Thomas Gauthier

Yûsuke Okuyama

Gabriel Vigny

APPROXIMATION OF NON-ARCHIMEDEAN LYAPUNOV EXPONENTS AND APPLICATIONS OVER GLOBAL FIELDS by

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map f of P 1 of degree d > 1 over K, in terms of the multipliers of n-periodic points of f , with an explicit control in terms of n, f and K. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over K.

Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show:

a quantified version of Silverman's and Ingram's recent comparison between the critical height and any ample height on the moduli space M d ( Q), two improvements of McMullen's finiteness of the multiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period, a characterization of non-affine isotrivial rational maps defined over a the function field C(X) of a normal projective variety X in terms of the growth of the degree of the multipliers.

in k[T ], where the product over Fix * (f n ) is taken with multiplicity and σ * 0,n ≡ 1 by convention.

Fnally, the critical set Crit(f ) of f is the set

of all (classical) critical points of f in P 1 ( k).

Locally uniform approximation of the Lyapunov exponent and degeneration over a non-archimedean field

Let K be an algebraically closed field of characteristic 0 that is complete with respect to an absolute value | • |, which we assume is non-trivial in that |K| ̸ = {0, 1}.

Recall that K is said to be non-archimedean if the strong triangle inequality |z + w| ≤ max{|z|, |w|} holds for any z, w ∈ K, and otherwise, to be archimedean; K ∼ = C if and only if K is archimedean.

Assume first K is either archimedean or non-archimedean. Let f be any rational map of degree d > 1 defined over K. For every lift F = (F 0 , F 1 ) of f with F 0 (p 0 , p 1 ) = ∑ d j=0 a j p d-j 0 p j 1 and F 1 (p 0 , p 1 ) =

Recall that the resultant of such a map F is given by Res(F ) :=

a d , b 0 , . . . , b d ] 2d , and that Res(F ) ̸ = 0, see, e.g., [Sil2, §2.4] for more details. Following Kawaguchi-Silverman [KS, Definition 2], we say F is minimal if |F | = 1. Such a minimal lift F of f is unique up to multiplication in {z ∈ K : |z| = 1}, so that the resultant of f Res(f ) := Res(F ) ∈ K *

Introduction

Fix an integer d > 1 and let Rat d be the space of degree d rational maps of P 1 . This is an irreducible affine variety of dimension 2d + 1 which is defined over Q. The group linear algebraic SL 2 acts on the space Rat d by conjugacy. The dynamical moduli space M d of degree d rational maps on P 1 is the geometric quotient Rat d /SL 2 , which is an irreducible

The first and third authors' research is partially supported by the ANR grant Lambda ANR-13-BS01-0002. The second author's research is partially supported by JSPS Grant-in-Aid for Scientific Research (C), 15K04924.

affine variety of dimension 2d -2 and defined over Q, and is singular if and only if d ≥ 3 (see Silverman [START_REF] Joseph | The space of rational maps on P 1[END_REF] for more precise statements).

In [GOV], the authors established an approximation formula for the Lyapunov exponent with respect to the equilibrium measure of any complex rational maps in terms of the absolute values of the multipliers of periodic points which is locally uniform on the complex manifold Rat d (C). The aim of establishing such an approximation property was to understand the equidistribution of centers of hyperbolic components of disjoint type towards the bifurcation measure µ bif of the Moduli space M d (C) and also to give a precise asymptotic count of such components, also related to the bifurcation measure.

Since the proof of the (complex) approximation formula essentially relies on onedimensional potential theoretic tools and on Fatou's upper bound of attracting cycles of complex rational maps, it is natural to wonder whether a similar formula is true over any complete algebraically closed field of characteristic 0.

In the present article, we address this question and give a certain number of consequences over local and global fields. We first present the formula and all the involved tools as well as a degeneration property over a local non-archimedean field. In a second time, we present applications over global fields.

Notations. -We introduce here some notations we will use in the text which are well-defined over any field. We refer to [START_REF] Joseph | The arithmetic of dynamical systems[END_REF] for more details.

Fix a field k and let k be an algebraic closure of k (we use K to denote local fields in the sequel). Fix a rational map f ∈ k(z) of degree d > 1.

For any n ∈ N * , we denote by Fix(f n ) the set of all periodic points of f of period dividing n, i.e. Fix(f n ) := {z ∈ P 1 ( k) : f n (z) = z}. We let Fix * (f n ) the set of points z ∈ P 1 ( k) such that either z ∈ Fix(f n ) \ ∪ m|n, m<n Fix(f m ), or there is m|n with m < n such that z ∈ Fix(f m ) and (f m ) ′ (z) is a primitive n/m-root of unity. For any n ∈ N * , we let

d n := ∑ m|n µ(n/m)(d m + 1).
The set Fix * (f n ) contains d n points counted with multiplicity (as a point of the divisor {f n (z) = z}). For any 1 ≤ j ≤ d n , we let σ * j,n (f ) be the j-th elementary symmetric polynomial associated to the unordered d n -tuple {(f n ) ′ (z)} z∈Fix * (f n ) , where points are listed with multiplicity. The map Λn : f → (σ * 1,n (f ), . . . , σ * dn,n (f )) actually defines a morphism from the space of all degree d rational maps to the affine space A dn which is defined over Q and descends to a morphism M d → A dn .

We also denote by p d,n (f, T ) ∈ k[T ] the multiplier polynomial, i.e. the unique monic degree d n /n polynomial with coefficients in k such that

( p d,n (f, T ) ) n = ∏ z∈Fix * (f n ) ( (f n ) ′ (z) -T ) = dn ∑ j=0 σ * j,n (f ) • (-T ) dn-j ,
where F is a minimal lift of f , is defined up to multiplication in {z ∈ K : |z| = 1}. We may identify Rat d (K) with the Zariski open set Rat d (K) = P 2d+1 (K) \ {Res = 0}. When K is non-archimedean, by the strong triangle inequality, | Res F | ≤ 1 = |F |. Let P 1 := P 1 (K) be the Berkovich projective line over K. The Lyapunov exponent of f with respect to the equilibrium (or canonical) measure µ f of is defined by L(f ) := ∫ P 1 log f # µ f ∈ R, where f # is the continuous extension to P 1 of the chordal derivative of f (see Section 2.3 for more details). For every n ∈ N * , every r ∈]0, 1] and every f ∈ Rat d (K), set

L n (f, r) := 1 nd n ∑ z∈Fix * (f n ) log max{r, |(f n ) ′ (z)|}. For n ∈ N * , let σ 2 (n) := ∑ m∈N:m|n m 2 . Note that σ 2 (n) = O ( n 2 log log n ) as n → ∞.
Suppose now K is non-archimedean. For every integer d > 1 and every n ∈ N * , following [BIJL], we also set

ϵ d n := min { |m| d n : m ∈ {1, 2, . . . , d n } } ∈ |K * | ⊂]0, 1].
Our first result, from which we derive several applications, is the following nonarchimedean counterpart of [START_REF] Gauthier | Hyperbolic components of rational maps: Quantitative equidistribution and counting[END_REF]Theorem 3.1], where we adapt our strategy to the non-archimedean setting.

Theorem A. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value | • |, and fix an integer d > 1. Then for every f ∈ Rat d (K), every n ∈ N * , and every r ∈]0, ϵ d n ],

L n (f, r) -L(f ) ≤ 8(d -1) 2 ( |L(f )| - 4d 2 -2d -1 d(2d -2) log |Res(f )| + | log r| ) σ 2 (n) d n .
In particular, |Res(f )| can be replaced with inf h∈[f ] |Res(h)| in the above inequality.

This type of results has a long history. For a fixed f defined over a number field, Szpiro and Tucker [ST] proved a qualitative version of this approximation formula. In [START_REF] Okuyama | Quantitative approximations of the Lyapunov exponent of a rational function over valued fields[END_REF], the second author proved a quantitative version of this formula with an error term which does not depend nicely on the map.

As usual, let O(D K ) be the ring of K-analytic functions on D K = {z ∈ K : |z| < 1}, i.e. a function f ∈ O(D K ) if it can be expanded as a power series f (t) = ∑ i≥0 a i t i with the condition |a i |ρ i → 0 for all ρ < 1 since K is non-archimedean. Observe that a function f belongs to O(D K )[t -1 ] iff it is a meromorphic function on D K with no pole in D K \ {0}. An element f t (z) ∈ O(D K )[t -1 ](z) is said to be a meromorphic family of rational functions of degree d parametrized by D K if we have f t ∈ Rat d (K) for every t ∈ D K \ {0}. The following consequence of Theorem A is a non-archimedean counterpart of a combination of [START_REF] Demarco | Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity[END_REF]Theorem 1.4] and [START_REF] Demarco | Bifurcations, intersections, and heights[END_REF]Proposition 3.1] (see also [START_REF] Gauthier | Hyperbolic components of rational maps: Quantitative equidistribution and counting[END_REF]Theorem 3.6]).

Theorem B (Degeneration). -Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value | • |, and fix an integer d > 1. Then for every meromorphic family f t (z) ∈ O(D K )[t -1 ](z) of rational functions of degree d parametrized by D K , there exists α ∈ R + such that L(f t ) = α log |t| -1 + o(log |t| -1 ) as t → 0. Note that Favre [F] has proved the same statement for meromorphic families of endomorphisms of P k (C) using the machinery of hybrid spaces. Note also that, in the case of families of polynomials, the much more stronger assertion that the (subharmonic) function t → L(f t ) -α log |t| -1 on D * K extends continuously around t = 0 has been established in [FG] by Favre and the first author. Note that the continuity statement is not true in full generality. Indeed, DeMarco and the second author [DO] provide a family of examples of meromorphic families of complex rational maps for which L(f t ) -α log |t| -1 is unbounded near 0.

Global fields and height functions

A global field is a field k which is canonically equipped with (i) a proper set M k of all places of k,

(ii) a family (| • | v ) v∈M k , where for each v ∈ M k , | • | v is a non-trivial absolute value of k representing v. A place v ∈ M k is said to be an infinite place of k if | • | v is archimedean; otherwise, v ∈ M k
is said to be a finite place of k, (iii) a family (N v ) v∈M k of positive integers such that for every z ∈ k × , we have |z| v = 1 for all but finitely many places v ∈ M k and the so called product formula holds:

∏ v∈M k |z| Nv v = 1.
Note in addition that if k ′ is an algebraic extension of k, then the set of places w ∈ M k ′ that extend places of v, denoted by w|v, is finite and that they satisfy the compatibility condition ∑

w|v

N w = [k ′ : k].
Number fields are the archetype of global fields. In this paragraph, we give applications of Theorem 3.1 over number fields.

For each v ∈ M k , let k v be the completion of k with respect to |•| v and C v the completion of an algebraic closure of k v with respect to | • | v , and we fix an embedding of k in C v which extends that of k. By convention, the dependence of any local quantity induced by

| • | v on each v ∈ M k is emphasized by adding the suffix v to it, e.g., we will denote L(f ) v for the Lyapunov exponent of f ∈ Rat d ( k) acting on P 1 (C v ).
Choose an integer N ≥ 1. The naive height function h P N ,k on P N (k) associated to k is defined for every x = [x 0 : . . . :

x n ] ∈ P N ( k) by h P N ,k (x) = 1 [k ′ : k] ∑ v∈M k ′ N v log max 0≤j≤N |x j | v ,
where k ′ is any finite extension of k such that x ∈ P N (k ′ ).

Let k be a global field, and fix an integer d > 1. We define a height function h d,k on Rat d ( k) as follows: we identify Rat d ( k) with

P 2d+1 ( k) \ {Res = 0}, which is a Zariski open subset of P 2d+1 (k). For every f = [a 0 , . . . , a d , b 0 , . . . , b d ] ∈ Rat d ( k), we let h d,k (f ) := h P 2d+1 ,k ([a 0 , . . . , a d , b 0 , . . . , b d ]). Pick f ∈ Rat d (k). The Call-Silverman canonical height function ĥf,k of f on P 1 (k) relative to k is defined by ĥf,k (z) := lim n→∞ h P 1 ,k (f n (z)) d n ∈ R, z ∈ P 1 (k).
The critical height function h crit,k on Rat d (k) relative to k is defined by

h crit,k (f ) := 1 [k ′ : k] ∑ c∈Crit(f ) ĥf,k ′ (c), f ∈ Rat d (k), where k ′ is any finite extension of k such that f ∈ Rat d (k ′ ) and Crit(f ) ⊂ P 1 (k ′ ). The function h crit,k on Rat d (k) descends to a function on M d (k), which is still denoted by h crit,k .
Although the critical height functions defined above are named as "height" functions, it is a priori not clear at all whether they are height functions, in any possible sense.

Applications over number fields

Fix d > 1. Silverman conjectured [START_REF] Joseph | Moduli spaces and arithmetic dynamics[END_REF]Conjecture 6.29] that the absolute critical height h crit,Q is commensurable to any ample height on M d away from the flexible Lattès locus. Ingram [I] recently proved this conjecture. We give here a quantitative version of Ingram's result.

Let us be more specific. Recall that f ∈ Rat d (C) is a Lattès map if there exists a complex elliptic curve E and a homomorphism ϕ : E → E and a finite morphism π :

E → P 1 (C) such that f • π = π • ϕ on E.
We say f is flexible if ϕ is given by ϕ(P ) = [m]P + P 0 for some m ∈ Z and some P 0 ∈ E. Let L d ⊂ M d be the flexible Lattès locus; it is an affine curve which is defined over Q.

We also fix an embedding ι : M d → A N of the moduli space in some affine space and let M d be the Zariski closure of ι(M d ) in the ambient projective space P N . The line bundle

D := O P N (1)| M d defines an ample line bundle on M d . Let also h M d ,D := h P N ,Q | ι(M d ) = h P N ,Q • ι + O(1),
and set

C 1 (d, D) := 1 8(d -1) ( ∥µ bif ∥ M d deg D (M d ) ) 1/(2d-2) and C 2 (d, D) = 2(d -1) ∥T bif ∥ M d ,D deg D (M d ) ,
where T bif and µ bif are the bifurcation current and the bifurcation measure (see Definition 6.4 below) and ∥T bif ∥ M d ,D and ∥µ bif ∥ M d denote their respective mass. Of course, the mass of T bif depends on the choice of the embedding. We prove Theorem C. -For every integer d > 1, there exists a constant A ≥ 0 such that

C 1 (d, D)h M d ,D -A ≤ h crit,Q ≤ C 2 (d, D)h M d ,D + A on (M d \ L d )( Q).
Moreover, the constant A can be computed explicitly.

The strategy of the proof is similar to that of Ingram [I]. In his proof, he relates the critical height to the height of a single multiplier. Instead, using our quantitative approximation formulas of L(f ) v for each place, we relate the critical height with the average of the heights of all multipliers of cycles of period dividing a given n. Recall that Λn :

M d → A d n +1
denote the map which, to any conjugacy class [f ] associates the symmetric functions of the multipliers of periodic of period dividing n.

One of the interests of our strategy is that our relation between the critical and multiplier heights can be used in both ways. As an example of this idea, we get that the multipliers of formal exact period n define a moduli height for all n large enough: we prove that for all n ≥ n 1 , there exists a computable constant C depending only on n and d such that Theorem D improves McMullen's finiteness theorem [START_REF] Mcmullen | Families of rational maps and iterative root-finding algorithms[END_REF]Corollary 2.3] which states that, for every integer d > 1, if n is large enough, the map

C 1 (d, D) 2 h M d ,D -C ≤ 1 nd n h P dn ,Q • Λn ≤ ( C 2 (d, D) + C 1 (d, D) 2 ) h M d ,D + C, on (M d \ L d )( Q).
Λ n : M d (C) → C d n +1
given by the symmetric functions of the multipliers of periodic points

z ∈ Fix(f n ) of period dividing n is finite-to-one on (M d \ L d )(C).

Applications over global function fields

Let X be a connected smooth projective curve define over C and let k := C(X) be the function field of X. In this case, there is a one-to-one correspondence between closed points of X and places of the field k. If v ∈ M k corresponds to the point x, we choose | • | v to be defined by |g| v = e -ordx(g) for all g ∈ C(X). Moreover, we can take N v = 1 for every v ∈ M k . The naive height function on P N (k) associated to k can be computed as

h P N ,k (x) = max 0≤j≤N deg(x j ) deg(π) , x = [x 0 : • • • : x N ],
where π : Y → X is any finite morphism such that that x ∈ P N (C(Y )). In particular, the canonical height function ĥf of a rational map f defined over k can be computed as

x → ĥf (x) = lim n→∞ 1 d n deg(f n (x)) deg(π x ) ,
where π x : Y → X is any finite branched cover such that x ∈ P 1 (C(Y )).

We say f ∈ Rat d (k) is isotrivial if there exists a finite branched cover Y → X and a Möbius transformation M ∈ PGL(2) which is defined over the finite extension k ′ = C(Y ) of k such that the rational map M • f • M -1 is actually defined over C. This is equivalent to the fact that the specialization (M

• f • M -1 ) t ∈ Rat d (C) are independent of t ∈ Y .
We say that f ∈ Rat d (k) is affine if there exists a non-isotrivial elliptic curve E defined over k and a homomorphism ϕ : E → E given by ϕ(P ) = [m]P + P 0 for some m ∈ Z and some P 0 ∈ E and a degree 2 morphism

E → P 1 k such that f • π = π • f on E.
Equivalently, we say f is affine if for every t ∈ X, the specialization f t ∈ Rat d (C) is a Lattès map, i.e. if there exists a complex elliptic curve E t , a double branched cover π t : E t → P 1 (C) and an integer m ∈ Z and P 0,t ∈ E t such that π t ([m]

P + P 0,t ) = f t • π t (P ) on E. In particular, d = m 2 .
Applying Theorem A to the action of f on P 1 (C v ) for all v ∈ M k , together with previously known results, we easily establish the following quantitative characterization of those exceptional properties for such a rational map f . Theorem E. -Let X be a connected smooth projective curve defined over C, and fix an integer d > 1. For every f ∈ Rat d (C(X)), the following assertions are equivalent:

(i) f is either isotrivial or affine, (ii) max 1≤j≤dn deg(σ * j,n (f )) = o(nd n ) as n → ∞, (iii) max 1≤j≤dn deg(σ * j,n (f )) = O(nσ 2 (n)) as n → ∞.
Note finally that a similar statement over the function field of a normal irreducible projective variety defined over C can be obtained, applying the same strategy word by word, see 5.3.

Structure of the paper

The paper is organized as follows: In Section 2 we recall classical facts about the dynamics of rational maps over any metrized field and give preliminary technical results. Section 3 is dedicated to the proof of Theorem A and B. We then focus on estimates over global function fields in Section 4. Section 5 is concerned with the proof of Theorem E. The applications of the archimedean approximation formula are given in Section 6 and Section 7 and 8 are dedicated to the proofs of Theorem C and D.

Acknowledgment

We would like to thank Patrick Ingram for useful discussions and for providing the reference [BG], and Nguyen-Bac Dang for useful discussions concerning Siu's numerical bigness criterion. The second author also thanks the LAMFA and the Université de Picardie Jules Verne, where he was visiting in autumn 2017 and this work grew up. Let ∥ • ∥ be the maximal norm on K 2 (when K is non-archimedean) or the Hermitian norm on K 2 (when K is archimedean, i.e., K ∼ = C). In the rest of this section, for simplicity we denote P 1 (K) by P 1 . With the wedge product (p 0 , p 1 ) ∧ (q 0 , q 1 ) := p 0 q 1 -p 1 q 0 on K, the normalized chordal metric [z, w] on P 1 is defined by [z, w]

Background and key algebraic estimates

:= |p ∧ q| ∥p∥ • ∥q∥
, where p ∈ π -1 (z), q ∈ π -1 (w), on P 1 (and as a function on P 1 ×P 1 ). The subgroup U K in the linear fractional transformation group PGL(2, K) defined by PGL(2, O K ) (when K is non-archimedean) or PSU(2, K) (when K is archimedean) acts isometrically on (P 1 , [z, w]).

Let us recall some details on the Berkovich projective line P 1 = P 1 (K). When K is archimedean, then P 1 coincides with P 1 , so we assume K is non-archimedean until the end of this paragraph. Then P 1 is a compact, Hausdorff, and uniquely arcwise connected topological space augmenting P 1 , and is also a tree in the sense of Jonsson

) of K-closed disks in K such that ∩ n B n = ∅).
The tree structure on P 1 is induced by the partial ordering of all those disks in K by inclusions. The topology of P 1 is nothing but the weak topology of P 1 as a tree. For every P ∈ K[z], the function |P | on K extends continuously to P 1 \ {∞} so that

|P |(S) = sup B S |P (•)| on P 1 \ {∞};
in particular |P |(S can ) = max{|coefficients of P |} by the strong triangle inequality.

When K is non-archimedean, the chordal distance [z, w] on P 1 (×P 1 ) uniquely extends to a (jointly) upper semicontinuous and separately continuous function [S, S ′ ] can on P 1 × P 1 , which is called the generalized Hsia kernel on P 1 with respect to S can (for more details, see [START_REF] Favre | Equidistribution quantitative des points de petite hauteur sur la droite projective[END_REF]§3.4], [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]§4.4]); for every z ∈ P 1 , [S can , z] can = 1, and for any S, S ′ , S ′′ ∈ P 1 , we still have the strong triangle inequality

[S, S ′ ] can ≤ max{[S, S ′′ ] can , [S ′ , S ′′ ] can }
(see e.g. [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]Proposition 4.10(C)]). When K is archimedean, it is convenient to define the kernel function [z, w] can on P 1 (×P 1 )by [z, w] itself.

For every S ∈ P 1 , let δ S be the Dirac measure on P 1 at S. Let us introduce the probability Radon measure

Ω can := { δ Scan when K is non-archimedean ω when K is archimedean on P 1 ,
where ω is the Fubini-Study area element on P 1 normalized as ω(P 1 ) = 1 when K is archimedean, and for every S ∈ P 1 . The Laplacian ∆ on P 1 is normalized so that for each

S ′ ∈ P 1 , ∆ log[•, S ′ ] can = δ S ′ -Ω can on P 1
(for non-archimedean K, see [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]§5.4], [START_REF] Favre | Théorie ergodique des fractions rationnelles sur un corps ultramétrique[END_REF]§2.4]; in [BR] the opposite sign convention on ∆ is adopted). The action on P 1 of each f ∈ K(z) extends to a canonical continuous action on P 1 . If in addition deg f > 0, then the canonical action on P 1 of f is also surjective, open, and finite. Moreover, the local degree function a → deg a f of f on P 1 also canonically extends to an upper semicontinuous function P 1 → {1, 2, . . . , deg f } such that for every domain

V ⊂ P 1 and every component U of f -1 (V ), the function S → ∑ S ′ ∈f -1 (S)∩U deg S ′ (f ) is constant on V .
The pullback action f * induced by f on the space of Radon measures on P 1 is defined for every S ∈ P 1 by f

* δ S = ∑ S ′ ∈f -1 (S) deg S ′ (f ) • δ S ′ on P 1
, and satisfies the functoriality f * • ∆ = ∆ • f * (see e.g. [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]§9]).

For every f ∈ K(z) of degree > 1, the f -equilibrium (or canonical) measure µ f on P 1 is defined by such a unique probability Radon measure ν on P 1 that

f * ν = (deg f ) • ν on P 1 and ν(E(f )) = 0, where E(f ) := {a ∈ P 1 : # ∪ n∈N f -n (a)
< ∞} is at most countable and, if K has characteristic 0, then #E(f ) ≤ 2. The (Berkovich) Julia set J(f ) is defined by the support of µ f ; a point S in P 1 is in J(f ) if and only if for any sequence (n j ) in N tending to ∞ as j → ∞, we have [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]§10], [START_REF] Chambert-Loir | Mesures et équidistribution sur les espaces de Berkovich[END_REF]§2], [START_REF] Favre | Théorie ergodique des fractions rationnelles sur un corps ultramétrique[END_REF]§3.1] for more details). The (Berkovich) Fatou set F(f ) is the open subset P 1 \ J(f ) in P 1 , and each component of

∩ U : open in P 1 and contains S ( ∪ j∈N f n j (U ) ) = P 1 \ E(f ) (see
F(f ) is called a (Berkovich) Fatou component of f . If W is a (Berkovich) Fatou component of f , then so is f (W ),
and we say W is cyclic under f if f N (W ) = W for some N ∈ N * . We also say that W is an immediate Berkovich attracting basin of period N for f if there is a (super)attracting fixed point a of f N in W ∩ P 1 with lim n→∞ (f N ) n (S) = a for any S ∈ W .

The dynamical Green function

A continuous weight g on P 1 is a continuous function on P 1 such that µ g := ∆g + δ Scan is a probability Radon measure on P 1 . For a continuous weight g on P 1 , the g-potential kernel on P 1 (or the negative of an Arakelov Green kernel function on P 1 relative to µ g [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]§8.10] or an Ω can -quasipotential on P 1 of µ g ) is defined by

Φ g (S, S ′ ) := log[S, S ′ ] can -g(S) -g(S ′ ) on P 1 × P 1 , the g-equilibrium energy V g of P 1 (in fact V g ∈ R) is the supremum of the g- energy functional ν → ∫ P 1 ×P 1 Φ g d(ν ×ν) ∈ [-∞, +∞
) over all probability Radon measures ν on P 1 , and we call a probability Radon measure ν on P 1 at which the above g-energy functional attains the supremum V g a g-equilibrium mass distribution on P 1 ; in fact, µ g is the unique g-equilibrium mass distribution on P 1 , and

S → ∫ P 1 Φ g (S, •)dµ g ≡ V g
on P 1 (for non-archimedean K, see [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]Theorem 8.67,Proposition 8.70]). A normalized weight g on P 1 is a continuous weight on P 1 satisfying V g = 0; for every continuous weight g on P 1 , g := g + V g /2 is the unique normalized weight on P 1 such that µ g = µ g on P 1 .

From now on, pick f ∈ K(z) of degree d > 1. For every lift F of f , the function For every n ∈ N * , F n is a lift of f n , and by a telescoping sum argument (cf. [START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF]§10], [FRL2, §6.1]), the following limit exists

T F := log ∥F (•)∥ -d • log ∥ • ∥ on K 2 \ {0}
g F := lim n→∞ T F n d n = lim n→∞ n-1 ∑ j=0 T F • f j d j+1 on P 1 . (2.1)
More precisely, for every n ∈ N * , we have the following uniform error estimate sup

P 1 g F - T F n d n ≤ sup P 1 |T F | d n (d -1) , (2.2)
and for each α ∈ K \ {0},

T αF = T F + log |α| and g αF = g F + log |α| d -1 . (2.3)
on P 1 . The function g F is called the dynamical Green function on P 1 of F , and which is a continuous weight on P 1 satisfying µ g F = µ f on P 1 ; indeed, µ g F = ∆g F + Ω Scan = lim n→∞ ((f n ) * Ω Scan /d n ) weakly on P 1 , and µ g F satisfies the defining two properties of µ f . We note the following pull-back formula

g F • f = d • g F -T F on P 1 (2.4) ([Oku3, Proof of Lemma 2.4]
) and the following energy formula

V g F = - log | Res F | d(d -1)
(for a simple proof, see e.g. [Ba, Appendix A] or [START_REF] Okuyama | Potential theory and a characterization of polynomials in complex dynamics[END_REF]Appendix]. For K ∼ = C, this is equivalent to DeMarco's capacity formula [START_REF] Demarco | Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity[END_REF]). More intrinsically, the dynamical Green function g f of f on P 1 is defined by the unique normalized weight on P 1 satisfying µ g f = µ f on P 1 ; for every lift F of f , we have In this article, we are saying F to be minimal (rather than "normalized") if it satisfies the latter condition |F |, and we would avoid to use the (more ambiguous) terminology "a normalized lift of f " for both normalizations of F .

g f = g F - log | Res F | d(2d -2) on P 1 , (2.5) so that Φ g f = Φ g F + log | Res F | d(d -1) on P 1 × P 1 . Since Res F ∈ K \ {0}
We recall a standard fact from elimination theory.

Fact 2.1. -Let R be an integral domain. Then for any two homogeneous polynomials

F 0 (X, Y ) = ∑ d j=0 a j X d-j Y j , F 1 (X, Y ) = ∑ d k=0 b k X d-k Y k in R[X, Y ] d , there exist G 1 , G 2 , H 1 , H 2 ∈ R[X, Y ] d-1 such that F 1 (X, Y )G 1 (X, Y ) + F 2 (X, Y )G 2 (X, Y ) =(Res F )X 2d-1 and F 1 (X, Y )H 1 (X, Y ) + F 2 (X, Y )H 2 (X, Y ) =(Res F )Y 2d-1 in R[X, Y ] and that all the 4d coefficients of G 1 , G 2 , H 1 , H 2 ∈ R[X, Y ] d-1 (determined by (a 0 , . . . , a d , b 0 , . . . , b d )) are in Z[a 0 , . . . , a d , b 0 , . . . , b d ] 2d-1 (see e.g. [Sil2, Proposition 2.13]).
The following algebraic estimates on T F and g F are crucial for our purpose.

Lemma 2.2. -There exists a constant

A 1 (d, K) ≥ 0 depending only on d and K such that for every lift F of any f ∈ Rat d (K), on P 1 , log | Res F | -(2d -1) log |F | -A 1 (d, K) ≤ T F ≤ log |F | + A 1 (d, K) log | Res F | -(2d -1) log |F | -A 1 (d, K) ≤ (d -1)g F ≤ log |F | + A 1 (d, K). Moreover, A 1 (d, K) = 0 when K is non-archimedean. Proof. -Fix f ∈ Rat d (K) and a lift F = (F 0 , F 1 ) of f , and write as F 0 (X, Y ) = ∑ d j=0 a j X j Y d-j and F 1 (X, Y ) = ∑ d k=0 b k X k Y d-k and let G 1 , G 2 , H 1 , H 2 ∈ K[X 0 , X 1 ] d-1 be the homogeneous polynomials associated to F 0 , F 1 as in Fact 2.1. Set A := { 1 when K is non-archimedean, √ 2(d + 1) when K is archimedean. Fix p ∈ K 2
. By the (strong) triangle inequality, we have

∥F (p)∥ ≤ A|F | • ∥p∥ d .
When K is non-archimedean, using the strong triangle inequality and Fact 2.1 gives

max{|G 1 (p)|, |G 2 (p)|, |H 1 (p)|, |H 2 (p)|} ≤ |F | 2d-1 ∥p∥ d-1 .
When K is archimedean, the triangle inequality and Fact 2.1 give

max{|G 1 (p)|, |G 2 (p)|, |H 1 (p)|, |H 2 (p)|} ≤ A ′ |F | 2d-1 ∥p∥ d-1
where A ′ depends only on d is given by 2d times the maximum of the absolute value of the integer coefficients that appear in the expression of the coefficients of

G 1 , G 2 , H 1 , H 2 as polynomials in a 0 , . . . , a d , b 0 , . . . , b d .
Using the equalities in Fact 2.1, we get

| Res F | • ∥p∥ d ≤ |F | 2d-1 ∥F (p)∥ if K is non-archimedean and | Res F | • ∥p∥ d ≤ A ′′ |F | 2d-1 ∥F (p)∥ if K is archimedean, where A ′′ = 2 √ 2A ′
and the lemma follows by definition of T F and g F .

The chordal derivative and a lemma à la Przytycki

Pick f ∈ K(z) of degree d > 1. The chordal derivative of f is the function z → f # (z) := lim P 1 ∋w→z [f (w), f (z)] [w, z] ∈ R + on P 1 (K), so that f # (z) = |f ′ (z)| if f (z) = z.
For every lift F of f , by a computation involving Euler's identity, we have [START_REF] Jonsson | Sums of Lyapunov exponents for some polynomial maps of C 2 . Ergodic Theory Dynam[END_REF]Theorem 4.3]). We can note that the map f :

f # (z) = 1 |d| ∥p∥ 2 ∥F (p)∥ 2 | det DF (p)|, where p ∈ π -1 (z), (2.6) on P 1 (cf.
P 1 → P 1 is M 1 (f )-Lipschitz continuous for the metric [•, •],
where

M 1 (f ) := { 1/| Res(f )| when K is non-archimedean (by Rumely-Winburn [RW]), sup P 1 (f # ) when K is archimedean, and let us set M (f ) := M 1 (f ) 2 (≥ 1).
The following is a non-archimedean counterpart to [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF]Lemma 3.1] (see also [START_REF] Gauthier | Hyperbolic components of rational maps: Quantitative equidistribution and counting[END_REF]Lemma 2.2]).

Lemma 2.3. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value

| • |. For any f ∈ K(z) of degree d > 1, any c ∈ Crit(f ) and any n ∈ N * , if 0 < [f n (c), c] < 1/M 1 (f ) n+1 , then [J(f ), c] can ≥ 1/M 1 (f ) n+1 . Moreover, c lies in the Berkovich immediate basin of a (super)attracting fixed point a ∈ P 1 (K) of f n such that [a, c] ≤ [f n (c), c]. Proof. -Fix f ∈ K(z) of degree > 1 and c ∈ Crit(f ). We claim that for every z ∈ P 1 , if [z, c] < 1/M 1 (f ), then [f (z), f (c)] ≤ M 1 (f ) 2 • [z, c] 2 ; (2.7) indeed, fixing such A, B ∈ PGL(2, O K ) that A(f (c)) = B -1 (c) = 0 and setting g := A • f • B, we have g(0) = g ′ (0) = 0, so that h(z) := g(z)/z extends analytically to z = 0 and that h(0) = 0, and M 1 (f ) = M 1 (g). If |z| < 1/M 1 (g)(≤ 1), then we have |g(z)| = [g(z), 0] = [g(z), g(0)] ≤ M 1 (g) • [z, 0] = M 1 (g)|z| < 1,
and in particular |h(z)| ≤ M 1 (g). Hence by the (non-archimedean) Schwarz lemma (see [START_REF] Rivera-Letelier | Dynamique des fonctions rationnelles sur des corps locaux[END_REF]Corollaire 1.4

& Lemme de Schwarz]), if 0 < |z| < 1/M 1 (g), then g(z) z = |h(z) -h(0)| ≤ sup |w|<1/M 1 (g) |h(w)| 1/M 1 (g) |z -0| ≤ M 1 (g) 2 |z|, that is, if [z, 0](= |z|) < 1/M 1 (g), then [g(z), 0] = |g(z)| ≤ M 1 (g) 2 |z| 2 = M 1 (g) 2 [z, 0] 2 .
Hence the claim holds.

Then by (2.7), the separate continuity of [S, S ′ ] can on P 1 × P 1 , the density of P 1 in P 1 , and the continuity of the action on P 1 of f , for every

S ∈ P 1 , if [S, c] can < 1/M 1 (f ), then we still have [f (S), f (c)] can ≤ M 1 (f ) 2 • [S, c] 2 can .
Similarly, for every S ∈ P 1 and every

n ∈ N, we have [f n (S), f n (c)] can ≤ M 1 (f ) n-1 • [f (S), f (c)] can . Fix n ∈ N so that 0 < [f n (c), c] < 1/M 1 (f ) n+1 . We claim that [J(f ), c] can ≥ 1/M 1 (f ) n+1 ; otherwise, for every S ∈ P 1 , if [S, c] can < ϵ 0 := max{[f n (c), c], [J(f ), c] can }(∈ (0, 1/M 1 (f ) n+1 )), then by (2.7), we have [f n (S), f n (c)] can ≤ M 1 (f ) n-1 • [f (S), f (c)] can ≤ M 1 (f ) n+1 • [S, c] 2 can ≤ [S, c] can < ϵ 0 , so that [f n (S), c] can ≤ max{[f n (S), f n (c)] can , [f n (c), c]} < ϵ 0 . Hence setting B(c, ϵ 0 ) := {S ∈ P 1 : [S, c] can < ϵ 0 (< 1)}, which is a domain in P 1 satisfying #(P 1 \ B(c, ϵ 0 )) > 2, we must have f n•j (B(c, ϵ 0 )) ⊂ B(c, ϵ 0 ) for every j ∈ N, so that B(c, ϵ 0 ) ⊂ P 1 \ J(f ).
However, by the choice of ϵ 0 , we have B(c, ϵ 0 ) ∩ J(f ) ̸ = ∅, which is a contradiction. Hence the claim holds.

Replacing the above ϵ 0 with ϵ

1 := [f n (c), c], setting B(c, ϵ 1 ) := {S ∈ P 1 : [S, c] can < ϵ 1 (< 1)}, we still have f n (B(c, ϵ 1 )) ⊂ B(c, ϵ 1 ) ⊂ F(f ). Let W be the Berkovich Fatou component of f containing B(c, ϵ 1 ), which is cyclic since we have f n (W ) = W . Fix such C ∈ PGL(2, O K ) that C(c) = 0. Then {z ∈ K : |z| ≤ ϵ 1 } = C(B(c, ϵ 1 ) ∩ P 1 ) ⊃ (C • f n • C -1 )({z ∈ K : |z| ≤ ϵ 1 }).
Hence by the (non-archimedean) Schwarz lemma (see [START_REF] Rivera-Letelier | Dynamique des fonctions rationnelles sur des corps locaux[END_REF]Corollaire 1.4

]), C • f n • C -1 has a fixed point b in {z ∈ K : |z| ≤ ϵ 1 } = {z ∈ K : |z -b| ≤ ϵ 1 }, so that |(C • f n • C -1 ) ′ (b)| ≤ 1, and indeed |(C • f n • C -1 ) ′ (b)| < 1 since C -1 (0) = c ∈ Crit(f n ). Hence a := C -1 (b) ∈ B(c, ϵ 1 ) ∩ P 1 ⊂ W ∩ P 1 is a (super) attracting fixed point of f n and [a, c] = [C(a), C(c)] = [b, 0] = |b| ≤ ϵ 1 = [f n (c), c].
By an argument similar to the above shows a more effective version of [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF]Lemma 3.1], which would be needed below.

Lemma 2.4. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to an archimedean absolute value. For every

f ∈ K(z) of degree > 1, every c ∈ Crit(f ), and every n ∈ N * , if 0 < [f n (c), c] < 1/(64M 1 (f ) n+1 ), then [J(f ), c] ≥ 1/(32M 1 (f ) n+1
) and, moreover, c lies in the immediate basin of a (super)attracting fixed point

a ∈ P 1 (K) of f n such that [a, c] ≤ 2[f n (c), c].
The following algebraic upper estimate on f # is a keystone in the sequel.

Lemma 2.5. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-tivial absolute value | • |. There exists a constant A 2 (d, K) ≥ 0 depending only on d and K such that for every f ∈ Rat d (K) and every lift

F of f , log(f # ) ≤ -log |d| -2 log | Res F | + 4d log |F | + A 2 (d, K) on P 1 . Moreover, A 2 (d, K) = 0 when K is non-archimedean. Proof. -Fix f ∈ Rat d (K) and a lift F = (F 0 , F 1 ) of f . Then det DF (X, Y ) ∈ K[X, Y ] 2(d-1) and, writing F 0 = ∑ d j=0 a j X j Y d-j and F 1 = ∑ d k=0 b k X k Y d-k , all the coefficients of det F are in Z[a 0 , . . . , a d , b 0 , . . . , b d ] 2 . By the (strong) triangle inequality, we have | det DF (p)| ≤ C|F | 2 on {p ∈ K 2 : ∥p∥ ≤ 1}, where we set C := 1 when K is non-archimedean or C := 2d 2 (d + 1) 2 when K is archimedean. Combined with (2.6), this yields log(f # ) • π(p) ≤ -log |d| -2T F • π + 2 log |F | + log C on {p ∈ K 2 : ∥p∥ ≤ 1}, so that also by Lemma 2.2, we have log(f # ) ≤ -log |d|-2T F +2 log |F |+log C ≤ -log |d|-2 log | Res F |+4d log |F |+A 2 (d, K)
on P 1 , and in turn on P 1 by the continuity of the chordal derivative f # on P 1 and the density of P 1 in P 1 .

The Lyapunov exponent

From now on, let us also assume that K has characteristic 0. Pick

f ∈ K(z) of degree d > 1 and choose C F 1 , . . . , C F 2d-2 ∈ K 2 \{0} so that the Jacobian determinant det DF ∈ K[p 0 , p 1 ] of a lift F of f factors as det DF (p 0 , p 1 ) = 2d-2 ∏ j=1 ((p 0 , p 1 ) ∧ C F j ) in K[p 0 , p 1 ] and setting c j := π(C F j ) for each j ∈ {1, . . . , 2d -2}(so that {c j : j ∈ {1, . . . , 2d -2}} = Crit(f )), by (2.6), the function f # on P 1 extends continuously to P 1 so that log(f # ) = -log |d| + 2d-2 ∑ j=1 (log[•, c j ] can + log ∥C F j ∥) -2T F = -log |d| + 2d-2 ∑ j=1 (Φ g F (•, c j ) + g F (c j ) + log ∥C F j ∥) + 2g F • f -2g F on P 1 .

Definition 2.6 ([FRL2]

). -The Lyapunov exponent of f with respect to µ f is

L(f ) := ∫ P 1 log(f # )dµ f ∈ R.
Note that L(f ) can be written as

L(f ) = -log |d| + 2d-2 ∑ j=1 (g F (c j ) + log ∥C F j ∥), (2.8) where F is a lift of f satisfying | Res F | = 1, so that log(f # ) = L(f ) + ∑ c∈Crit(f ) Φ g f (•, c) + 2g f • f -2g f on P 1 , (2.9)
and where the sum over Crit(f ) takes into account the multiplicity of each c ∈ Crit(f ) [START_REF] Okuyama | Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics[END_REF]Lemma 3.6] or [START_REF] Okuyama | Quantitative approximations of the Lyapunov exponent of a rational function over valued fields[END_REF]Lemma 2.4]). Note that L(f ) needs not be positive in full generality.

The following lower estimate is crucial for our purpose.

Proposition 2.7. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial absolute value | • |, and fix an integer d > 1. Then for every f ∈ Rat d (K) and every lift F of f , we have

L(f ) ≥ -log |d| + ∑ c∈Crit(f ) g F (c) - 2 d log | Res F | + log |C F |,
where

C F ∈ K \ {0} is the leading coefficient of det DF (1, z) ∈ K[z].
Proof. -Fix a lift F of f and pick C F 1 , . . . , C F 2d-2 as above. Then by (2.8) and by the homogeneity property (2.5) of g F , we have

L(f ) = -log |d| + 2d-2 ∑ j=1 (g F (c j ) + log ∥C F j ∥) - 2 d log | Res F | ≥ -log |d| + ∑ c∈crit(f ) g F (c j ) + ∑ j:c j ̸ =∞ log |C F j ∧ (0, 1)| + ∑ j:c j =∞ log |C F j ∧ (1, 0)| - 2 d log | Res F |,
where we used that ∥(0,

1)∥ = ∥(1, 0)∥ = 1. The fact that log |C F | = ∑ j:c j ̸ =∞ log |C F j ∧ (0, 1)| + ∑ j:c j =∞ log |C F j ∧ (1, 0)| completes the proof.

Locally uniform approximation of the Lyapunov exponent

In the whole section, K is an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial absolute value | • |. Fix an integer d > 1. For every n ∈ N * and every r > 0, define the function

f → L n (f, r) := 1 nd n ∑ z∈Fix * (f n ) log max{r, |(f n )(z)|} (3.1)
on Rat d (K), where the sum over Fix * (f n ) is taken wit multiplicity.

When K is archimedean, i.e., K ∼ = C, the function L n (•, r) is nothing but the function f → 1 d n ∫ 2π 0 log |p d,n (f, re iθ )| dθ 2π
on Rat d (K). When K is non-archimedean, for every f ∈ Rat d (K) and every n ∈ N * , the function w → |p d,n (f, w)| on K extends continuously to P 1 \ {∞}. Denote by |p d,n (f, •)| this extension. Moreover, for every r > 0, let S 0,r be the point in P 1 such that B S 0,r = {z ∈ K : |z| ≤ r}. By Gauss lemma, when K is non-archimedean, for every f ∈ Rat d (K), every n ∈ N * , and every r > 0, we have

L n (f, r) = 1 d n ∫ P 1 log |p d,n (f, •)|dδ S 0,r = 1 nd n log max 0≤j≤dn |σ * j,n (f )| • r dn-j . (3.2)

The non-archimedean locally uniform approximation formula

The following is a more precise version of Theorem A.

Theorem 3.1. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value, and fix an integer d > 1.

Then for every f ∈ Rat d (K), every n ∈ N * , and every r ∈ (0,

ϵ d n ], L n (f, r) -L(f ) ≤ 2(2d -2) 2 • ( |L(f )| + log M (f ) + sup P 1 |g f | + | log r| ) σ 2 (n) d n .
Proof.

-Fix an integer d > 1, n ∈ N * , and r ∈ (0, ϵ d n ]. In the following, for each f ∈ Rat d (K) and every m ∈ N * , the sums over subsets in Crit(f m ), Fix(f m ), or Fix * (f m ) take into account the multiplicities of their elements.

Fix f ∈ Rat d (K) satisfying Fix(f n ) ∩ Crit(f ) = ∅.
By the Möbius inversion, we have

d n • L n (f, r) = 1 n ∑ z∈Fix * (f n ) log max{r, (f n ) # (z)} = 1 n ∑ m|n µ ( n m ) ∑ z∈Fix(f m ) n m log max{r m/n , (f m ) # (z)} = ∑ m|n µ ( n m ) • 1 m ∑ z∈Fix(f m ) log max{r m/n , (f m ) # (z)}. (3.3)
For every m ∈ N * dividing n, taking the sums of both sides in (2.9) over Fix(f m ), by

∑ z∈Fix(f m ) 1 = d m + 1 and [OS2, Lemma 3.5], we have 1 m ∑ z∈Fix(f m ) log((f m ) # (z)) = ∑ z∈Fix(f m ) log((f # )(z)) =(d m + 1) • L(f ) + ∑ c∈Crit(f ) ∑ z∈Fix(f m ) Φ g f (z, c) =(d m + 1) • L(f ) + ∑ c∈Crit(f ) Φ g f (f m (c), c) and similarly, 1 m ∑ z∈Fix(f m ):(f n ) # (z)<r log((f m ) # (z)) = Card{z ∈ Fix(f m ) : (f n ) # (z) < r} • L(f ) + ∑ c∈Crit(f ) ∑ z∈Fix(f m ):(f n ) # (z)<r Φ g f (z, c), so that (3.4) 1 m ∑ z∈Fix(f m ) log max{r m/n , (f m ) # (z)} = ( d m + 1 -#{z ∈ Fix(f m ) : (f n ) # (z) < r} ) • L(f ) + ∑ c∈Crit(f ) ( Φ g f (f m (c), c) - ∑ z∈Fix(f m ):(f n ) # (z)<r Φ g f (z, c) ) + #{z ∈ Fix(f m ) : (f n ) # (z) < r} n • log r.
We claim that, for every c ∈ Crit(f ) and every m ∈ N * dividing n,

(3.5) log[f m (c), c] - ∑ z∈Fix(f m ):(f m ) # (z)<r log[z, c] ≤ #{z ∈ Fix(f m ) : (f n ) # (z) < r} • (2m + 1) log M 1 (f ); for, if [f m (c), c] ≥ 1/M 1 (f ) m+1 , then log[f m (c), c]- ∑ z∈Fix(f m ):(f m ) # (z)<r log[z, c] ≥ -(m+ 1) log M 1 (f ) + 0(≤ 0). On the other hand, for every z ∈ Fix(f m ), we have [f m (c), c] ≤ max{[f m (c), f m (z)], [z, c]} ≤ M 1 (f ) m • [z, c], so that log[f m (c), c] - ∑ z∈Fix(f m ):(f m ) # (z)<r log[z, c] ≤ 0 + #{z ∈ Fix(f m ) : (f n ) # (z) < r} • (2m + 1) log M 1 (f ).
Hence the claim holds in this case.

If (0 <)[f m (c), c] < 1/M 1 (f ) m+1 , then by Theorem 2.3, we have [c, J(f )] can ≥ 1/M 1 (f ) m+1 and c lies in the Berkovich immediate basin of a (super)-attracting fixed point a ∈ P 1 of f m such that [a, c] ≤ [f m (c), c]. In particular, 0 ≤ log[f m (c), c] -log[a, c], so that 0 ≤ log[f m (c), c] - ∑ z∈Fix(f m ):(f m ) # (z)<r log[z, c].
On the other hand, we not only have

[f m (c), c] ≤ max{[f m (c), f m (a)], [a, c]} ≤ M 1 (f ) m • [a, c]
but, noting that for each classical attracting fixed point z of f m other than a, the unique arc joining c and z in P 1 intersects J(f ), also have

inf z∈Fix(f m )\{a}:(f m ) # (z)<r [c, z] ≥ [c, J(f )] can ≥ 1/M 1 (f ) m+1 , so that log[f m (c), c] - ∑ z∈Fix(f m ) (f m ) # (z)<r log[z, c] ≤ #{z ∈ Fix(f m ) : (f n ) # (z) < r} • (m + 1) log M 1 (f ).
Hence the claim also holds in this case. Once (3.3), (3.4), and (3.5) are at our disposal, we have

L n (f, r) - ( 1 - ∑ m∈N:m|n µ(n/m)#{z ∈ Fix(f m ) : (f n ) # (z) < r} d n ) L(f ) ≤ 2d -2 d n • ( ∑ m∈N:m|n m • #{z ∈ Fix(f m ) : (f n ) # (z) < r} • (2m + 1) log M 1 (f ) + ∑ m∈N:m|n (2 + #{z ∈ Fix(f m ) : (f n ) # (z) < r}) • sup P 1 |g f | ) + ∑ m∈N:m|n #{z ∈ Fix(f m ) : (f n ) # (z) < r} n • d n • | log r|.
Now, we recall that, by Benedetto-Ingram-Jones-Levy's non-archimedean counterpart to Fatou's upper bound of the number of (super)attracting cycles [START_REF] Benedetto | Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps[END_REF]Theorem 1.5], for every m ∈ N * and every 0

< r ≤ ϵ d m , #{z ∈ Fix(f m ) : (f m ) # (z) < r} ≤ #Crit(f ) • m ≤ (2d -2) • m. (3.6)
Hence we have the desired inequality under the assumption that Fix(

f n ) ∩ Crit(f ) = ∅.
By (2.2) and Lemma 2.2, the function (f, z) → g f (z) on Rat d (K) × P 1 is continuous, and then by (2.9), the function f → L(f ) on Rat d (K) is also continuous. Moreover, by (3.2), the function f → L n (f, r) on Rat d (K) is continuous. Hence the left hand side in the desired inequality depends continuously on f ∈ Rat d (K), and also by the definition of M (f ), the right hand side depends upper semicontinuously on f ∈ Rat d (K). Since {f ∈ Rat d (K) : Fix(f n ) ∩ Crit(f ) ̸ = ∅} is a proper Zariski closed subset in Rat d (K), the desired inequality still holds without assuming Fix(f n ) ∩ Crit(f ) = ∅.

An argument similar to the above shows a little more effective version of [GOV, Theorem 3.1], which is needed below. Theorem 3.2. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to an archimedean absolute value | • |, and fix an integer d > 1. Then for any f ∈ Rat d (K), any n ∈ N * , and any r ∈ (0, 1], we have

L n (f, r) -L(f ) ≤ 2(2d -2) 2 • ( L(f ) + 16 log M (f ) + sup P 1 |g f | + | log r| ) σ 2 (n) d n .

The key algebraic estimates and the end of the proof of Theorem A

Theorem A follows from Theorem 3.1 and the following.

Lemma 3.3. -Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial absolute value | • |, and fix an integer d > 1. Then for every f ∈ Rat d (K) and every lift F of f , we have

log | Res(f )| = log | Res F | -2d log |F | ≤ 2A 1 (d, K), |L(f )| ≤ -log |d| -2 log | Res F | + 4d log |F | + { A 2 (d, K) when K is archimedean 2 log |F | -log |C F | when K is non-archimedean log M (f ) ≤ -2 log |d| -4 log | Res F | + 8d log |F | + 2A 2 (d, K), sup P 1 |g f | ≤ 2d -1 d(2d -2) ( 2d log |F | -log | Res F | ) + 3d -2 d(d -1) A 1 (d, K),
where the constants A 1 (d, K) and A 2 (d, K) are given by Lemmas 2.2 and 2.5 and

C F ∈ K \ {0} is the leading coefficient of det DF (1, z) ∈ K[z].
Proof. -Fix f ∈ Rat d (K) and a lift F of f . The first estimate follows from Lemma 2.2 and the definition of Res(f ). We now prove the second inequality. By Lemma 2.5, we have

L(f ) := ∫ P 1 log(f # )µ f ≤ -log |d| -2 log | Res F | + 4d log |F | + A 2 (d, K).
When L(f ) ≥ 0 (in particular when K is archimedean), the conclusion follows. When L(f ) < 0, the field K is non-archimedean and, by Proposition 2.7, and by the lower estimate on g F from Lemma 2.2, and the first inequality, we also have Finally, we prove the last estimate. By Lemma 2.2 and (2.5), and using that

L(f ) ≥ -log |d| - ( 1 - 1 d ) (-2 log | Res F | + 4d log |F |) -(2 log |F | -log |C F |) ≥ log |d| -(-2 log | Res F | + 4d log |F |) -(2 log |F | -log |C F |),
-log | Res F | + 2d log |F | + 2A 1 (d, K) ≥ 0, we have g f - 1 d A 1 (d, K) ≤ (2d -1) 2d(d -1) (2d log |F | -log | Res F | + 2A 1 (d, K)) on P 1 , ending the proof.
Proof of Theorem A. -According to Theorem 3.1, it is sufficient to bound

|L(f )| + log M (f ) + sup P 1 |g f | from above. Since K is non-archimedean, we have log M (f ) = -2 log |Res(f )|
and the first and last inequalities from Lemma 3.3 give sup

P 1 |g f | ≤ - 2d -1 d(2d -2) log |Res(f )|.
This ends the proof by Theorem 3.1.

Degeneration of the Lyapunov exponent

Fix an integer d > 1. Recall that the integral domain O(D K )[t -1 ] consists of all meromorphic functions on D K having no poles on D * K .

Proof of Theorem

B. -Let f t (z) ∈ Rat d (O(D K )[t -1 ]
) be a meromorphic family of rational functions (over K) of degree d parametrized by D K , and fix a lift

F t = (F t,1 , F t,2 ) of f t with F t,i ∈ O(D K )[t -1 ][p 1 , p 2 ]. Note that the function t → C Ft belongs to O(D K )[t -1
] and is not identically 0. In particular, there exists δ > 0 such that C Ft ̸ = 0 for all 0 < |t| < δ.

Fix n ∈ N * and 0 < |t| < δ. By Theorem A, we have

|L n (f t , ϵ d n ) -L(f t )| ≤ 8(d -1) 2 ( |L(f t )| - 4d 2 -2d -1 d(2d -2) log | Res(f t )| + | log ϵ d n | ) σ 2 (n) d n = 8(d -1) 2 ( |L(f t )| + 4d 2 -2d -1 d(2d -2) (2d log |F t | -log | Res F t |) + | log ϵ d n | ) σ 2 (n) d n ,
where

|L(f )| ≤ -log |d| -2 log |Res(F t )| + (4d + 2) log |F t | -log |C Ft |.
Note that, since the coefficients of

F t lie in O(D K )[t -1 ], it is clear that log | Res F t | = O(log |t| -1 ), log |F t | = O(log |t| -1 ) and log |C Ft | = O(log |t| -1
) as t → 0. Moreover, for every n ∈ N * , the limit α n := lim t→0 L n (f t , ϵ d n )/(log |t| -1 ) ∈ N also exists by (3.2). Hence there is C ≥ 0 such that for every n ∈ N * , lim sup

t→0 L(f t ) log |t| -1 -C • σ 2 (n) d n ≤ α n ≤ lim inf t→0 L(f t ) log |t| -1 + C • σ 2 (n) d n .
In particular, as n → ∞, we find

lim sup t→0 L(f t ) log |t| -1 ≤ lim inf n→∞ α n ≤ lim sup n→∞ α n ≤ lim inf t→0 L(f t ) log |t| -1 < ∞ since σ 2 (n) = O(n 2 log log n) = o(d n ) as n → ∞, ending the proof.

The key global algebraic estimates

Let k be a global field and fix an integer d > 1. For the sake of completeness, we include a proof of the following.

Lemma 4.1. -For every f ∈ Rat d (k), ∑ v∈M k ′ N v L(f ) v = ∑ c∈Crit(f ) ĥf,k ′ (c) = [k ′ : k]h crit,k (f ),
where k ′ is any algebraic extension of k such that f ∈ k ′ (z).

Proof. -Fix f ∈ Rat d (k) and a lift F of f , let k ′ , k ′′ be any algebraic extensions of k such that f ∈ k ′ (z) and Crit(f ) ⊂ P 1 k ′′ , respectively, and choose points

C F 1 , . . . , C F 2d-2 ∈ k ′′2 \{0} such that det DF factors as det DF (p 0 , p 1 ) = ∏ 2d-2 j=1 ((p 0 , p 1 ) ∧ C F j ) in k ′′ [p 0 , p 1 ]
. By (2.8) and the product formula, we have

∑ v∈M k ′ N v L(f ) v = ∑ v∈M k ′ N v 2d-2 ∑ j=1 (g f,v (π(C F j )) + log ∥C F j ∥) = 1 [k ′′ : k ′ ] 2d-2 ∑ j=1 ∑ v∈M k ′′ N v ( g f,v (π(C F j )) + log ∥C F j ∥
) .

By Lemma 2.2 and the product formula, there is a finite subset E in M k ′′ such that for every v ∈ M k ′′ \ E, we have T F,v ≡ 0 on P 1 , and ∥C F j ∥ v = 1 for every j ∈ {1, . . . , 2d -2}. Hence for every j ∈ {1, . . . , 2d -2}, recalling the definition (2.1) of g F , we have

∑ v∈M k ′′ N v g f,v (π(C F j )) = ∑ v∈M k ′′ N v g F,v (π(C F j )) = ∑ v∈E N v lim n→∞ n-1 ∑ j=0 T F,v (f j (π(C F j )) d j+1 = lim n→∞ ∑ v∈E N v n-1 ∑ j=0 T F,v (f j (π(C F j )) d j+1 = lim n→∞ ∑ v∈M k ′′ N v n-1 ∑ j=0 T F,v (f j (π(C F j )) d j+1 = lim n→∞ 1 d n ∑ v∈M k ′′ N v log ∥F n (C F j )∥ v - ∑ v∈M k ′′ N v log ∥C F j ∥ v , so that ∑ v∈M k ′′ N v ( g f,v (π(C F j )) + log ∥C F j ∥ v ) = ĥf,k ′′ (π(C F j )) = [k ′′ : k ′ ] • ĥf,k ′ (π(C F j )),
which completes the proof.

Recall that the constants A 1 (d, K), A 2 (d, K) ≥ 0 given by Lemmas 2.2 and 2.5 respectively, vanish when K is non-archimedean, or depend only on d but not on K, and can be computed explicitly when K is archimedean. From now on, we denote by

A 1 (d) (resp. A 2 (d)) the constant A 1 (d, K) (resp. A 2 (d, K)) for archimedean K.
Recall that a global field k has no infinite places if k is a function field, and if k is a number field, then

∑ v∈M k :infinite N v = [k : Q]. We thus set C(k) := { [k : Q] if k is a number field, 0 if k is a function field. Lemma 4.2. -For every f ∈ Rat d ( k), we have 1 [k ′ : k] ∑ v∈M k ′ N v |L(f ) v | ≤ (4d + 2) • h d,k (f ) + C(k) ( A 2 (d) + log(2d 3 ) ) 1 [k ′ : k] ∑ v∈M k ′ N v log M (f ) v ≤ 8d • h d,k (f ) + 2C(k)A 2 (d) 1 [k ′ : k] ∑ v∈M k ′ N v sup P 1 (Cv) |g f,v | ≤ 2d -1 d -1 • h d,k (f ) + 3d -2 d(d -1) C(k)A 1 (d),
where k ′ is any algebraic extension of k such that f ∈ k ′ (z).

Proof. -Fix f ∈ Rat d (k) and an algebraic extension

k ′ of k such that f ∈ k ′ (z). Fix a lift F of f ∈ Rat d (k ′ ) and let C F ∈ k ′ \ {0} be the leading coefficient of det DF (1, z) ∈ k ′ [z].
First, note that, whenever v is archimedean, we have

2 log |F | v -log |C F | v ≥ -log(2d 3 ).
Using Lemma 3.3, we have

|L(f ) v | ≤ -log |d| v -2 log | Res F | v + 4d log |F | v + { A 2 (d) when v is infinite 2 log |F | v -log |C F | v when v is finite, = -log |d| v -2 log | Res F | v + (4d + 2) log |F | v -log |C F | v + { -(2 log |F | v -log |C F | v ) + A 2 (d) when v is infinite 0 when v is finite, ≤ -log |d| v -2 log | Res F | v + (4d + 2) log |F | v -log |C F | v + { log(2d 3 ) + A 2 (d) when v is infinite 0 when v is finite.
Taking the sum of the both sides over all v ∈ M k ′ , by the product formula, we obtain ∑

v∈M k ′ N v |L(f ) v | ≤ (4d + 2) ∑ v∈M k ′ N v log |F | v + C(k ′ ) ( A 2 (d) + log(2d 3 ) ) . Since C(k ′ ) = [k ′ : Q] = [k ′ : k][k : Q] = [k ′ : k]C(k)
when k is a number field and since

∑ v∈M k ′ N v log |F | v = [k ′ : k]h d,k (f )
, this gives the first estimate. The other two estimates also hold using similarly Lemma 3.3.

Applications over a function field

In this section, we let X be a smooth irreducible projective curve defined over C and set k = C(X). Fix an integer d > 1. Recall that any place v ∈ M k is represented by a closed point x of X and that, in the present case, N v = 1 for all v ∈ M k . Recall also that for any integers d > 1 and n ≥ 1, we have ϵ d n ,v = 1 for all places v ∈ M k .

The critical and multiplier heights over a function field

Pick f ∈ Rat d (k). Note that, by definition, ĥf (x) ≥ 0 and that ĥf (x) = 0 if and only if deg(f n (x)) = O(1) as n → ∞. Moreover, unless f is isotrivial, for every x ∈ P 1 (k), we have ĥf,k (x) = 0 if and only if the forward orbit of x is finite, i.e. x is preperiodic under iteration of f (see, e.g., [Be]).

Proposition 5.1. -For every f ∈ Rat d (k) and every integer n ≥ 1, we have

1 nd n max 0≤j≤dn deg(σ * j,n (f )) -h crit,k (f ) ≤ 8d(12d 2 -8d -3) σ 2 (n) d n • h d,k (f ).
Proof. -Let us set

E n := 1 nd n max 0≤j≤dn {deg(σ * j,n (f ))} -h crit,k (f ). Recall that h crit,k (f ) = ∑ v∈M k L(f ) v by Lemma 4.1 adn that max 0≤j≤dn deg(σ * j,n (f )) = h P dn ,k ([σ * 0,n (f ) : • • • : σ * dn,n (f )]).
Applying Theorem 3.1 at each v ∈ M k and using the equality (3.2), we get

|E n | = 1 nd n h P dn ,k ([σ * 0,n (f ) : • • • : σ * dn,n (f )]) - ∑ v∈M k L(f ) v = ∑ v∈M k 1 nd n log max 0≤j≤dn {|σ * j,n (f )| v } - ∑ v∈M k L(f ) v = ∑ v∈M k (L n (f, 1) v -L(f ) v ) ≤ 8(d -1) 2 ∑ v∈M k ( |L(f ) v | + log M (f ) v + sup P 1 (Cv) |g f,v | ) σ 2 (n) d n .
By Lemma 4.2 and

d n = d n - ∑ k|n,k<n d k ≥ (1 -d -1 )d n , this completes the proof. Corollary 5.2. -For every f ∈ Rat d (k), lim inf n→∞ 1 n max { deg((f n ) ′ (z)) [k((f n ) ′ (z)) : k] : z ∈ Fix * (f n ) } ≥ h crit,k (f ).
Proof. -For every n ∈ N * , by Proposition 5.1 and the definition of L n,v (f, 1), we have

h crit,k (f ) + o(1) ≤ 1 nd n max 0≤j≤dn {deg(σ * j,n (f ))} = ∑ v∈M k L n (f, 1) v ≤ 1 nd n ∑ z∈Fix * (f n ) deg((f n ) ′ (z)) [k((f n ) ′ (z)) : k] ≤ 1 nd n • d n • max { deg((f n ) ′ (z)) [k((f n ) ′ (z)) : k] : z ∈ Fix * (f n ) } , since Fix * (f n ) contains exactly d n points. Together with the fact that σ 2 (n) = o(d n ) and d n ∼ d n , it is sufficient to make n → ∞ to end the proof. Example 5.3. -Let f (z) := z d + t ∈ Rat d (C(t)), so that h d,C(t) (f ) = 1 and Crit(f ) = {0, ∞}.
Since the multiplicity of ∞ as a critical point of f equals d -1 and f (∞) = ∞, we have deg(f n (∞)) = 0 for every n ∈ N * so that ĥf,C(t) (∞) = 0. On the other hand, for every n ∈ N * , we have deg(

f n (0)) = d n-1 . Hence h crit,C(t) (f ) = (d -1) ĥf,C(t) (∞) + (d -1) ĥf,C(t) (0) = (d -1) lim n→∞ deg(f n (0)) d n = d -1 d .
On the other hand, by Eremenko-Levin [START_REF] Erëmenko | Estimation of the characteristic exponents of a polynomial[END_REF]Theorem 1.6], for every n ≥ 1,

min zt∈Fix * (f n t ) 1 n log |(f n t ) ′ (z t )| = d -1 d log |t| + o(1) as t → ∞,
which directly implies that for every z ∈ Fix * (f n ),

1 n deg((f n ) ′ (z)) [k((f n ) ′ (z)) : k] = d -1 d .
In particular, the inequality in Corollary 5.2 is sharp in that for any integer d > 1, there exists f ∈ Rat d (C(t)) such that (i) h d,C(t) (f ) > 0 and h crit,C(t) (f ) > 0, and (ii) for every integer n ≥ 1 and every z ∈ Fix * (f n ),

1 n deg((f n ) ′ (z)) [k((f n ) ′ (z)) : k] = h crit,C(t) (f ).

Proof of Theorem E

The following is our main application of Theorem A over function fields. The equivalence between the first three points and the final one is new, and we deduce it from the other entries and Proposition 5.1.

Theorem 5.4. -Let X be an irreducible smooth projective curve defined over C and let k = C(X). Fix an integer d > 1, let f ∈ Rat d (k) and S ⊂ M k be the finite set of places of bad reduction of f . If f is not affine, the following assertions are equivalent:

(i) max 1≤j≤dn deg(σ * j,n (f )) = o(nd n ) as n → ∞, (ii) max 1≤j≤dn deg(σ * j,n (f )) = O(nσ 2 (n)) as n → ∞, (iii) lim inf n→∞ 1 n max{[k((f n ) ′ (z)) : k] -1 deg((f n ) ′ (z)) : z ∈ Fix * (f n )} = 0, (iv) deg((f n ) ′ (z)) = 0 for all z ∈ Fix * (f n ) and all n ∈ N * , (v) h crit,k (f ) = 0, (vi) for every c ∈ Crit(f ), deg(f n (c)) = O(1) as n → ∞, (vii) f is J-stable as a holomorphic family parametrized by X \ S, (viii) f is isotrivial.
Proof. -By Proposition 5.1, the assertions (i), (ii) and (v) are equivalent. Moreover, items (v) and (vi) are equivalent by definition of the critical height. Note that (iv) implies (iii) trivially and that (iii) implies (v) by Corollary 5.2. Now, if (viii) holds, there exists a finite branched cover Y → X such that k

′ := C(Y ) is a finite extension of k and M ∈ SL 2 (k ′ ) such that all the specializations (M • f • M -1 ) t ∈ Rat d (C) are independent of t ∈ Y . In other words, f = M -1 • g • M where g ∈ Rat d (k ′ ) has coefficients in C.
In particular, for every n ∈ N * , every z ∈ Fix * (f n ) can be written as M -1 y where y ∈ Fix * (g n ), thus, by the chain rule, (

f n ) ′ (z) = (g n ) ′ (y) ∈ C, whence deg((f n ) ′ (z)) = 0 for all z ∈ Fix * (f n ), which is item (iv).
By e.g. [START_REF] Demarco | Bifurcations, intersections, and heights[END_REF]Theorem 1.4] item (v) is equivalent to the passivity of all critical points on X \ S, i.e. to (vii) by [MSS]. Finally, since we assumed that f is not affine (vii) implies (viii) by Lemma 2.1 and Theorem 2.2 of [Mc].

To conclude this section, we prove Theorem E.

Proof of Theorem E. -Note first that

h P dn ,k ( [σ * 0,n (f ) : • • • : σ * dn,n (f )] ) = max 0≤j≤dn deg ( σ * j,n (f )
) , so that Theorem E follows immediately from Theorem 5.4, once we check that a map f which is affine satisfies assertions (i) -(vii) of the above Theorem. Pick f : P 1 k → P 1 k which is affine. Then there exists an elliptic curve E defined over k, an isogeny ϕ : E → E given by ϕ(P ) = [m](P + P 0 ) for some P 0 ∈ E and some m ∈ Z with m 2 = d and a double branched cover π :

E → P 1 k such that π • ϕ = f • π on E. Then, for all n ∈ N * ,(f n ) ′ (z) = m n for all periodic point z ∈ Fix * (f n ), whence that assertions (i)-(iv) are satisfied.
Finally, for all c ∈ Crit(f ), the orbit of c is finite whence (v)-(vii) hold trivially.

The case of function fields of higher dimensional varieties

Assume that X is a normal irreducible projective variety defined over C of any dimension and let k := C(X) be its function field. The results exposed in the present section can be adapted to this context. Let us explain how following the exposition of [START_REF] Bombieri | Heights in Diophantine geometry[END_REF]Chapter 1].

First, pick an ample line bundle L on X and denote by deg(Z) the degree of a cycle Z with respect to L. For any g ∈ k × , let ord Z (g) be the order of g at Z and let

|g| Z := e -deg(Z)ord Z (g) , g ∈ C(X) × .
Note that for two distinct prime divisors Z and Z ′ , the absolute values | • | Z and | • | Z ′ are non-trivial and not equivalent. In addition, since the degree of a principal divisor is 0, we have

∑ Z deg(Z)ord Z (g) = 0 for all g ∈ k ×
, where the sum ranges over all prime divisors of X, whence the sum is actually finite. The field k equipped with (| • | Z ) Z and and N Z = 1 for all Z is thus a global field. Equivalently, we have the product formula

∏ Z |g| Z = 1, g ∈ k × .
We then may define the naive height function h P N ,k on P N (k) by

h P N ,k (x) := - ∑ Z deg(Z) min 0≤j≤N ord Z (x j ) if x = [x 0 : • • • : x N ] ∈ P N (k). In particular, if x ∈ k × is a rational function on X, then h P 1 ,k (x) = h P 1 ,k ([x : 1]) = - ∑ Z deg(Z) min ( 0, ord Z (x)
) .

It thus can be computed as the degree of the divisor x -1 {∞}. In particular, h P 1 ,k (x) = 0 if and only if x is constant.

Let us come back to our dynamical setting: the adaptation of Theorem 5.4 remains valid in the present context, since we did not use any where that X has dimension 1 in the proof.

In particular, we have the Theorem 5.5. -let X be a normal irreducible projective variety defined over C of any dimension and let k := C(X) be its function field. Pick a rational map f ∈ Rat d (k) of degree d > 1 which is not affine. The following assertions are equivalent:

(i) h P dn ,k ([1, σ * 1,n (f ) : • • • : σ * dn,n (f )]) = o(nd n ) as n → ∞, (ii) h P dn ,k ([1, σ * 1,n (f ) : • • • : σ * dn,n (f )]) = O(nσ 2 (n)) as n → ∞, (iii) h crit,k (f ) = 0, (iv) f is isotrivial.

Multiplier maps and bifurcation currents

Symmetric powers of P 1

Pick an integer N ≥ 2. The symmetric group S N acts on (P 1 ) N coordinate-wise: for each σ ∈ S N , and each (x 1 , . . . , x N ), we have σ • (x 1 , . . . , x N ) = (x σ(1) , . . . , x σ(N ) ). We denote by S N P 1 the N -th symmetric power of the projective line, i.e. the quotient of (P 1 ) N by the action of the symmetric group S N by permutation. This quotient is a projective variety.

Denote by P N : (P 1 ) N → S N P 1 the quotient map and by π i : (P 1 ) N → P 1 the projection onto the i-th coordinate.

Let H N be the divisor on

S N P 1 such that P * N H N = ∑ N i=1 π * i O P 1 (1)
. This divisor is ample and we let h S N P 1 be the associated height function on (S N P 1 )( Q) (which is welldefined up to an additive constant), i.e. we let

h S N P 1 ,Q (P N (x 1 , . . . , x N )) := N ∑ j=1 h P 1 ,Q (x j ),
for all (x 1 , . . . , x N ) ∈ (P 1 ( Q)) N . When working over the field C, we also let ω N be the positive smooth (1, 1)-form representing c 1 (H N ) which satisfies

P * N ω N = N ∑ j=1 π * j (λ S 1 ),
where λ S 1 is the curvature form of the standard metrization of O P 1 (1) associated with the height h P 1 ,Q , i.e. the Lebesgue measure on the unit circle normalized by λ S 1 (C) = 1. We include a proof of the following for the sake of completeness.

Lemma 6.1. -The variety S N P 1 is isomorphic to P N and H N ≃ O P N (1). In particular,

h S N P 1 ,Q = h P N ,Q + O(1) on P N ( Q).
Proof. -For every 0 ≤ j ≤ N , set

η j ([z 1 : t 1 ], . . . , [z N : t N ]) := ∑ I N ∏ ℓ=1 z I ℓ ℓ • t 1-I ℓ ℓ on (P 1 ) N ,
where the sum is taken over all N -tuples I ∈ {0, 1} N with

I 1 + • • • + I N = N -j.
The map PN = [η 0 , . . . , η N ] : (P 1 ) N → P N has the topological degree N ! and is surjective, and each fiber of PN is invariant under the action of S N . Hence PN descends as an isomorphism S N P 1 → P N .

To see that H N ≃ O P N (1), it is sufficient to prove that (H N N ) = 1 since the ample cone of P N is an open half line. By the change of variable formula and Fubini, we compute

(H N N ) = ∫ P N (C) ω N N = ∫ (P 1 (C)) N 1 N ! (P * N ω N ) N = 1 N ! ∫ (P 1 (C)) N   N ∑ j=1 π * j λ S 1   N = ∫ (P 1 (C)) N N ∧ j=1 π * j λ S 1 = N ∏ j=1 ∫ P 1 (C) λ S 1 = 1,
as required. Finally, the comparison of height functions follows from fonctoriality properties of height functions [HS].

Remark 6.2. -Fix a field K. When x 1 , . . . , x N ∈ A 1 (K), it is easy to see that

P N (x 1 , . . . , x N ) = [σ N : • • • : σ 1 : 1] ∈ A N (K),
where σ j is the degree j elementary symmetric polynomial in (x 1 , . . . , x N ) (recall that A N (K) denotes the affine space).

Currents, DSH functions and the bifurcation currents

We refer to [DS, Appendix A] for more details on currents and DSH functions. Pick any affine variety X defined over C and choose any embedding ι :

X → C N . Let D := O P N (1)| X
and ω D be the restriction of the ambient normalized Fubini-Study form to X. For any positive closed current T of bidimension (k, k) defined on X and any Borel set A ⊂ X, we denote by ∥T ∥ A,D the number

∥T ∥ A,D := ∫ A T ∧ ω k D .
This is the mass of the current T in A relatively to ω D . Let Ψ be a (ℓ, ℓ)-form in X. We say that Ψ is DSH if we can write dd c Ψ = T + -T - where T ± are positive closed currents of finite mass in X. We also set

∥Ψ∥ * DSH,D := inf ( ∥T + ∥ X,D + ∥T -∥ X,D ) ,
where the infimum is taken over all closed positive currents T ± such that For all R > 1, we define the auxiliary function Ψ R :

dd c Ψ = T + -T - (note that ∥T + ∥ X,D = ∥T -∥ X,
A N (C) → R by letting Ψ R (Z) := 1 log R min {log max{∥Z∥, R} -2 log R, 0} for all Z ∈ A N (C). For all 1 ≤ j ≤ k := dim X, define the (k -j, k -j)-DSH form on X Φ R,X,j := Ψ R | X • (ω D ) k-j . Lemma 6.3. -The (k -j, k -j)-form Φ R,X,j is DSH and continuous on X with support in B(0, R 2 ) ∩ X and if dd c Φ R,X,j = T + -T -, we have ∥T ± ∥ X,D ≤ deg D (X) log R .
Proof. -First, note that the function Ψ R is DSH and continuous on X and satisfies

   Ψ R ([f ]) = -1 if ι([f ]) ∈ B(0, R), -1 ≤ Ψ R ([f ]) ≤ 0 if ι([f ]) ∈ B(0, R 2 ) \ B(0, R), 0 if ι([f ]) / ∈ B(0, R 2 ).
As ω D is positive and closed form on X, this implies Φ R,X,j is DSH continuous with support in B(0, R 2 ) ∩ X. Moreover, if dd c Φ R,X,j = T + -T -, then

T ± = (S ± | X ) ∧ ω k-j D
, where S ± are such that dd c Ψ R = S ± and, using Bézout's theorem, we have

∥T ± ∥ X,D = ∥(S ± | X ) ∧ ω k-j D ∥ X,D ≤ ∥S ± ∥ • deg D (X),
where the mass

∥S ± ∥ = ∫ S ± ∧ ω N -1
FS,P N is computed with respect to the Fubini Study form of P N . By construction of Ψ R , we immediately get The measure µ bif is a finite positive measure on M d of strictly positive total mass (see [BB]). We also mention some other remarkable facts below, which will not be used explicitly in this article.

∥S ± ∥ = ∫ A N (C) S ± ∧ ω N -1 D = 1 R
By DeMarco [START_REF] Demarco | Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity[END_REF], the support of T bif coincides with the bifurcation locus of the moduli space M d (C). If p > 1, the current T p bif detects, in a certain sense, stronger bifurcations than T bif . Indeed, its topological support admits several dynamical characterizations similar to that of the bifurcation locus: for example, it is the closure of parameters admitting p distinct neutral cycles or p critical points preperiodic to repelling cycles (see [START_REF] Bassanelli | Bifurcation currents in holomorphic dynamics on P k[END_REF][START_REF] Dujardin | The supports of higher bifurcation currents[END_REF][START_REF] Gauthier | Higher bifurcation currents, neutral cycles, and the Mandelbrot set[END_REF]). For more precise (even quantitative) equidistribution results, see [GOV].

Complex analytic properties of the multiplier maps

We now pick an integer n ≥ 1. Definition 6.5. -The period n multiplier map Λ n : Rat d → P d n +1 is defined by

Λ n ([f ]) := P d n +1 ( (f n ) ′ (z 1 ), . . . , (f n ) ′ (z d n +1 ) ) ,
for any representative f of [f ], where z 1 , . . . , z d n +1 is any ordering of Fix(f n ).

Similarly, the period n formally multiplier map Λn : Rat d → P dn is defined by

Λn ([f ]) := P dn ( (f n ) ′ (z 1 ), . . . , (f n ) ′ (z dn ) ) ,
for any representative f of [f ], where z 1 , . . . , z dn is any ordering of Fix * (f n ).

As mentioned before, the maps Λ n and Λn descend to maps on M d that are defined over Q and we still denote them by Λ n and Λn respectively. Remark 6.2 implies that for any field K, any n ≥ 1 and any

[f ] ∈ M d (K), we have Λ n ([f ]) = [σ d n +1,n ([f ]) : • • • : σ 1,n ([f ]) : 1] ∈ A d n +1 (K) and Λn ([f ]) = [σ * dn,n ([f ]) : • • • : σ * 1,n ([f ]) : 1] ∈ A dn (K).
In particular, the maps Λ n and Λn actually define morphisms from M d to the corresponding affine space.

Let us begin with a few lemmas using classical pluripotential theory. Similarly to L, for every r > 0, the function L n (•, r) descends to a continuous and psh function on M d (C). Lemma 6.6. -For any integer n ≥ 1, we have

1 nd n Λ * n (ω dn ) = dd c L n (•, 1
) and

1 n(d n + 1) Λ * n (ω d n +1 ) = ∑ k|n d k d n + 1 dd c L k (•, 1).
In particular, for any 1 ≤ j ≤ 2d -2 and any n ≥ 1, we have

(Λ * n ω d n +1 ) j ≥ ( Λ * n ω dn ) j .
Proof. -The first equality follows from the definition of the form ω dn . To get the second equality, it is now sufficient to remark that Λ * n (ω

d n +1 ) = ∑ k|n n k Λ * k (ω d k )
and to recall that, by definition of d n , we have

d n + 1 = ∑ k|n d k .
Using the two equalities, the continuity and the plurisubharmonicity of the L k 's gives

(Λ * n (ω d n +1 )) j =   ∑ k|n nd k dd c L k   j ≥ (nd n dd c L n ) j = ( Λ * n ω dn ) j , see, e.g. [K, Corollary 3.4.9]. Lemma 6.7. -Let d > 1. Pick integers n ≥ 1 and 1 ≤ k ≤ 2d -2. Let X ⊂ M d (C) be an irreducible subvariety of dimension k. Let X n ⊂ X (resp. Xn ⊂ X) be the algebraic subvariety of all [f ] ∈ X such that dim ( X ∩ Λ -1 n {Λ n ([f ])} ) > 0 (resp. dim ( X ∩ Λ-1 n { Λn ([f ])} ) > 0). Then, we have ∫ Xn (Λ * n ω d n +1 ) k = ∫ Xn ( Λ * n ω dn ) k = 0.
Proof. -Fix an integer n ≥ 1 and let Ẽn ⊂ X be the set of [f ] ∈ X such that there exists an analytic map φ : D → X with φ(0) = [f ] and such that L n (φ(•), 1) :

D → R is harmonic. It is clear that, by definition of Ẽn , since nd n dd c L n (•, 1) = Λ * n ω dn , we have {[f ] ∈ X : dim X ∩ Λ-1 n { Λn ([f ])} > 0} ⊂ Ẽn .
By Lemma 6.6, the positive measure μn = ( Λ * n ω dn ) k | X has continuous potential on X. In particular, we have μn ( Ẽn ) = 0, by e.g. [START_REF] Sibony | Dynamique des applications rationnelles de P k . In Dynamique et géométrie complexes[END_REF]Corollary A.10.3] and the conclusion follows for Xn . The same proof also works for X n .

Lemma 6.8. -Let X ⊂ M d (C) be any complex algebraic subvareity of dimension k ≥ 1. Assume X ∩ L d (C) is a strict subvareity of X. Then ∫ X T k bif > 0. Proof. -Let us set Per n (w) := {[f ] ∈ M d (C) ; p d,n ([f ], w) = 0}.
Up to taking a finite branched cover of X, we may assume that (f λ ) λ∈X is a family of rational maps endowed with 2d -2 marked critical points, i.e. that there exist morphisms c 1 , . . . , c 2d-2 :

X → P 1 such that Crit(f λ ) = {c 1 (λ), . . . , c 2d-2 (λ)} counted with multiplic- ity. Since dim X > 0 and X ∩ L d (C) is a strict subvariety of X, the bifurcation locus of the quasi-projective vareity Y := X \ X ∩ L d (C) is non-empty by [Mc, Theorem 2.2].
By [MSS], this implies that there exists m 1 ≥ 1 and

θ 1 ∈ R \ Q such that Per m 1 (e 2iπθ 1 ) is a non-empty proper subvariety of Y . We repeat the argument to find m 2 > m 1 and θ 2 ∈ R \ Q such that Per m 2 (e 2iπθ 2 ) ∩ Per m 1 (e 2iπθ 1 ) ̸ = ∅ has codimension 2 in X. Applying this argument inductively gives m 1 < • • • < m k and θ 1 , . . . , θ k ∈ R\Q such that Per m 1 (e 2iπθ 1 )∩ • • • ∩ Per m k (e 2iπθ k ) is a non-empty finite subset of Y .
We define a quasi-projective curve in Y by C := ∩ k-1 j=1 Per m j (e 2iπθ j ). As there exists a parameter λ0 ∈ C for which f λ0 has a neutral cycle, non-persistent in C, the bifurcation locus Bif(C) = supp(dd c L| C ) of C is non-empty. Pick λ 0 ∈ Bif(C) and let U be a small neighborhood of that λ 0 in X. By Montel Theorem, there exists

λ 1 ∈ U ∩ C, 1 ≤ i 1 ≤ 2d -2 and q 1 , r 1 ≥ 1 such that f q 1 λ 1 (c i 1 (λ 1 )) = f q 1 +r 1 λ 1 (c i 1 (λ 1 )) and f q 1 λ 1 (c i 1 (λ 1
)) is a repelling periodic point of f λ 1 of exact period r 1 , and such that this relation is not persistent through C (see e.g. [START_REF] Dujardin | Distribution of rational maps with a preperiodic critical point[END_REF]Lemma 2.3]). Let

C 1 := {λ ∈ U ; f q 1 λ (c i 1 (λ)) = f q 1 +r 1 λ (c i 1 (λ))} ∩ k-2 ∩ j=1 Per m j (e 2iπθ j ) ⊂ U.
By the same argument as above, we find

λ 2 ∈ U ∩ C 1 very close to λ 1 , 1 ≤ i 2 ≤ 2d - 2 distinct from i 1 , q 2 ≥ 1 and r 2 > r 1 such that f q 2 λ 2 (c i 2 (λ 2 )) = f q 2 +r 2 λ 2 (c i 2 (λ 2 )) and f q 2 λ 2 (c i 2 (λ 2 )
) is a repelling periodic point of f λ 2 of exact period r 2 , and such that this relation is not persistent through C 1 . By a finite induction, we find a parameter λ k-1 ∈ Y , positive integers q 1 , . . . , q k-1 and r 1 < r 2 < . . . < r k-1 and pairwise distinct indices i 1 , . . . , i k-1 such that for all 1 ≤ j ≤ k -1:

λ k-1 ∈ Y j := {λ ∈ Y ; f q j λ (c i j (λ)) = f q j +r j λ (c i j (λ))}, f q j λ k-1 (c i j (λ k-1
)) is a repelling periodic point for f λ k-1 of exact periods r j and the intersection of the Y j 's is proper at λ k-1 . Moreover, the intersection

Y 1 ∩ • • • ∩ Y k-1 ∩ Per m k (e 2iπθ k )
is proper, hence we may proceed as above to find

λ k ∈ Y 1 ∩ • • • ∩ Y k-1 , positive integers q k and r k and i k / ∈ {i 1 , . . . , i k-1 } such that λ k ∈ Y k := {λ ∈ Y ; f q k λ (c i k (λ)) = f q k +r k λ (c i k (λ))}
and, for all 1

≤ j ≤ k -1, f q j λ k (c i j (λ k )
) is a repelling periodic point for f λ k of exact periods r j and the intersection of the Y j 's is proper at λ k . By [G1, Theorem 6.2], we have

λ k ∈ supp(T k bif ∧ [X]
) and the proof is complete.

Multiplier maps and the mass of bifurcation currents

From now on, we fix an embedding i : M d → A N into some affine space and we let M d be the Zariski closure of i(M d ) in P N . Set

D := O P N (1)| M d and let ω M d ,D be the form ω FS,P N | M d (C)
, where ω FS,P N is the normalized Fubini-Study form of P N (C). Recall that D is an ample line bundle on M d and that we then have

(D 2d-2 ) = ∫ M d ω 2d-2 M d ,D = deg D (M d ).
The multiplier map Λ n and exact multiplier map Λn define rational mappings M d P d n +1 M d P dn which are defined over Q that we still denote by Λ n and Λn respectively. We prove here complex analytic properties of the multiplier maps that we will rely on in the sequel. The first ingredient we use follows from [GOV] and can be summarized as follows.

Proposition 6.9. -There exists a constant C > 0, depending only on d and deg D (M d ), such that for any irreducible subvariety X ⊂ M d (C) of dimension k ≥ 1 and any integers 1 ≤ j ≤ k and n ≥ 1, we have

1 (nd n ) j ∥( Λ * n ω dn ) j ∥ X,D -∥T j bif ∥ X,D ≤ ( ∥T bif ∥ X,D + C deg D (X) σ 2 (n) d n ) j -∥T bif ∥ j X,D ,
and

1 n j (d n + 1) j ∥(Λ * n ω d n +1 ) j ∥ X,D -∥T j bif ∥ X,D ≤ ( ∥T bif ∥ X,D + C deg D (X) ∑ ℓ|n σ 2 (ℓ) d n ) j -∥T bif ∥ j X,D .
For any h ∈ Rat d (C), we let ũ(h) := L(h) + 32 log

( sup P 1 (C) h # ) + sup P 1 (C) |g h | ≥ 0,
u([f ]) ≤ C 1 log + ∥[f ]|∥ A N (C) + C 2 , (6.1) for all [f ] ∈ M d (C).
The proof of this Claim is an adaption of the argument in the proof of [START_REF] Gauthier | Hyperbolic components of rational maps: Quantitative equidistribution and counting[END_REF]Lemma 6.5] and we postpone it at the end of the paragraph.

Proof of Proposition 6.9. -We follow closely the proof of [GOV, Theorem A and Theorem C]. We treat both inequalities at the same time, since it follows the same strategy. Recall the definition of the auxiliary function Ψ R on A N (C) for any R > 0 in Subsection 6.2. Fix X, hence k, and 1 ≤ j ≤ k as required.

As before, for all R > 0, we define Ψ R :

A N → R by letting Ψ R (Z) := 1 log R min {log max{∥Z∥, R} -2 log R, 0}
for all Z ∈ A N (C). Fix X, k ≥ 1, and 1 ≤ j ≤ k as required. Pick an integer n and let Sn,j := 1 (nd n ) j ( Λ * n ω dn ) j -T j bif and Φ R,X,j :=

( Ψ R • (ω M d ,L ) k-j ) X , R > 0.
Assume first j = 1. By (6.1) and by Theorem 3.2 and Lemmas 6.3 and 6.6, we get

⟨ Sn,1 , Φ R,X,1 ⟩ ≤ σ 2 (n) d n (C 1 log R + C 2 ) ∥T + ∥ X,D ≤ 2 σ 2 (n) d n deg L (X) ( C 1 + C 2 log R ) ,
where T ± are closed and positive currents such that dd

c Φ R,X,1 = T + -T -and C 1 , C 2 > 0 depend only on d and deg D (M d ) but not on R. Moreover, when R → +∞, ⟨ Sn,1 , Φ R,X,1 ⟩ -→ 1 nd n ∥ Λ * n (ω dn )∥ X,D -∥T bif ∥ X,D .
so that

1 nd n ∥ Λ * n (ω dn )∥ X,D -∥T bif ∥ X,D ≤ C σ 2 (n) d n deg D (X), (6.2)
where C > 0 is independent of n. To conclude in that case, we just need to recall that

d n ≥ (1 -d -1 )d n .
Assume now j ≥ 2. Lemma 6.6 and an easy computation give Sn,j = (dd c L n (•, 1) -

dd c L) ∧ j-1 ∑ r=0 1 (nd n ) n ( Λ * n (ω dn )) r ∧ T j-r-1 bif .
The same strategy as in the case j = 1 implies that, for a given R > 0, we have

⟨ Sn,j , Ψ R,X,j ⟩ ≤ σ 2 (n) d n (C 1 log R + C 2 ) j-1 ∑ r=0 ∫ X (T + + T -) ∧ 1 (nd n ) r ( Λ * n ω dn ) r ∧ T j-r-1
bif Applying Bézout's Theorem and Lemma 6.3, we get

⟨ Sn,j , Ψ R,X,j ⟩ ≤ σ 2 (n) d n ( C 1 + C 2 log R ) 2 j-1 ∑ r=0 deg D (X) 1 (nd n ) r ∥ Λ * n ω dn ∥ r X,D ∥T bif ∥ j-r-1 X,D .
Making R → +∞ , we find that

1 (nd n ) j ∥( Λ * n ω dn ) j ∥ X,D -∥T j bif ∥ X,D (6.3) is bounded above by C 4 σ 2 (n) d n deg D (X) j-1 ∑ r=0 1 (nd n ) r ∥ Λ * n ω dn ∥ r X,D ∥T bif ∥ j-r-1 X,D ,
where C 4 = max{2C 1 , C} is again a universal constant. Finally, (6.2) gives

1 (nd n ) r ∥ Λ * n ω dn ∥ X,D ≤ ∥T bif ∥ X,L + C 4 deg D (X) σ 2 (n) d n .
In particular, the above upper bound on (6.3) can be rewritten as

( ∥T bif ∥ X,D + C 4 deg D (X) σ 2 (n) d n ) j -∥T bif ∥ j X,D ,
and the proof is complete in that case.

To end the proof, observe that, if we let

S n,j := 1 n j (d n + 1) j (Λ * n ω d n +1 ) j -T j bif , since we have Λ * n ω d n +1 = ∑ ℓ|n n ℓ Λ * ℓ (ω d ℓ ) and d n + 1 = ∑ ℓ|n d ℓ , we have 1 n(d n + 1) Λ * n ω d n +1 = ∑ ℓ|n d ℓ n(d n + 1) 1 d ℓ n ℓ Λ * ℓ (ω d ℓ )
and an easy computation gives

S n,j = ∑ ℓ|n d ℓ d n + 1 (dd c L ℓ (•, 1) -dd c L) ∧ j-1 ∑ r=0 1 n r (d n + 1) r (Λ * n ω d n +1 ) r ∧ T j-r-1 bif .
We then apply the same strategy as in the previous case to conclude.

Proof of the Claim. -This follows from the same strategy of proof as that of Lemma 6.5 of [GOV]. Any f ∈ Rat d (C) is conjugated to a (non-zero) finite number of rational maps g admitting a lift G of the form

G(z, w) =   Z d + d-1 ∑ j=1 a j Z j W d-j + W d , d-1 ∑ j=1 b j Z j W d-j   . with (a 1 , . . . , a d-1 , b d-1 , . . . , b 1 ) ∈ C 2d-2 . Let Z := {Res(G) = 0} ⊂ P 2d-2 (C) and let V := P 2d-2 (C) \ Z.
This is a Zariski open set and, applying the same proof as in [START_REF] Gauthier | Hyperbolic components of rational maps: Quantitative equidistribution and counting[END_REF]Lemma 6.5] in the above family parametrized by The map j : g ∈ V → [g] ∈ M d (C) extends as a meromorphic map P 2d-2 (C) M d (C) and is holomorphic on V . In particular,there exists C, C ′ , α > 0 such that

V gives C 1 , C 2 , C 3 > 0 such that ũ(g) ≤ C 1 |log |Res(G)|| + C 2 log + ∥g∥ + C 3 for all g ∈ V ,
d P N (j(g), ∂M d ) ≤ Cd P 2d-2 (g, Z) α + C ′
for all g ∈ V . This ends the proof by surjectivity of j on V .

The multiplier map is generically finite for n large enough

We now come to our main result in this section. We want here to quantify the fact that multipliers determine the conjugacy class outside the flexible Lattès locus, up to finitely many choices.

Namely, we prove the following. Proof. -Let us begin with proving item (i). We proceed by contradiction and assume that for arbitrary large n ≥ 1, the map Λn (resp. Λ n ) is not finite-to-1 on any Zariski open subset of M d (C).

In particular, we have ( Λ * n ω dn ) 2d-2 = 0 for all n ≥ 1 by Lemma 6.7. Applying Proposition 6.9 (to X = M d and k = 2d -2) , we thus get

∫ M d µ bif ≤ ( ∥T bif ∥ M d ,D + C deg D (M d ) σ 2 (n) d n ) 2d-2 -∥T bif ∥ 2d-2 M d ,D ,
for arbitrary large n ≥ 1 where C is a constant that depends only on d and deg D (M d ). This is a contradiction, since the bifurcation measure has positive mass and the right hand side of the above inequality is O(σ 2 (n)d -n ). Finally, if ( Λ * n ω dn ) 2d-2 is non-zero, by Lemma 6.6, we have (Λ * n ω d n +1 ) 2d-2 ≥ ( Λ * n ω dn ) 2d-2 > 0 and the conclusion follows again using Lemma 6.7.

We now come to the proof of item (ii). By (i), for any n ≥ n 0 , there exists a nonempty Zariski open set (C) and that for all integers n, k ≥ 1, we have

U n = M d \ Z n of M d (C) such that Λ n is finite-to-one on U n . Let Z n := M d (C) \ U n , it is clear that Z n is a proper subvariety of M d
L d ⊂ Z n and Z kn ⊂ Z n .
Fix now n ≥ n 0 . Let Y 1 , . . . , Y l be the irreducible components of Z n which are distinct from L d . Pick 1 ≤ j ≤ l and let k := dim Y j ≥ 1. As above, by Lemma 6.7, the map Λ qn is generically finite to one on Y j if and only if the measure (Λ * qn ω d qn +1 ) k is non-zero. Assume that for arbitrary large q ≥ 1, the map Λ qn is not generically finite on Y j . Lemma 6.9 gives

0 ≤ ∫ Y j T k bif ≤ C ′ σ 2 (qn) d qn
for arbitrary large q ≥ 1 and some constant C ′ > 0 independent of q, whence T k bif = 0 on the variety Y j . Since Y j ∩ L d is a strict subvareity of Y j , Lemma 6.8 implies T k bif > 0 on Y j . This is a contradiction.

The multiplier height

Recall that we have fixed an embedding ι : M d → A N into some affine space, that M d is the closure of M d in P N and that D = O(1)| M d is an ample line bundle. Recall the definition of the multiplier map Λ n and the constants C 1 (d, D) and C 2 (d, D) from the introduction and that these constants depend only on d and on the embedding ι we chose.

In this section we prove the following.

Theorem 7.1. -There exists an integer n 1 ≥ 1 depending only on d such that for all m ≥ n 1 , there exists a multiple n of m such that the following holds:

2C 1 (d, D)h M d ,D -A n ≤ 1 n(d n + 1) h P d n +1 ,Q • Λ n ≤ 2C 2 (d, D)h M d ,D + A n on (M d \ L d )( Q)
for some constant A n ≥ 0 depending only on d and n.

Proof.

-Choose an integer n 1 ≥ n 0 (d) where n 0 is given by Proposition 6.10, and such that the following holds for all

m ≥ n 1    C deg D (M d ) ∑ ℓ|m σ 2 (ℓ) d m < 1 2 ∥T bif ∥ M d ,D , ( ∥T bif ∥ M d ,D + C deg D (M d ) ∑ ℓ|m σ 2 (ℓ) d m ) 2d-2 < ∥T bif ∥ 2d-2 M d ,D + 1 2 ∥µ bif ∥ M d .
Choose now m ≥ n 1 and let n be the multiple of m given by the second point of Proposition 6.10 so that the map Λ n is finite to one outside of the flexible Lattès locus L d .

Recall that H d n +1 ≃ O P d n +1 (1) and that h S d n +1 P 1 is an ample height associated with

H d n +1 and that h S d n +1 P 1 = h P d n +1 + O(1). Let E n be the restriction of H d n +1 to the Zariski closure X n of Λ n (M d ) in P d n +1 . By the choice of n, dim X n = 2d
(2d -2)N 2 (2 deg D (M d )) 1/(2d-2) ≤ N 1 ∥µ bif ∥ 1/(2d-2) M d and let M 1 , M 2 ≥ 1 be any integers satisfying M 1 deg D (M d ) ≥ 3(d -1)M 2 ∥T bif ∥ M d ,D .
Then the following divisors are big:

n(d n + 1)M 1 D -M 2 E n and N 1 E n -n(d n + 1)N 2 D.
Let us first finish the proof of Theorem 7.1. Take M 1 and M 2 satisfying the hypothesis of the above lemma. We follow closely the strategy of [START_REF] Joseph | Height estimates for equidimensional dominant rational maps[END_REF]§1] [START_REF] Hindry | Diophantine geometry[END_REF]Theorem B.3.2(e)], this implies

: since M 1 n(d n +1) D-M 2 E n is big, there exists an integer k ≥ 1 such that kn(d n +1)M 1 D-kM 2 E n is effective. According to
h M d ,kn(d n +1)M 1 D-kM 2 En (x) ≥ O(1) for all x ∈ ψ -1 M d ( Q), since the base locus of kn(d n + 1)M 1 D -kM 2 E n is (at worst) contained in M d \ ψ -1 (M d ).
Furthermore, functorial properties of height functions [START_REF] Hindry | Diophantine geometry[END_REF]] then imply

h M d ,kn(d n +1)M 1 D-kM 2 En (x) = h M d ,kn(d n +1)M 1 D (x) -h M d ,kM 2 En (x) + O(1) = kn(d n + 1)M 1 h M d , D (x) -kM 2 h M d , En ([f ]) + O(1) = kn(d n + 1)M 1 h M d , D (x) -kM 2 h S d n +1 P 1 • Λ n (x) + O(1) for all x ∈ ψ -1 (M d \ L d )( Q), since Λ n is finite exactly on ψ -1 (M d \ L d ). All the above summarizes as M 1 h M d , D (x) ≥ 1 n(d n + 1) M 2 h S d n +1 P 1 • Λ n (ψ(x)) + O(1) for all x ∈ ψ -1 (M d \ L d )( Q).
Since ψ is a surjective morphism, Λ n is well-defined and has finite fibers on (

M d \ L d )( Q) and since Λ n = Λ • ψ on ψ -1 ((M d \ L d )( Q)), the definition of D implies M 1 h M d ,D ([f ]) ≥ 1 n(d n + 1) M 2 h S d n +1 P 1 • Λ n ([f ]) + O(1) for all [f ] ∈ (M d \ L d )( Q).
In particular, we choose M 1 and M 2 large enough so that

3(d -1)∥T bif ∥ M d ,D deg D (M d ) ≤ M 1 M 2 ≤ 4(d -1)∥T bif ∥ M d ,D deg D (M d )
and the fact that h S d n +1 P 1 = h P d n +1 + O(1) gives the wanted inequality.

Applying the same strategy to the big divisor

N 1 E n -n(d n + 1)N 2 L gives N 1 h S d n +1 P 1 • Λ n ([f ]) ≥ n(d n + 1)N 2 h M d ,D ([f ]) + O(1), for all [f ] ∈ (M d \ L d )( Q).
As above, we choos N 1 and N 2 so that 2(d -1)

( 2 deg D (M d ) ∥µ bif ∥ M d ) 1/(2d-2) ≤ N 1 N 2 ≤ 4(d -1) ( deg D (M d ) ∥µ bif ∥ M d ) 1/(2d-2)
and the conclusion follows.

It now remains to prove Lemma 7.2. For that we rely on Siu's Theorem which states that if D and E are nef divisors on a projective variety X of dimension k, and if

(D k ) > k(D k-1 • E), (7.1)
then the divisor D -E is big (see, e.g. [START_REF] Lazarsfeld | Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Theorem 2.2.15]).

We now treat the other case. Recall that, by the projection formula, we have

( D 2d-2 ) = deg D (M d ).
Moreover, the current Λ * n ω d n +1 has continuous potentials and the form ψ

* ω M d ,D is smooth, so the measure ( Λ * n ω d n +1 ) ∧ (ψ * ω 2d-3 M d ,D
) does not give mass to the divisor at infinity M d \ ψ -1 (M d ). By the change of variable formula, we get

( D 2d-3 • E n ) = ∫ ψ -1 (M d ) Λ * n (ω d n +1 ) ∧ (ψ * ω M d ,D ) 2d-3 = ∥Λ * n ω d n +1 ∥ M d ,D ,
and, as above, Proposition 6.9 and the choice of n give

( D 2d-3 • E n ) < 3 2 n(d n + 1)∥T bif ∥ M d ,D . Let M 1 , M 2 ≥ 1 be any integers such that M 1 deg D (M d ) ≥ 3(d -1)M 2 ∥T bif ∥ M d ,D and let us set D 1 := M n(d n + 1) D and E ′ n := M 2 E n .
As a consequence of the above, we find (2d -2)

( ( D 1 ) 2d-3 • E ′ n ) < (2d -2)M 2d-3 1 n 2d-2 (d n + 1) 2d-2 M 2 3 2 ∥T bif ∥ M d ,D < 3(d -1)M 2d-3 1 M 2 n 2d-2 (d n + 1) 2d-2 ∥T bif ∥ M d ,D < M 2d-2 1 n 2d-2 (d n + 1) 2d-2 deg D (M d ) = ( D 2d-2 1 ).
As above, Siu's Theorem implies that D 1 -E ′ n is big.

The critical height and applications

To simplify the notations and since there is no possible ambiguity, we omit the subscript Q for height functions in the whole section.

Comparing the multiplier height with the critical height

Our main goal here is to prove the equivalent of Proposition 5.1 over number fields. Since there are both finite places having positive residue characteristic and infinite places, we get a less precise control on the error term.

Proposition 8.1. -There exists a constant C n , depending only on d and n such that for any f ∈ Rat d ( Q) and any n ≥ 1, we have

1 nd n h S dn P 1 • Λn (f ) -h crit (f ) ≤ ( 1056d 2 -1024d -24 ) (d -1)h d (f ) • σ 2 (n) d n + C n . Moreover, C n can be computed explicitly. Proof. -Pick n ≥ 1 and f ∈ Rat d ( Q). Let K n be a finite extension of Q such that f ∈ Rat d (K n ) and Fix * (f n ) ∪ Crit(f ) ⊂ P 1 (K n ). By Lemma 4.1, we have h crit (f ) = ∑ c∈Crit(f ) ĥf (c) = 1 [K n : Q] ∑ v∈M Kn N v L v (f ).
As above, we find

E n := 1 nd n ∑ z∈Fix * (f n ) h P 1 ((f n ) ′ (z)) -h crit (f ) = 1 [K n : Q] ∑ v∈M Kn N v (L n (f, 1) v -L v (f )) . Setting δ n (f ) := 1 [K n : Q] ∑ v∈M Kn N v nd n ∑ z∈Fix * (f n ) log max{|(f n ) ′ (z)| v , 1} max{|(f n ) ′ (z)| v , ε d n ,v } ,
we find

E n = 1 [K n : Q] ∑ v∈M Kn N v (L n (f, ε d n ,v ) v -L v (f )) + δ n (f ). We write v ≤ d n if v ∈ M Kn is non-archimedean and if its residue characteristic p satisfies p ≤ d n . Recall that ε d n ,v = min{|ℓ| d n v : 1 ≤ ℓ ≤ d n } satisfies ε d n ,v < 1 if and only if v ≤ d n and that ε d n ,v > d -nd n in any case.
Since for all v ∈ M Kn and all z ∈ Fix * (f n ), we have

log max{|(f n ) ′ (z)| v , 1} -log max{|(f n ) ′ (z)| v , ε d n ,v } ≤ log ε -1 d n ,v
and since Card(Fix * (f n )) = d n and since for all prime p, [K n

: Q] = ∑ v|p N v , we find |δ n (f )| ≤ 1 [K n : Q] ∑ v≤d n N v nd n ∑ z∈Fix * (f n ) log ε -1 d n ,v ≤ 1 [K n : Q] ∑ v≤d n N v nd n ∑ z∈Fix * (f n ) nd n log d ≤ ∑ 0<p≤d n d n log d ≤ d 2n log d.
Combining Theorem 3.1 and Theorem 3.2 at all places of K n gives

[K n : Q]|E n | ≤ ∑ v∈M Kn N v |L n (f, ε d n ,v ) v -L(f ) v | + [K n : Q]d 2n log d ≤ ∑ v∈M Kn AN v ( |L(f ) v | + sup P 1 Cv |g f,v | v + 16 log M (f ) v -log ε d n ,v ) σ 2 (n) d n + [K n : Q]d 2n log d
where A := 8(d -1) 2 > 0 and where we used that M (f ) v ≥ 1 for all v. On the one hand, reasoning as above, we find

∑ v∈M 0 Kn -N v log(ε d n ,v ) σ 2 (n) d n ≤ [K n : Q] d d -1 nσ 2 (n)d n log d.
On the other hand, Lemma 4.2 then gives

∑ v∈M Kn N v ( |L(f ) v | + sup P 1 Cv |g f,v | v + 16 log M (f ) v ) ≤ 132d 2 -128d -3 d -1 h d,Kn (f ) + [K n : Q]C(d), with C(d) = 3d-2 d(2d-2) A 1 (d) + 33A 2 (d) + log(2d 3 ),
(f ) = [K n : Q]h d (f ), this concludes the proof.
We now deduce easily the following.

Corollary 8.2. -There exists a constant Cn > 0, depending only on d and n such that for any f ∈ Rat d ( Q) and any n ≥ 1, we have

1 n(d n + 1) h S d n +1 P 1 ( Λ n (f ) ) -h crit (f ) ≤ ( 1056d 2 -1024d-24 ) (d-1)h d (f ) ∑ k|n σ 2 (k) d n + 1 + Cn
where Cn can be computed explicitly.

Proof.

-Pick an integer n and f ∈ Rat d ( Q). By definition of L n (f, 1) v , see (3.1), the chain rule, and since d n + 1 = ∑ m|n d m , we have

R n := 1 n(d n + 1) ∑ z∈Fix(f n ) h P 1 ((f n ) ′ (z)) -h crit (f ) = ∑ m|n d m d n + 1 E m ,
which with Proposition 8.1 completes the proof.

The minimal height, quantified

Recall According to [START_REF] Joseph | Moduli spaces and arithmetic dynamics[END_REF]Lemma 6.32], there exists constants A 1 , A 2 > 0 and B 1 , B 2 such that

A 1 h M d ,D -B 1 ≤ h min ≤ A 2 h M d ,D + B 2 on M d ( Q). (8.1)
In fact, we only need the bound from above and we can quantify the multiplicative constant. Namely, we can prove the next lemma. Proof. -As in the proof of Lemma 7.2, we use the numerical criterion of Siu, see formula (7.1). We proceed in two steps: First, we treat the case when f has at least three distinct fixed points. Second, we treat the case when f has at least one multiple fixed point.

Let us define X as the set of rational maps f having a lift of the form

F (Z, W ) =   Z d + d-1 ∑ j=1 a j Z j W d-j , d-1 ∑ j=0 b j Z j W d-j   with ∑ j a j =
∑ ℓ b ℓ , so that f has 0, 1 and ∞ as fixed points. This variety identifies with a Zariski open set of a linear subspace of P 2d+1 of dimension 2d -2, i.e. we may compactify X as P 2d-2 .

Let Π : X → M d be the restriction of the canonical projection. It is clear that the fiber of Π over any [f ] ∈ M d \ Per 1 (1), is non-empty and finite with a uniform bound on its cardinality. In particular, Π defines a rational map Π : P 2d-2 M d which is dominant and generically finite. Let Y be a normal projective variety defined over Q and π 1 , π 2 be a morphism such that (2d-2) • (H 2d-2 ).

Y π 1 π 2 # # G G G G G G G G G P 2d-2 Π / / _ _ _ M d
We note that deg(Π) deg D (M d ) > 1 and we define H := (2d -2)H.

All the above summarizes as (2d -2)

( H2d-3 • E ) = (2d -2) 2d-2 (H 2d-3 • E) < (2d -2) 2d-2 (deg(Π) deg D (M d )) 1/(2d-2) (H 2d-3 • E)
< (2d -2) 2d-2 (H 2d-2 ) = ( H2d-2 ). By Siu's numerical criterion, this implies H -E is a big divisor on Y . In particular, by an argument similar to that in the proof of Lemma 7.2, we find

(2d -2)h M d ,D • Π ≥ h P 2d-2 + O(1) = h d,Q + O(1)
on X( Q) and in turn we have the desired estimate

h min ([f ]) ≤ h d (f ) ≤ (2d -2)h M d ,D ([f ]) + O(1)
for all f ∈ X( Q).

We now come to the case of rational maps having a multiple fixed point. Let Z be the set of rational maps f having a lift of the form

F (z, w) =   Z d + d-1 ∑ j=1 a j Z j W d-j + W d , Z d-1 W + d-2 ∑ j=1 b j Z j W d-j   .
The family Z is again a Zariski open set of a linear subspace of P 2d+1 of dimension 2d -3. Moreover, any rational map with a multiple fixed point is conjugated to such a map f ∈ Z; and fibers of the restriction of the canonical projection Π : Z → Per 1 (1) ⊂ M d over any [f ] ∈ Per 1 (1) are non-empty, finite with uniformly bounded cardinality. The same proof as above gives 1)

h min ([f ]) ≤ h(f ) ≤ (2d -3)h M d ,D ([f ]) + O(
≤ (2d -2)h M d ,D ([f ]) + O(1)
for all f ∈ Z( Q). Since any class [f ] ∈ M d ( Q) admits either a representative in X( Q) or a representative in Z( Q), this ends the proof.

The critical height is a moduli height, quantified

We now want to give the proof of Theorem C and of Theorem D. We now prove that Λn is finite to one on (M d \ L d )( Q). Assume first there exists x ∈ P dn ( Q) such that there exists an irreducible component X of Λ -1 n,ex {x} with positive and such that X( Q) ∩ L d ( Q) is finite. Let V := X \ X ∩ L d , so that V is an irreducible quasi-projective variety of positive dimension defined over a number field K. By the first step of the proof, we have

h M d ,D ≤ 2 C 1 (d, D)nd n ( h P dn (x) + C ′ n ) < +∞ on V ( K).
On the other hand, using again the functorial properties of height functions, we see that the height h M d ,D restricts to V as a height function associated with an ample line bundle on X. This is a contradiction, since h M d ,D is uniformly bounded on V ( K).

To conclude, it is sufficient to remark that, since Λ n,ex has finite fibers on (M d \ L d )( Q), it also has finite fibers on (M d \L d )(K) for any extension K (algebraic or not) of Q, ending the proof.

Remark 8.4 (Quadratic case). -(i) When d = 2, the moduli space is canonically isomorphic to the affine space A 2 and the isomorphism is given by (σ 1,1 , σ 1,2 ) by [START_REF] Milnor | Geometry and dynamics of quadratic rational maps[END_REF][START_REF] Joseph | The space of rational maps on P 1[END_REF]. We can choose D to be the line bundle O(1). In particular, deg D (M 2 ) = 1 and the equation (8.3) reads as 1 4 ∥µ bif ∥ 1/2 > 8608 σ 2 (n) 2 n . According to [GOV, Corollary D], the mass of µ bif is then

∥µ bif ∥ = 1 3 - 1 8 ∑ n≥1 ϕ(n) (2 n -1) 2 ≥ 0.1875.
where ϕ is the Euler totient function. In particular, the condition is fulfilled for n ≥ 27, whence Λn is finite to one for all n ≥ 27.

(ii) This is explicit but not satisfactory, since it is easy to prove that the map Λ n has finite fibers for all n ≥ 1. Indeed, for n = 1, the map Λ1 = Λ 1 is known to be an isomorphism from A 2 to its image. Moreover, as seen in the proof of Proposition 6.10, Λ n is finite to one on a Zariski open set A 2 (C) \ Z n for all n ≥ 1 and the set Z n satisfies Z n ⊂ Z 1 = ∅ for all n > 1, as required.

2. 1 .

 1 The Berkovich projective line and the dynamics of rational maps Let K be an algebraically closed field that is complete with respect to a non-trivial absolute value | • |. When K is non-archimedean, let O K be the ring of K-integers O K := {z ∈ K : |z| ≤ 1}. The residual characteristic of K is by definition the characteristic of the residue field of O K .

  and by the homogeneity of F , up to multiplication in {z ∈ K : |z| = 1}, there is a unique lift F of f such that g F = g f on P 1 , or equivalently, that | Res F | = 1. Caution. -Normalizing a lift F of f as | Res F | = 1 is as natural as normalizing F as |F | = 1.

  since |d| ≤ 1 and -2 log | Res F | + 4d log |F | ≥ 0 in this case, which completes the proof. When K is non-archimedean, since log M (f ) = -2 log | Res(f )| = -2(log | Res F | -2d log |F |) ≥ 0, the third estimate is obvious. When K is archimedean, since log M (f ) = 2 log sup P 1 (f # ), the third estimate follows from Lemma 2.5.

,

  as required. Since two conjugated rational maps have the same Lyapunov exponent, the Lyapunov function f → L(f ) on Rat d (C) descends to a continuous and psh function L : M d → R; for more details including the complex analytic properties of the quotient map Rat d (C) → M d (C), we refer to [BB, §6]. Definition 6.4. -For any integer 1 ≤ p ≤ 2d -3, the p-bifurcation current on M d is given by T p bif := (dd c L) p , and the bifurcation measure on M d is by µ bif := T 2d-2 bif = (dd c L) 2d-2 . Finally we simply let T bif := T 1 bif .

  and, for any [f ] ∈ M d (C), we let u([f ]) := inf ũ(h), where the infimum is taken over all h ∈ Rat d (C) which are conjugated to f . Claim. -There exists C 1 , C 2 > 0 depending only on d and deg L (M d ) such that

  where ∥g∥ is the Euclidian norm of the vector (a 1 , . . . ,a d-1 , b d-1 , . . . , b 1 ) ∈ C 2d-2 . Since Res ∈ C[V ]and V is a Zariski open subset of P 2d-2 (C), this gives constants C 4 , C 5 > 0 such that for all g ∈ V , we have ũ(g) ≤ -C 4 log d P 2d-2 (g, Z) + C 5 .

Proposition 6 .

 6 10. -Fix an integer d > 1. There exists an integer n 0 (d) ≥ 1 depending only on d such that the following holds (i) for all n ≥ n 0 (d), the maps Λ n and Λn are generically finite-to-one, (ii) for all n ≥ n 0 (d), there exists a multiple N of n such that the map Λ N is finite-to-one outside of the flexible Lattès curve L d ⊂ M d .

  -2 and, by definition, E n is an ample divisor on X n . The morphism Λ n : M d → P d n +1 is generically finite, and defines a rational map Λ n : M d X n , which is dominant and generically finite, by the choice of n. Moreover, Λ n | M d is a morphism. In particular, there exists a normal projective variety M d , a birational morphism ψ : M d → M d and a generically finite morphism Λ n : M d → X n such that the following diagram commutes ψ restricts as an isomorphism ( M d ) reg → (M d ) reg . As the divisors D and E n are ample, and as Λ n and ψ are morphisms, the divisors E n := Λ * n E n and D := ψ * D and big and nef divisors of M d . We have the Lemma 7.2. -Let N 1 , N 2 ≥ 1 be any integers satisfying

  that we have chosen an embedding M d → A N and that we have setD = O P N (1)| ι(M d ) . As in [Sil4, §6.2], for any [f ] ∈ M d ( Q), let h min ([f ]) := min { h(g) ; g ∼ f and g ∈ Rat d ( Q) } .

  Lemma 8.3. -There exists A > 0 depending only on d and deg D (M d ) such thath min ≤ (2d -2)h M d ,D + A, on M d ( Q).

  commutes and the map π 1 is an isomorphism onto X. Set H := π * 2 D and E := π * 1 O P 2d-2 (1). As π 1 and π 2 are surjective morphisms which are generically finite, the divisors H and E are big and nef. Moreover, the projection formula gives(H 2d-2 ) = deg(π 2 )(D 2d-2 ) = deg(Π) deg D (M d ) and (E 2d-2 ) = deg(π 1 )(O P 2d-2(1) 2d-2 ) = 1. We now choose an embedding Y → P M and denote by deg(Y ) the degree degO P M (1) (Y ).As in the proof of Lemma 7.2, we may identify π * 2 ω M d ,D with the restriction to Y of a form α and π * 1 ω 2d-2 with the restriction of a current S. B zout's Theorem gives(H 2d-2 ) = ∫ Y (π * 2 ω M d ,D ) 2d-2 = deg(Y ) • ∥α∥ 2d-1 ω 2d-2 ) 2d-2 = deg(Y ) • ∥S∥ 2d-2. Combined with the above use of the projection formula, this gives ( ∥S∥ ∥α∥) 2d-2 deg(Π) deg D (M d ) = 1.Using again Bézout, we find(H 2d-3 • E) = ∫ Y (π * 2 ω M d ,D ) 2d-3 ∧ (π * 1 ω 2d-2 ) = deg(Y ) • ∥α∥ 2d-3 ∥S∥ = deg(Y ) • ∥α∥ 2d-2 ∥S∥ ∥α∥ = ∥S∥ ∥α∥ (H 2d-2 ) = (deg(Π) deg D (M d )) -1/

  Proof of Theorem C. -Corollary 8.2 implies that 1 n(d n + 1)h P d n +1 • Λ n ([f ]) -h crit ([f ]) ≤ C(d) • h min ([f ]) • ∑ k|n σ 2 (k) d n + C n , (8.2) for all n ≥ 1 and all [f ] ∈ M d ( Q),where C n > 0 is a constant depending only on d and n and C(d) = (1056d 2 -1024d -24)(d -1).Pick now n so that Theorem 7.1 applies and such that(2d -2)C(d) ∑ k|n σ 2 (k) d n ≤ C 1 (d, D),whereC 1 (d, D) is defined in the introduction. Combining Lemma 8.3 with (8.2) gives 1 n(d n + 1) h P d n +1 • Λ n ([f ]) -h crit ([f ]) ≤ C 1 (d, D)h M d ,D ([f ]) + C ′ n , for all [f ] ∈ M d ( Q). Since C 1 (d, D) ≤ C 2 (d,D), Theorem 7.1 ends the proof. Proof of Theorem D. -Fix n ∈ N * . Le us first prove that, when n is large enough, h S dn/n P 1 • Λ n,ex is a height function on (M d \ L d )( Q). By Proposition 8.1 and Lemma 8.3, 1 nd nh S dn P 1 • Λn -h crit ≤ (2d -2)C(d) σ 2 (n) d n h M d ,D + C ′ n ,where C(d) = 1056d 3 -1024d 2 -24d and C ′ n is a constant that depends only in d and n. By Theorem C, for all[f ] ∈ (M d \ L d )( Q), we have C 1 (d, D)h M d ,D -A ≤ h crit ≤ C 2 (d, D)h M d ,D + A,where A depends only on d and C 1 (d, D) and C 2 (d, D) are defined in the introduction. In particular, as soon as1 2 C 1 (d, D) > (2d -2)C(d) σ 2 (n) d n , (8.3) the function 1 ndn h S dn P 1 • Λn is a Weil height on (M d \ L d )( Q).More precisely, there exists a constant C ′′ n such that 1 2 C 1 (d, D)h M d ,D -C ′′ n d ,D + C ′′ n , on (M d \ L d )( Q).

  There exists an integer n 1 ≥ 1 which depends only on d, the mass ∥µ bif ∥ M d and ∥T bif ∥ M d ,D and the degree deg D (M d ) and such that for all n ≥ n 1 , the exact multiplier map Λn : M d (C) → C dn is finite-to-one outside of the curve of flexible Lattès locus L d (C).

Using classical finiteness properties of ample height functions, we prove

Theorem D (Multipliers of exact period n as moduli). -

  := {z ∈ K : |z -a| ≤ r} in K itself), and setting diam S := r, the point S is of type I if and only if diam S = 0 (or equivalently S ∈ K = P 1 \ {∞}), and is of type II (resp. III) if and only if diam S ∈ |K * | (resp. diam S ∈ R >0 \ |K * |). The point ∞ ∈ P 1 is also of type I, and the canonical (or Gauss) point S can in P 1 is of type II and is represented by O K (similarly, type IV points are represented by a decreasing sequence (B n

	[J2, Definition
	2.2]. A point S ∈ P 1 is of one and only one of types I, II, III, and IV. Any type I,
	II, or III point S ∈ P 1 \ {∞} is represented by the multiplicative supremum seminorm
	| • | {z∈K:|z-a|≤r} on K[z] (or, by a K-closed disk
	B

S

  descends to P 1 and in turn continuously extends to P 1 , and the function T F /d is a continuous weight on P 1 ; indeed, ∆(T F /d)+Ω can = (f * δ Ωcan )/d is a probability Radon measure on P 1 (see, e.g.,[START_REF] Okuyama | Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-Archimedean dynamics[END_REF] Definition 2.8] for non-archimedean K).

  D since they are cohomologuous). This is not exactly the usual DSH norm but just a semi-norm. Nevertheless, one has ∥Ψ∥ * DSH ≤ ∥Ψ∥ DSH , where ∥Ψ∥ DSH := ∥Ψ∥ * DSH + ∥Ψ∥ L 1 . The interest of those DSH-norms lies in the fact that they behave nicely under change of coordinates. Furthermore, when Ψ is C 2 with support in a compact set K, there is a constant C > 0 depending only on K such that ∥Ψ∥ DSH ≤ C∥Ψ∥ C 2 .

  where A 1 (d) and A 2 (d) are given by Lemma 2.2 and 2.5 respectively and depends only on d. Since h d,Kn

Thomas Gauthier, Yûsuke Okuyama & Gabriel Vigny

Proof of Lemma 7.2. -Observe first that the map Λ n = Λ n • ψ : M d → X n is surjective and has finite fibers on a Zariski open set of M d . In particular, E n is big and nef. Moreover,

where we used the change of variable formula and that deg(ψ) = 1. According to Proposition 6.9, the choice of n guarantees that

so that the above gives

On the other hand, since ψ is surjective finite and proper and deg(ψ) = 1, D is big and nef and the projection formula gives

Fix an embedding j :

Recall that, if T is a closed positive (1, 1)-current (resp. form) on M d , it can be identified with the restriction of a closed positive (1, 1)-current (resp. form) on P q , so that we may identify Λ * n ω d n +1 with the restriction to M d of a current S n and ψ * ω M d ,D with the restriction of a form α.

Let ∥T ∥ be the mass of any current T with respect to the Fubini Study form ω of P q and let deg( M d ) be the degree of the embedding j( M d ). Bézout's Theorem gives

and

By the above, the divisors N 1 E n and N 2 D satisfy (2d -2)

By Siu's Theorem, this implies N 1 E n -n(d n + 1)N 2 D is big, as required.