2z. Since |a2|=1, we still have |p|t=1, and the equality a2=1 is now satisfied. By Lemma 5.6, we get supB|Q|<1 with Q:=Pn−P˜, so that   (Pnk)′(p)=∏i=0k−1(Pn)′(Pni(p))=(P˜k)′(p˜). But the multiplier of p is equal to λ∈ℂ. Hence it is equal to the multiplier of a repelling periodic orbit of some quadratic polynomial (namely P˜) having a preperiodic critical orbit, as was to be shown.     ▪ Proof of Lemma 5.6. The point 0 is critical for P hence a1=0. Since the Gauss point is fixed by Pn, we have maxi≥2|ai|=1. Let d≥2 be the maximum over all integers i such that |ai|=1. The number of critical points of Pn lying in the closed unit ball (counted with multiplicity) is precisely equal to d−1. Since the exact period of xg is n, and the other point escapes to infinity, the ball B contains a unique critical point of Pn namely 0. It follows that d=2, and |a2|=1>maxi≥3|ai|. Finally 0 is preperiodic by Pn, hence the complex quadratic polynomial Pn˜ has a preperiodic critical orbit.     ▪ Proof of Lemma 5.7. Since the multiplier of p is λ∈ℂ, its t-adic norm is 1, hence a small ball U centered at p of positive radius is included in the filled-in Julia set of P. By [31, Corollary 4.8], U is eventually mapped into B, hence the claim.     ▪ 6. A Polynomial on a Special Curve Admits a Symmetry We fix K a number field, and s0,s1 two positive integers such that s0 and s1 are coprime. We shall say that a cubic polynomial P:=Pc,a with c,a in a finite extension L of K satisfies the condition (P) if the following holds: ( gQc,a,v=gPc,a,v) For any place v of L, we have s0gP,v(c0)=s1gP,v(c1). ( P2) Given any place v of L, if min{gP,v(Pn(c0)),gP,v(Pn(c1))}>Gv(P)+τv for some integer n≥1, then   φP,v(Pn(c0))s0φP,v(Pn(c1))s1 is a root of unity lying in K. Recall the definition of the constant τv:=τ(Lv) from Proposition 2.3. Observe that if the condition in (P2) never occurs, then Gv(c,a)=0 for all places v of L, hence P is post-critically finite (see e.g., [21, Theorem 3.2]). We prove here the following Theorem 6.1. Suppose P=Pc,a is a cubic polynomial defined over a number field L satisfying the assumptions (P) which is not post-critically finite and such that min{gP,v(Pq(c0)),gP,v(Pq(c1))}>Gv(P)+τv for some integer q and some place v of L. Then there exists a root of unity ζ∈K, an integer q′≤C(K,q), and an integer m≥0 such that the polynomial Q(z):=ζPm(z)+(1−ζ)c2 commutes with all iterates Pk such that ζ3k=ζ, and either Q(Pq′(c0))=Pq′(c1), or Q(Pq′(c1))=Pq′(c0).     □ Remark. We shall prove along the way that there exists an integer k≥1 with ζ3k=ζ so that the commutativity statement is non-empty.     □ 6.1. Algebraization of adelic branches at infinity The material of this section is taken from [45]. Let K be a number field. For any place v on K, denote by Kv the completion of K w.r.t. the v-adic norm. We cover the line at infinity H∞ of the compactification of the affine space AK2=SpecK[x,y] by ℙK2 by charts Uα=SpecK[xα,yα] centered at α∈H∞(K) such that α={(xα,yα)=(0,0)}, H∞∩Uα={xα=0}, and xα=1/x, yα=y/x+c for some c∈K (or xα=1/y, yα=x/y). Fix S a finite set of places of K. By definition, an adelic branch s at infinity defined over the ring OK,S is a formal branch based at a point α∈H∞(K) given in coordinates xα,yα as above by a formal Puiseux series   yα=∑j≥1ajxαj/m∈OK,S[[xα1/m]] such that ∑j≥1ajxj is an adelic series. Observe that for any place v∈S, then the radius of convergence is a least 1. In the sequel, we set rs,α,v to be the minimum between 1 and the radius of convergence over Kv of this Puiseux series. Any adelic branch s based at α at infinity thus defines an analytic curve in an (unbounded) open subset of Av2,an:   Zv(s):={(xα,yα)∈Uα(Kv);yαm=∑j≥1ajxαj,0<|xα|v<rs,α,v}. Theorem 6.2 (Xie). Suppose s1,…,sl are adelic branches at infinity, and let {Bv}v∈MK be a set of positive real numbers such that Bv=1 for all but finitely many places. Assume that there exists a sequence of distinct points pn=(xn,yn)∈A2(K) such that for all n and for each place v∈MK then either we have max{|xn|v,|yn|v}≤Bv or pn∈∪i=1lZv(si). Then there exists an algebraic curve Z defined over K such that any branch of Z at infinity is contained in the set {s1,…,sl} and pn belongs to Z(K) for all n large enough.     □ 6.2 Construction of an invariant correspondence Our aim is to prove the following statement. Theorem 6.3. Suppose P=Pc,a is a cubic polynomial satisfying the assumptions (P). Then there exists a (possibly reducible) algebraic curve ZP⊂A1×A1 such that: ϕ(ZP)=ZP with ϕ(x,y):=(P(x),P(y)); for all n large enough, we have (Pn(c0),Pn(c1))∈ZP; any branch at infinity of ZP is given by an equation φP(x)s0=ζ⋅φP(y)s1 for some root of unity ζ∈K.     □ Proof. The proof is a direct application of Xie's theorem. Recall that the set UK of roots of unity that is contained in the number field K is finite. Recall that for each place v over L, we let gP,v:=limn13nlog+|Pn|v be the Green function of P, and write Gv(P)=max{gP,v(c0),gP,v(c1)}. Lemma 6.4. For any ζ∈UK, there exists an adelic branch cζ based at a point q∈H∞(L) such that for any place v the analytic curve Zv(cζ) is defined by the equation {φP,v(x)s0=ζ⋅φP,v(y)s1} in the range min{|x|v,|y|v}>exp(Gv(P)+τv).     □ Define (xn,yn):=(Pn(c0),Pn(c1))∈A2(L), and consider the family of all adelic curves cζ given by Lemma 6.4 for all ζ∈UK. We shall first check that all hypotheses of Xie's theorem are satisfied. To do so pick any integer n and any place v on L. Suppose first that gP,v(c0)=0. Since gP,v(Pn(c0))=3ngP,v(c0)=0, we get |xn|v≤eCv=:Bv by Lemma 2.4. The same upper bound applies to |yn|v since gP,v(c1)=0 by (P1) so that max{|xn|v,|yn|v}≤Bv in this case. Observe that Bv=1 for all but finitely many places v of L by Lemma 2.4. Suppose now that gP,v(c0)>0 so that gP,v(c1)>0 by (P1). Fix N large enough such that gP,v(PN(c0))>Gv(P)+τv and gP,v(PN(c1))>Gv(P)+τv. Then PN(c0) and PN(c1) lie in the domain of definition of the Böttcher coordinate by Proposition 2.3. Since   gP,v(Pn(c0))=3n−NgP,v(PN(c0))≥gP,v(PN(c0))>Gv(P), we may also evaluate φP at xn for all n≥N. The same holds for yn and we get that   φP(xn)s0φP(yn)s1 is a root of unity ζ∈K by (P2) hence (xn,yn) belongs to Zv(cζ) for all n≥N. Xie's theorem thus applies to the sequence {(xn,yn)}n≥N, and we get a (possibly reducible) curve Z1⊂A1×A1 that contains infinitely many points (xn,yn) and such that each of its branch at infinity is equal to cζ for some ζ∈UK. Recall that ϕ(x,y)=(P(x),P(y)), and pick any integer n≥1. Let Z be an irreducible component of Z1. Then ϕn(Z) is an irreducible curve defined over L whose branches at infinity are the images under ϕn of the branches at infinity of Z. Fix ζ∈UK and pick (x,y)∈Zv(cζ). Then (x′,y′)=(P(x),P(y)) satisfies   φP(x′)s0φP(y′)s1=φP(x)3s0φP(y)3s1=ζ3, hence ϕ(cζ)=cζ3. We conclude that any branch at infinity of ϕn(Z) is of the form cζ for some ζ∈UK. Since two irreducible curves having a branch at infinity in common are equal, we see that Z is pre-periodic for the morphism ϕ so that ϕl+k(Z)=ϕk(Z) for some l,k>0. Setting ZP:=∪j=kl+k−1ϕj(Z), we obtain a (possibly reducible) curve defined over L such that ϕ(ZP)=ZP and (xn,yn)∈ZP for all n≥k. This concludes the proof of the theorem, since Z1 has only finitely many irreducible components.     ▪ Proof of Lemma 6.4. Recall from Lemma 2.2 that   φP(z)=ω(z−c2)+∑k≥1akzk, is an adelic series at infinity in the sense of Section 3.1 , and therefore   φP−1(z)=1ωz+c2+∑k≥1bkzk, by Lemma 3.1. We may assume that ak,bk∈OK,S. Recall from Proposition 2.3 that φP,v induces an analytic isomorphism between {z,gP,v(z)>Gv(P)+τv} and {z′,|z′|v>exp(Gv(P)+τv)}. By Lemma 3.1 the formal map φP−1 defines an adelic series at infinity in the terminology of Section 3.1. For each place v, this series coincides with the inverse map of φP on the complement of the closed disk of radius exp(Gv(P)+τv) hence its domain of convergence is exactly {z′,|z′|v>exp(Gv(P)+τv)}. It follows that   Zv:={(x,y),φP(x)s0=ζφP(y)s1} defines an analytic curve in the domain min{gP,v(x),gP,v(y)}>Gv(P)+τv, whose image under the isomorphism (x′,y′):=(φP,v(x),φP,v(y)) is given by   Z′v:={(x′,y′),(x′)s0=ζ(y′)s1} where min{|x′|v,|y′|v}>exp(Gv(P)+τv). Pick any ξ∈ℚ¯ such that ξs1ζ=1. Let cζ be the adelic branch at infinity defined by the formal Laurent series (φP−1(t−s1),φP−1(ξt−s0)). For all places v, the analytic curve Zv(cζ) is included in Zv. Since s0 and s1 are coprime, for any pair (x′,y′) with (x′)s0=ζ(y′)s1 and min{|x′|v,|y′|v}>exp(Gv(P)+τv), there exists 0<|t|v<exp(−Gv(P)+τvmin{s0,s1}) such that x′=t−s1 and y′=ξt−s0. This proves that Zv(cζ)=Zv for all place as required.     ▪ 6.3 Invariant correspondences are graphs Let Z0,…,Zp−1 be the irreducible components of ZP such that ϕ(Zi)=Zi+1 (the index computed modulo p). Since we assumed P not to be post-critically finite, it is non-special in the sense of [40]. We may thus apply Theorem 4.9 of op. cit. (or [39, Theorem 6.24]) to the component Z0 of ZP that is ϕp-invariant. It implies that after exchanging x and y if necessary, Z0 is the graph of a polynomial map, that is, Z0={(Q(t),t)} for some Q∈L[t] such that Q∘Pp=Pp∘Q. Observe that by [30] the two polynomials P and Q share a common iterate when deg(Q)≥2 since we assumed P not to be post-critically finite. We now work at an Archimedean place. Recall that the branch at infinity of Z0 is of the form φP(x)s0=ζφP(y)s1 for some ζ∈UK. Since s0 and s1 are coprime, it follows that s0=1 and s1=deg(Q), and therefore s1 is a power of 3, say s1=3m. We get   φP(Q(t))=ζφP(t)3m=ζφP(Pm(t)). (13) for all t of large enough norm. By Lemma 2.2, we get that φP(t)=ω(t−c2)+o(1) so that   ω(Q(t)−c2)=ωζ(Pm(t)−c2)+o(1) (14) which implies Q(t):=ζPm(t)+(1−ζ)c2 since a polynomial which tends to 0 at infinity is identically zero. At this point, recall our assumption that min{gP,v(Pq(c0)),gP,v(Pq(c1))}>Gv(P)+τv for some integer q and some place v of L. Then by (P2) φP(Pq(c0))s0=ξφP(Pq(c1))s1 for some root of unity ξ∈K which implies φP(Pq+n(c0))=ξ3nφP(Pq+n(c1))3m. Since for some n large enough the point (Pq+n(c0),Pq+n(c1)) belongs to Z0, we get ξ3n=ζ. Now observe that the least integer n such that ξ3n=ζ is less that the cardinality of UK. We get the existence of q′≤C(K,q) such that φP(Pq′(c0))=ζφP(Pq′(c1))3m. Since φP is injective, the equation (13) shows that Pq′(c0)=Q(Pq′(c1)). Observe that ζ3p=ζ. Indeed, since Z0 is ϕp-invariant and since ϕ(cζ)=cζ3, we get cζ3p=cζ, hence ζ3p=ζ. We now pick any integer k≥1 such that ζ3k=ζ. Then for all t large enough, we have   φP(Q∘Pk(t))=ζφP(Pk(t))3m=ζφP3m+k(t), whereas,   φP(Pk∘Q(t))=φP(Q(t))3k=ζ3kφP3k+m(t). Since φP is injective on a neighborhood of ∞, and since ζ3k=ζ by assumption, we conclude that Q∘Pk=Pk∘Q. This concludes the proof of Theorem 6.3. 7 Classification of Special Curves In this section, we prove Theorems C and A. Before starting the proofs, let us introduce some notation. Pick q,m≥0 and ζ a root of unity. We let Z(q,m,ζ) be the algebraic set of those (c,a)∈A2 such that the polynomial Qc,a:=ζPc,am+(1−ζ)c2 commutes with all iterates Pc,ak of Pc,a such that ζ3k=ζ, and either Qc,a(Pc,aq(c0))=Pc,aq(c1), or Qc,a(Pc,aq(c1))=Pc,aq(c0). Observe that, when m≥1, Q has degree 3m and when (c,a) belongs to a fixed normed field K then the Green function gQ:=limn13nmlog+|Qn| is equal to gc,a. Indeed since Q and Pk commute they have the same filled-in Julia set, KQ coincides with the filled-in Julia set KP of P. And gP (resp. gQ) is the unique continuous sub-harmonic function g on AK1,an that is zero on KP, harmonic outside, with a logarithmic growth at infinity gP(z)=log|z|+O(1) (resp. gQ(z)=log|z|+O(1)). As KP=KQ, this gives gc,a=gQ. 7.1. Proof of Theorem C The implication (1)⇒(3) is exactly point 1. of Theorem 4.1. The implication (3)⇒(4) follows from Corollary 4.2 when s0 and s1 are both non-zero. Indeed, by assumption the critical point c0 is pre-periodic for Pc,a if and only if c1 is. Moreover, we observe that c0 is active at at least one parameter of C, since the function G is proper, hence that the set for which it is pre-periodic is infinite, by e.g. [20, Lemma 2.3). When s1=0, then g0,v≡0 on C at all places. By [20, Theorem 2.5] there exist n>0 and k≥0 such that C is an irreducible component of {(c,a)∈A2;Pc,an+k(c0)=Pc,ak(c0)}. By Theorem 3.12 (applied to arbitrary weights) the family of functions {g1,v}v∈MK induces a semi-positive adelic metric on some ample line bundle on the normalization of the completion of C so that Thuillier–Yuan's theorem applies. This gives (4) by observing that g0+g1=g1. The case s0=0 is treated similarly. The implication (4)⇒(1) is obvious. To prove (2)⇒(1), we observe that if c0 is not persistently pre-periodic on C then it is active at at least one parameter by [20, Theorem 2.5] and that the set of parameters for which it is pre-periodic is infinite by e.g. [20, Lemma 2.3]. We now prove (3)⇒(2). We suppose c0 is not persistently pre-periodic on C. Pick some parameter (c,a)∈C and suppose c0 is pre-periodic. We need to show that Pc,a is post-critically finite. Since c0 is not persistently pre-periodic on C we have s1≠0 (again by [20, Theorem 2.5] applied at any Archimedean place). In the case s0=0 then c1 is persistently pre-periodic and Pc,a is clearly post-critically finite. We may thus assume that s0 and s1 are both non-zero and the functions g0,v,g1,v vanish on the same set in Cv,an for any place v of K. Observe that c0 being pre-periodic implies (c,a) to be defined over a number field. It follows that for all the Galois-conjugates (c′,a′) of (c,a) (over the defining field K of the curve C) we have Gv(c′,a′):=max{g0,v(c′,a′),g1,v(c′,a′)}=0. It follows from [29] or [21, Theorem 3.2] that Pc,a is post-critically finite. Let us now prove (3)⇒(5). We suppose C is special. If either c0 or c1 is persistently pre-periodic in C, the assertion (5) holds true with ζ=1 and i=j by [20, Theorem 2.5]. Assume from now on that we are not in this case. Replacing K by a finite extension we may assume that all roots of unity ζ appearing in Theorem 6.1 2. belong to K, since there are at most the number of branches at infinity of C of such roots of unity. Let B be the set of all (c,a)∈C(L) where L is a finite extension of K such that Pc,a is not post-critically finite. Given a place v of K we also define the subset Bv of B of parameters c,a such that g0,v(c,a)>0. This set is infinite since post-critically finite polynomials form a bounded set in Cv,an. Lemma 7.1. For any (c,a)∈B, the polynomial Pc,a satisfies (P1) and (P2).     □ Pick q large enough such that 3q>max{s0/s1,s1/s0}, and fix a place v of residual characteristic ≥5. Now choose any (c,a)∈Bv. Then g1,v(c,a) is also positive and min{gc,a,v(Pq(c0)),gc,a,v(Pq(c1))}>Gv(c,a) so that Theorem 6.1 applies by the previous lemma. We get a positive integer q′ (bounded from above by a constant C depending only on K and q), a root of unity ζ∈K and an integer m≥0 such that (c,a)∈Z(q′,m,ζ). Since gQ=gP, and Q(Pq′(c0))=Pq′(c1) we have   3mgP,v(Pq′(c0))=gP,v(Q(Pq′(c0)))=gP,v(Pq′(c1)) so that 3m=s0s1. We conclude that the algebraic set consisting of the union of the curves Z(q′,m,ζ) with 3m=s0s1, q′≤C and ζ ranging over all roots of unity lying in K contains Bv. It follows that C is an irreducible component of one of these curves. To end the proof of the theorem, we are left with proving (5)⇒(3). Suppose that C is an irreducible component of Z(m,q,ζ) for some m≥0 and q≥0 and some root of unity ζ. Observe that Z(q,m,ζ) hence C are defined over a number field say K. When m≥1, then for all place v of that number field we have gQc,a,v=gPc,a,v for all (c,a)∈C(L) for some finite extension L of K. In particular Qc,a(Pc,aq(ci))=Pc,aq(cj) implies 3mgi,v(c,a)=gj,v(c,a) which proves (3) (with s0=0 or s1=0 if i=j). When m=0 and ζ≠1 and C≠{c=0}, then Qc,a(c0)=(1−ζ)c/2≠0 hence i≠j. It follows that gPc,a,v∘Qc,a=gPc,a,v hence g0,v=g1,v. When C={c=0}, then c0=c1 so that again g0,v=g1,v. Finally when m=0 and ζ=1, then i≠j and Pc,aq(c0)=Pc,aq(c1) hence g0,v=g1,v at all places. Proof of Lemma 7.1. Pick (c,a)∈B. By the point 1. of Theorem 4.1, Pc,a satisfies (P1). To check (P2), we need to introduce a few sets. Fix any place v of K, and for any integer n≥0, define the open subset of Cv,an  Ωn,v:={(c′,a′),min{gc′,a′,v(Pc′,a′n(c0)),gc′,a′,v(Pc′,a′n(c1))}>Gv(c′,a′)+τv}. On Ωn,v one can define the analytic map   Mn(c′,a′):=φc′,a′(Pc′,a′n(c0))s0φc′,a′(Pc′,a′n(c1))s1. Observe that Ωn+1,v⊂Ωn,v, and Mk+l(c′,a′)=Mk(c′,a′)3l on Ωk,v for all integers k,l≥0. We also define the increasing sequence of open sets   Un,v:={(c′,a′),Gv(c′,a′)>τv3n−1}⊂Cv,an. Since Gv is subharmonic and proper on Cv,an, the set Un,v contains no bounded component by the maximum principle. Lemma 7.2. Suppose 3r≥max{s0/s1,s1/s0}. Then we have Ωn,v⊂Un,v and Un,v⊂Ωn+r,v.     □ By the point 2. of Theorem 4.1, one can find two integers q≥1 and N≥1 such that Mq is well-defined and constant equal to a root of unity lying in K in each component of UN,v. Let V be the connected component of Ωn,v containing (c,a). This open set might or might not be bounded. By the previous lemma, if n≥max{r+N,q}, then Un−r,v⊂Ωn,v so that Mn is well-defined on Un−r,v. Since all components of Un−r,v are unbounded, and Mn=Mq3n−q in UN,r, we conclude that Mn is locally constant in Un−r,v (hence on V) equal to a root of unity lying in K. When n≤n0=max{r+N,q}, then (Mn)3n0−n=Mn0 which we know is constant in V equal to a root of unity lying in K. We conclude that Mn is constant on V equal to a root of unity lying in a fixed extension K′ of K that only depends on the constants r,N and q. Since these constants are in turn independent of the place v, we conclude the proof of the lemma replacing K by K′.     ▪ Proof of Lemma 7.2. Pick (c′,a′)∈Ωn,v. We may suppose that Gv(c′,a′)=gc′,a′,v(c0) so that Gv(c′,a′)=gc′,a′,v(c0)=13ngc′,a′,v(Pc′,a′n(c0))>13n(Gv(c′,a′)+τv) which implies (c′,a′)∈Un,v. Conversely suppose (c′,a′)∈Un,v. As before we may suppose that Gv(c′,a′)=gc′,a′,v(c0) so that   gc′,a′,v(Pn+r(c0))≥gc′,a′,v(Pn(c0))=3ngc′,a′,v(c0) =3nGv(c′,a′)=Gv(c′,a′)+(3n−1)Gv(c′,a′)>Gv(c′,a′)+τv. Similarly we have   gc′,a′,v(Pn+r(c1))=3n+rs0s1gc′,a′,v(c1)≥3ngc′,a′,v(c0)>Gv(c′,a′)+τv hence (c′,a′)∈Ωn+r,v.     ▪ 7.2. Proof of Theorem A According to the implication (1)⇒(5) of Theorem C, any irreducible algebraic curve C of Poly3 containing infinitely many post-critically finite polynomials is a component of some Z(q,m,ζ) so that Theorem A reduces to the following. Proposition 7.3. The set Z(q,m,1) is equal to the union {Pc,am+q(c1)=Pc,aq(c0)}∪{Pc,am+q(c0)=Pc,aq(c1)}, hence contains an algebraic curve. Moreover, one has Z(1,0,1)=Z(0,0,1). The set Z(q,m,−1) is infinite if and only if m=0, and we have Z(q,0,−1)={12a3−c3−6c=0} for any q≥0. if ζ2≠1, the set Z(q,m,ζ) is finite.     □ We shall rely on the following observation. Denote by Crit(P) the set of critical points of the polynomial P. Lemma 7.4. Pick any (c,a)∈Z(q,m,ζ), and suppose that   Qc,a=ζPc,am+(1−ζ)c2 is a polynomial that commutes with Pc,ak and ζ is a (3k−1)-root of unity. Then we have   Qc,a(Crit(Pc,ak+m))=Qc,a(Crit(Pc,am))∪Crit(Pc,ak).     □ Proof. Write P=Pc,a and Q=Qc,a. Differentiate the equality Pk∘Q=Q∘Pk. Since Q′=ζ⋅(Pm)′, we get   Crit(Q∘Pk)=P−k(Crit(Q))∪Crit(Pk) =P−k(Crit(Pm))∪Crit(Pk)=Crit(Pk+m), and therefore   Crit(Pk+m)=Crit(Pk∘Q)=Crit(Pm)∪Q−1(Crit(Pk)), and we conclude taking the image of both sides by Q.     ▪ We now come to the proof of the Proposition. Proof of Proposition 7.3. We may and shall assume that all objects are defined over the field of complex numbers: 1. Suppose Z(q,0,ζ) contains an irreducible curve C. We shall prove that either ζ=±1, or C={c=0}. Observe that for any (c,a)∈Z(q,0,ζ), the polynomial Qc,a is an affine map which commutes with Pc,ak, hence gc,a(Qc,a(z))=gc,a(z) for all z∈ℂ. Without loss of generality, we may suppose that Qc,a(Pc,aq(c0))=Pc,aq(c1), hence G(c,a)=g0(c,a)=g1(c,a). Suppose that Z(q,0,ζ) contains an irreducible curve C. If g0 vanishes identically on C then g1 also, and this implies both critical points to be persistently pre-periodic so that all polynomials in C are post-critically finite. This cannot happen, so that we can find an open set U in C such that G(c,a)>0 for all (c,a)∈U. Pick any parameter (c,a) in U. We have Qc,a(Crit(Pc,ak))=Crit(Pc,ak) by Lemma 7.4, so that Qc,a(c0),Qc,a(c1)∈Crit(Pc,ak). Since   Crit(Pc,ak)=∪0≤j≤k−1Pc,a−j(Crit(Pc,a)) we get gc,a(α)=3−jgc,a(c0)<gc,a(c0)=G(c,a) for any α lying in Crit(Pc,ak) but not in Crit(Pc,a). However, gc,a(Qc,a(c0))=gc,a(Qc,a(c0))=G(c,a), therefore, we have Qc,a(c0),Qc,a(c1)∈Crit(Pc,a)={c0,c1}. This implies either (1−ζ)c=0, or (1+ζ)c=0, hence ζ=±1 or C={c=0} as required. 2. Suppose now that C is an irreducible curve included in Z(q,m,ζ) with m>0. We claim that either ζ=1, or C={c=0} as above. We proceed similarly as in the previous case. We suppose that Z(q,m,ζ) is infinite. For any (c,a)∈Z(q,m,ζ), the polynomial Qc,a commutes with Pc,ak for some k, and has degree 3m>1. In particular we have equality of Green functions gQc,a=gc,a. Without loss of generality we may (and shall) assume Qc,a(Pc,aq(c0))=Pc,aq(c1), which implies g0(c,a)=3mg1(c,a). Assume now by contradiction that ζ≠1. Proceeding as in the previous case, we can find an open set U⊂C such that G(c,a)=g0(c,a)>0 for all (c,a)∈U. Pick now (c,a)∈U.     □ Lemma 7.5. For any α∈Pc,a−m{c0}, we have Qc,a(α)∈{c0,Qc,a(c1)}.     □ Observe that   Qc,a(α)=ζPc,am(α)+(1−ζ)c2=ζc0+(1−ζ)c2=(1−ζ)c2. The equality Qc,a(c1)=Qc,a(α) therefore gives Qc,a(c1)=ζPc,am(c1)+(1−ζ)c2=(1−ζ)c2, and we find Pc,am(c1)=0=c0 so that C is a component of Z(1,m,1). The equality c0=Qc,a(α), implies (1−ζ)c2=0 so that either ζ=1, or C equals {c=0}. 3. We have Z(q,0,−1)=Z(0,0,−1) for all q≥0. Fix q≥0, and pick any (c,a)∈Z(q,0,−1). Observe that (−1)3=−1 hence Qc,a(z)=−z+c commutes with Pc,a by definition. A direct computation shows that this happens if and only if (c,a) belongs to the curve D1:={12a3−c3−6c=0}. One can also check that Qc,a(c0)=c1 for any parameter on D1, and this implies (Qc,a∘Pc,aq)(c0)=Pc,aq(Qc,a(c0))=Pc,aq(c1) for any q≥0. This implies the claim. 4. The irreducible curve D0={c=0} is included in Z(q,m,ζ) if and only if m=0 and ζ=1. Observe that any polynomial P:=P0,a in D0 is unicritical with a single critical point at 0, so that D0 is included in Z(q,0,1) for all q≥0. Observe also that g0=g1>0 on a non-empty open subset of D0. Suppose that D0 is included in Z(q,m,ζ) for some positive integer m>0. Then the Green function of Q:=Q0,a is equal to gP and the equation Q(Pq(c0))=Pq(c1) implies gP(c0)=3mgP(c1)>gP(c1) at least when P is close enough to infinity. This is absurd. Suppose now that D0 is included in Z(q,0,ζ) with ζ≠1 so that Q(z)=ζz. One checks by induction that for any integer k≥1 one has   Pk(z)=qkz3k+ska3z3k−3+l.o.t with qk,sk∈ℚ+*. Choose k minimal such that ζ3k=ζ. We get   Q−1∘Pk∘Q(z)=qkz3k+ska3ζ3k−4z3k−3+l.o.t≠Pk which yields a contradiction, and concludes the proof of our claim. 5. We may now prove the proposition. The first statement follows from the definition of Z(q,m,1), since in that case we have Q=Pm which always commutes with P. Moreover, the curve Z(1,0,1) is given by the equation   0=Pc,a(c0)−Pc,a(c1)=a3−(a3−c36)=c36, whence Z(1,0,1)={c=0}=Z(0,0,1). For the second statement, suppose first that Z(q,m,−1) is infinite. By the second step, we have m=0, or D0={c=0} is included in Z(q,m,−1). The fourth step rules out the latter possibility so that m=0. Conversely if m=0 we may apply the third step to conclude that Z(q,0,−1) is a curve equal to D1={12a3−c3−6c=0}. For the third statement, pick ζ≠±1 and suppose by contradiction that Z(q,m,ζ) is infinite. The first and second step imply that Z(q,m,ζ) contains D0 which is impossible by Step 4. This concludes the proof of the proposition.     ▪ Proof of Lemma 7.5. Take α∈Pc,a−m{c0}, and observe that α∈Crit(Pc,ak+m). According to Lemma 7.4, we have   Qc,a(α)∈Crit(Pc,ak)∪Qc,a(Crit(Pc,am)) and gc,a(Qc,a(α))=3mgc,a(α)=3m⋅3−mgc,a(c0)=gc,a(c0)=G(c,a)>0. Pick any point z∈Crit(Pc,ak)∪Qc,a(Crit(Pc,am)), and suppose it is equal to neither c0 nor Qc,a(c1). Then we are in one of the following (excluding) cases: z is a preimage of c0 under Pc,aj for some 1≤j≤k−1, and gc,a(z)<gP(c0); z is a preimage of c1 under Pc,aj for some 0≤j≤k−1, in which case gc,a(z)≤gc,a(c1)<gc,a(c0); z∈Qc,a(Crit(Pc,am)), so that gc,a(z)=3mgc,a(w) for some point w∈Crit(Pc,am)=∪0≤j≤m−1Pc,a−j(Crit(Pc,a)). In the last case two sub-cases arise. When w is a preimage of c0, we find   gc,a(z)=3mgc,a(w)≥3m13m−1gc,a(c0)>gc,a(c0). Otherwise w is a preimage of c1 distinct from c1 since z≠Qc,a(c1). And we find gc,a(z)=3mgc,a(w)≤3m−1gc,a(c1)<gc,a(c0). Since gc,a(Qc,a(α))=gc,a(c0) we conclude that z≠Qc,a(α) as required. Funding This work was supported by the ERC-starting grant project “Nonarcomp” [307856 to CF]., and was partially supported by ANR project “Lambda” [ANR-13-BS01-0002 to TG] Acknowledgements We thank Xavier Buff and Laura De Marco for discussions at a preliminary stage of this project, and the referee for his/her careful reading of this article and his/her constructive remarks. While finishing the writing of this article we have learned that Dragos Ghioca and Hexi Ye have independently obtained a proof of Theorem A. Their approach differs from ours in the sense that they directly prove the continuity of the metrizations induced by the functions g0,v and g1,v. We get around this problem by considering metrizations induced by max{s0g0,v,s1g1,v} for positive s0,s1 instead. We warmly thank D. Ghioca and H. Ye for sharing with us their preprint. 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