Thomas Gauthier

Yûsuke Okuyama

Gabriel Vigny

HYPERBOLIC COMPONENTS OF RATIONAL MAPS: QUANTITATIVE EQUIDISTRIBUTION AND COUNTING by

Let Λ be a quasi-projective variety and assume that, either Λ is a subvariety of the moduli space M d of degree d rational maps, or Λ parametrizes an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 . We prove the equidistribution of parameters having p distinct neutral cycles towards the bifurcation current T p bif letting the periods of the cycles go to ∞, with an exponential speed of convergence. Several consequences of this result are:

a precise asymptotic of the number of hyperbolic components of parameters admitting 2d -2 distinct attracting cycles of exact periods n1, . . . , n 2d-2 as minj nj → ∞ in term of the mass of the bifurcation measure and compute that mass in the case where d = 2. In particular, in M d , the number of such components is asymptotic to

, provided that minj nj is large enough. in the moduli space P d of polynomials of degree d, among hyperbolic components such that all (finite) critical points are in the immediate basins of (not necessarily distinct) attracting cycles of respective exact periods n1, . . . , n d-1 , the proportion of those components, counted with multiplicity, having at least two critical points in the same basin of attraction is exponentially small. in M d , we prove the equidistribution of the centers of the hyperbolic components admitting 2d -2 distinct attracting cycles of exact periods n1, . . . , n 2d-2 towards the bifurcation measure µ bif with an exponential speed of convergence. we have equidistribution, up to extraction, of the parameters having p distinct cycles of given multipliers towards the bifurcation current T p bif outside a pluripolar set of multipliers as the minimum of the periods of the cycles goes to ∞. As a by-product, we also get the weak genericity of hyperbolic postcritically finiteness in the moduli space of rational maps. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the log + of the modulus of the multipliers of periodic points.

Introduction

For a holomorphic family (f λ ) λ∈Λ of degree d > 1 rational maps on the Riemann sphere P 1 parametrized by a quasi-projective variety Λ, the bifurcation locus of (f λ ) λ∈Λ on Λ is the J-unstability locus in the sense of Mañe-Sad-Sullivan, i.e., the closure of the set of all parameters in Λ at which the Julia set J λ of f λ does not move continuously. It is now classical that this set is nowhere dense in Λ and admits several distinct topological descriptions, such as the closure of the set of parameters for which f λ admits a nonpersistent neutral cycle or the existence of an unstable critical dynamics (see e.g. [START_REF] Mañé | On the dynamics of rational maps[END_REF][START_REF] Yu | Some typical properties of the dynamics of rational mappings[END_REF][START_REF] Curtis | Complex dynamics and renormalization[END_REF]). From now on, pick any integer d > 1.

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.

The second author would also like to thank the LAMFA and the Université de Picardie Jules Verne, which he was visiting in spring 2016 and where this work grew up.

On the other hand, any (individual) rational map f of degree d on P 1 admits a unique maximal entropy measure µ f , whose support coincides with the Julia set J f of f , and the Lyapunov exponent of f with respect to µ f is defined by L(f ) := P 1 log |f |µ f and satisfies L(f ) ≥ 1 2 log d > 0. For a family (f λ ) λ∈Λ , the induced Lyapunov function L : λ ∈ Λ -→ L(f λ ) ∈ R is p.s.h and continuous on the parameter space Λ. We can define the bifurcation current of (f λ ) λ∈Λ on Λ as the closed positive (1, 1)-current

T bif := dd c L.
By DeMarco [START_REF] Demarco | Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity[END_REF], the support of dd c L coincides with the bifurcation locus of the family (f λ ) λ∈Λ . For any integer 1 ≤ p ≤ dim Λ, Bassanelli and Berteloot also defined the p-bifurcation currents T p bif as the p-th exterior product of T bif . It is a positive closed current of bidegree (p, p) so the bifurcation measure µ bif := (dd c L) dim Λ is a positive measure on Λ. If p > 1, the current T p bif detects, in a certain sense, stronger bifurcations than T bif = T 1 bif [START_REF] Bassanelli | Bifurcation currents in holomorphic dynamics on P k[END_REF]. 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] Dujardin | The supports of higher bifurcation currents[END_REF][START_REF] Gauthier | Higher bifurcation currents, neutral cycles, and the Mandelbrot set[END_REF]).

The group PSL 2 (C) of Möbius transformations acts on the space Rat d of degree d rational maps on P 1 , which is itself a holomorphic family of rational maps, by conjugacy. The moduli space M d of degree d rational maps on P 1 is the orbit space of PSL 2 (C) in Rat d , that is, the quotient of Rat d resulting from this action of PSL 2 (C). It is an irreducible affine variety of dimension 2d -2, and is singular if and only if d ≥ 3 (Silverman [START_REF] Joseph | The space of rational maps on P 1[END_REF]). The Lyapunov function f → L(f ) on Rat d descends to a continuous and psh function L : M d → R. For any integer 1 ≤ p ≤ 2d -2, the p-bifurcation current on M d is thus 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 ,
which is a finite positive measure on M d of strictly positive total mass (see [START_REF] Bassanelli | Bifurcation currents in holomorphic dynamics on P k[END_REF]).

One of the features of the bifurcation currents is to give measurable statements of the above density, or in general, accumulation properties. Let us be more precise. Let Λ be a quasi-projective variety such that, either Λ ⊂ M d , or parametrizing an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 . For any n ∈ N * and any w ∈ C \ {1}, let Per n (w) be the analytic hypersurface Per n (w) := {λ ∈ Λ : f λ has a cycle of multiplier w and the exact period n} in Λ and denote by [Per n (w)] the current of integration over Per n (w) on Λ. Since Λ is quasi-projective, the hypersurfaces Per n (w) are actually algebraic hypersurfaces of Λ (see e.g. [START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF]). By Bassanelli and Berteloot [BB2], the sequence (d -n [Per n (w)]) weighted by the Lebesgue measure on the disk of center 0 and radius |w| converges towards the bifurcation current T bif . Similar dynamically significant equidistribution properties towards the bifurcation current have been recently established in various contexts, as general holomorphic families of rational maps [START_REF] Dujardin | Distribution of rational maps with a preperiodic critical point[END_REF][START_REF] Bassanelli | Distribution of polynomials with cycles of a given multiplier[END_REF][START_REF] Okuyama | Equidistribution of rational functions having a superattracting periodic point towards the activity current and the bifurcation current[END_REF] or moduli spaces of polynomials [START_REF] Buff | Quadratic polynomials, multipliers and equidistribution[END_REF][START_REF] Gauthier | Equidistribution towards the bifurcation current I: multipliers and degree d polynomials[END_REF].

The proofs developed in op. cit. do not allow establishing equidistribution phenomena towards the bifurcation measure µ bif . Indeed, any of the above convergences obtained is essentially L 1 loc convergence of the potentials of currents, which does not guarantee continuity of the intersection.

One of the main purposes of the article is to prove the equidistribution of parameters having p non-repelling cycles towards the bifurcation current T p bif as the minimum of the periods of those cycles goes to ∞, with an exponential speed of convergence. We will then deduce several important consequences, notably in counting hyperbolic components of disjoint types in M d . Notice that such counting results are of combinatorial and algebraic nature and have a priori no relation to bifurcation measures. Furthermore, they are the first general results in that direction so far. For any p ∈ N * , any n = (n 1 , . . . , n p ) ∈ (N * ) p , and any ρ = (ρ 1 , . . . , ρ p ) ∈]0, 1] p , we set |n| := p j=1 n j , so that d |n| = p j=1 d n j and, in a similar way,

d |n| := p j=1 d j .
For any i ∈ {0, 1, 2} and any n ∈ N * , we also set σ i (n) := m|n m i , so in particular σ 0 ≤ σ 1 ≤ σ 2 on N * (beware that σ 2 (n) ≤ Cn 2 log log n for some constant C).

For any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 and any w = (w 1 , . . . , w 2d-2 ) ∈ C 2d-2 , let Stab(n) (resp. Stab(n, w)) be the set of all permutations of the indices {1, 2, . . . , 2d -2} that do not change the ordered (2d -2)-tuple (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 (resp. ((n 1 , w 1 ), . . . , (n 2d-2 , w 2d-2 )) ∈ (N * ×C) 2d-2 ), so in particular #Stab(n, w) ≤ #Stab(n) ≤ (2d -2)!.

For r > 0, we set D r = {|z| < r}, so that ∂D r = S r = {|z| = r}.

Statement of the main results

Let Λ be a quasi-projective variety such that, either Λ ⊂ M d , or parametrizing an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 . We refer to [Dem] for basics on positive closed currents and intersection theory on algebraic varieties. For any integer 1 ≤ p ≤ min{dim Λ, 2d -2}, any n = (n 1 , . . . , n p ) ∈ (N * ) p , and any ρ = (ρ 1 , . . . , ρ p ) ∈]0, 1] p , the following positive closed current (1)

T p n (ρ) := 1 d |n| [0,2π] p p j=1 [Per n j (ρ j e iθ j )] dθ 1 • • • dθ p (2π) p
on Λ is well-defined, and coincides with p j=1 T 1 n j (ρ j ) by the Fubini theorem (see e.g. [START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF]). We say a form Ψ on Λ is DSH if dd c Ψ = T + -T -for some positive closed currents T ± of finite masses on Λ. We refer to §2.1 for the precise definition of the semi-norm Ψ * DSH . One of our principal results is the following.

Theorem A. -Let Λ be a quasi-projective variety which either is a subvariety in M d or parametrizes an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 . Then for any compact subset K in Λ, there exists a constant C(K) > 0 such that for any integer 1 ≤ p ≤ min{dim Λ, 2d -2}, any n = (n 1 , . . . , n p ) ∈ (N * ) p , any ρ = (ρ 1 , . . . , ρ p ) ∈]0, 1] p , and any continuous DSH-form Ψ of bidegree (m -p, m -p) supported in K, we have

T p n (ρ) -T p bif , Ψ ≤ C(K) • max 1≤j≤p (1 + | log ρ j |) σ 2 (n j ) d n j • Ψ * DSH .
We first prove this theorem in the case where p = 1. To do that, we show in Section 3 a locally uniform version of the second author's result [START_REF] Okuyama | Quantitative approximations of the Lyapunov exponent of a rational function over valued fields[END_REF] on the quantitative approximation of the Lyapunov exponent of an (individual) f ∈ Rat d by the average of the logs of the moduli of the multipliers of all non-attracting n-periodic points of f (Lemma 3.3). This leads to an error term on the proximity between f n (c) and c for each critical point c of f and an error term on how close to 0 the multipliers of the periodic points of f are. To control those terms, we use a parametric version of a lemma of Przytycki [START_REF] Przytycki | Lyapunov characteristic exponents are nonnegative[END_REF]Lemma 1] proved by the first and third authors in [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF]. Intersection of currents and integrations by parts lead to the result for any p.

Theorem A is proved in Section 4 and has the following consequence.

Corollary B. -Let Λ be a quasi-projective variety such that, either Λ ⊂ M d , or parametrizing an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 . Pick an integer 1 ≤ p ≤ min{dim Λ, 2d -2}. Then for any sequence

(n k ) k∈N * of p-tuples n k = (n 1,k , . . . , n p,k ) in (N * ) p such that k max j (n -1 j,k ) < ∞, there exists a pluripolar subset E in C p such that for any w = (w 1 , . . . , w p ) ∈ C p \ E, p i=1 Per n i,k (w i ) is of pure codimension p in Λ for any k ∈ N * and T p bif = lim k→∞ 1 d |n k | p i=1 [Per n i,k (w i )]
in the weak sense of currents on Λ.

The techniques used in the proof of Corollary B also give that the current equidistributed on the set of parameters having p cycles of respective periods n 1,k , . . . , n p,k and multipliers w 1 , . . . , w p distributed by a PB measure on (P 1 ) p converges towards the bifurcation current T p bif when k → ∞, with the best possible order estimate O(max j (n -1 j )) as min j (n j ) → ∞. (see Theorem 4.1 below).

Remark. -Let us also observe that, as in [START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF], Theorem A gives another proof that Shishikura's upper bound 2d -2 of the number of distinct cycles of Fatou components of a given rational map of degree d is sharp (see [Sh]). In fact, provided min j n j is large enough, we can construct a rational map having 2d -2 distinct attracting periodic points of respective period n j (we no longer need to take a subsequence and have no arithmetic restrictions on the periods). Now let us focus on the moduli spaces of rational maps and hyperbolic components. Recall that the hyperbolic locus in M d is the set of all conjugacy classes of hyperbolic maps that are uniformly expanding on their Julia sets. It is an open subset of M d and a connected component of this hyperbolic locus is called a hyperbolic component in

M d . Definition 1.1. -A rational map f ∈ Rat d is said to be hyperbolic of type n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 if f has 2d -2 distinct attracting cycles of respective exact periods n 1 , . . . , n 2d-2 . A hyperbolic component Ω in M d is said to be of type n ∈ (N * ) 2d-2 if, for any [f ] ∈ Ω, f is hyperbolic of type n. A hyperbolic component in M d is of disjoint type if it is of type n for some n ∈ (N * ) 2d-2 . Definition 1.2. -For any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 , let N (n) denote the number of hyperbolic components of type n in M d .
A striking application of Theorem A is the following asymptotic on the global counting of hyperbolic components.

Theorem C. -As min j n j → +∞, #Stab(n) • N (n) d |n| = M d µ bif + O max j σ 2 (n j ) d n j .
In particular, N (n) > 0 if min j n j is large enough. Theorem C gives a combinatorial interpretation of the mass of the bifurcation measure. In the case d = 2, as a consequence of Theorem C together with the precise estimates of N (n 1 , n 2 ) by Kiwi and Rees [KR], we can determine the (total) mass of the bifurcation measure on M 2 .

Corollary D. -Let φ be the Euler totient function on N * . Then

M 2 µ bif = 1 3 - 1 8 n≥1 φ(n) (2 n -1) 2 .
In the proof of Theorem C, it is crucial that the estimate in Theorem A involves only the DSH-semi-norm • * DSH of the observable. Notice also that the mass of a limit of positive measures is not greater than the limit of the masses, so it could be possible that a proportion of components is lost passing to the limit as they would accumulate at the boundary of the moduli space. Theorem C says that it is not the case. The proof of Theorem C also relies crucially on the fact that the multipliers of attracting cycles parametrize the hyperbolic components of disjoint type of M d . Though this is essentially classical, there seems to be no available proofs in the literature so we include a proof of it in Section 6.2. The proof relies on the transversality of periodic critical orbit relations, which we show in Section 5, following the argument of Epstein [BE,[START_REF] Epstein | Transversality principles in holomorphic dynamics[END_REF].

It is comparable to the common situation in dynamical systems where the existence of e.g. repelling periodic orbit of large period follows from an equidistribution property, see [START_REF] Briend | Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP k[END_REF][START_REF] Briend | Deux caractérisations de la mesure d'équilibre d'un endomorphisme de P k (C)[END_REF] for holomorphic endomorphisms on P k .

As a consequence of Theorem C, we also establish the weak genericity of hyperbolic postcritically finite maps in M d (see Theorem F below), which is stronger than the Zariski density of such maps in M d .

We finally establish a quantitative equidistribution of parameters in hyperbolic components in M d of disjoint type, having given multipliers. For any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 and any w = (w 1 , . . . , w 2d-2 ) ∈ D 2d-2 , let C n,w denote the (finite) set of all conjugacy classes [f ] ∈ M d of hyperbolic rational maps f ∈ Rat d of type n whose attracting cycle of exact period n j has the multiplier w j for any 1 ≤ j ≤ 2d -2, and set

µ n,w := #Stab(n, w) d |n| [f ]∈Cn,w δ [f ] .
For simplicity, we denote C n,(0,...,0) and µ n,(0,...,0) by C n and µ n , respectively, so that any element in C n is the center of a hyperbolic component in M d of type n.

The following in particular implies the weak convergence µ n,w → µ bif on M d , which is even new and it was one of our motivations to give a proof of this convergence.

Theorem E. -For any compact subset K in M d , there exists C K > 0 such that 1. for any test function Ψ ∈ C 2 (M d ) with support in K and any n ∈ (N * ) 2d-2 , µ n -µ bif , Ψ ≤ C K • max 1≤j≤2d-2 σ 2 (n j ) d n j • Ψ C 2 , 2. for any test function Ψ ∈ C 1 (M d ) with support in K, any n ∈ (N * ) 2d-2
, and any

w = (w 1 , . . . , w 2d-2 ) ∈ D 2d-2 , µ n,w -µ bif , Ψ ≤ C K • max 1≤j≤2d-2 -1 d n j log |w j | , σ 2 (n j ) d n j 1/2 • Ψ C 1 .
Observe that an interpolation between Banach spaces gives a speed of convergence for any C α -observable with 0 < α ≤ 2 in the case of centers and 0 < α ≤ 1 in general.

Even though Theorem E looks very close to known qualitative/quantitative equidistribution results for holomorphic/anti-holomorphic polynomial families, e.g., [START_REF] Favre | Equidistribution quantitative des points de petite hauteur sur la droite projective[END_REF][START_REF] Favre | Distribution of postcritically finite polynomials[END_REF][START_REF] Gauthier | Distribution of postcritically finite polynomials III: Combinatorial continuity[END_REF][START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF][START_REF] Okuyama | Nevanlina theory and equidistribution in the unicritical polynomials family[END_REF], the compactness of the support of the bifurcation measure was a crucial point in those earlier works. Such a compactness is not the case for M d . It might also be worth stressing that in [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF], the first and third authors considered the currents of bifurcation T c of marked critical points c so were looking at unstable critical dynamics. Although it seems to be similar, here we study directly the bifurcation current T bif so the unstability of cycles (see the introduction of [START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF]). A feature of that approach is that we don't need the n j to be distinct, which was necessary in the above works.

Section 8 is devoted to the study of the moduli space P cm d of critically marked degree d polynomials where we give various results similar to those previously proved.

To finish, let us mention that, as an application of our approximation formula of the Lyapunov exponent, we give a proof of the estimate of the degeneration of the Lyapunov exponent of f as f → ∂ Rat d along an analytic disk in the spirit of [F] (see Theorem 3.6).

Acknowledgement. -We would like to thank the anonymous referee for very careful scrutiny and helpful comments and suggestions.

Preliminaries

Currents and DSH functions

We refer to [DS3, Appendix A] for more details on currents and DSH functions. Pick any quasi-projective variety Λ. Let β be the restriction of the ambient Fubini-Study form to Λ. For any positive closed current T of bidimension (k, k) defined on Λ and any Borel set A ⊂ Λ, we denote by T A the number

T A := A T ∧ β k .
This is the mass of the current T in A. We simply write T for T Λ .

Let Ψ be an ( , )-form in Λ. 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 Λ. We also set Ψ * DSH := inf

T ± T + + T -,
where T ± ranges over all closed positive currents such that dd c Ψ = T + -T -(note that T + = T -since they are cohomologous). 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 .

Resultant and the space Rat d

We refer to [START_REF] Bassanelli | Bifurcation currents in holomorphic dynamics on P k[END_REF] and [START_REF] Joseph | The space of rational maps on P 1[END_REF] for the content of this paragraph.

Notations. -Let π : C 2 \ {0} → P 1 be the canonical projection, • be the Hermitian norm on C 2 , and set (z 0 , z 1 ) ∧ (w 0 , w 1 ) :

= z 0 w 1 -z 1 w 0 on C 2 × C 2 . A pair F = (F 1 , F 2 ) ∈ C[x, y] d × C[x, y] d C 2d+2
of homogeneous degree d polynomials can be identified with a degree d homogeneous polynomial endomorphism of C 2 . The homogeneous resultant Res = Res d is the unique homogeneous degree 2d polynomial over Z in 2d + 2 variables such that Res(F ) = 0 if and only if F is degenerate, i.e. F -1 ({0}) = {0}, and Res((x d , y d )) = 1. We thus identify the space of all degree d nondegenerate homogeneous polynomial endomorphisms of C 2 with C 2d+2 \ {Res = 0}.

A rational map f on P 1 of degree d admits a (non-degenerate homogeneous polynomial) lift, i.e. there exists a degree d homogeneous polynomial endomorphism F :

C 2 → C 2 such that Res(F ) = 0 and that π • F = f • π on C 2 \ {0}.
Moreover, any two homogeneous polynomial endomorphisms F, G of C 2 are lifts of the same f if and only if there exists α ∈ C * such that F = α • G. Let us denote by Rat d the set of all degree d rational maps on P 1 . Since Res is homogeneous, we can also identify Rat d with P 2d+1 \ {Res = 0}. In particular, it is a quasi-projective variety of dimension 2d + 1.

The dynamical Green function of a rational map on P 1

In the whole text, we denote by ω FS the Fubini-Study form on P 1 normalized so that ω FS = 1 and by [•, •] the chordal metric on P 1 given by

[z, w] = |z 0 w 1 -w 1 z 0 |/ |z 0 | 2 + |z 1 | 2 |w 0 | 2 + |w 1 | 2
for any z = [z 0 : z 1 ], w = [w 0 : w 1 ] ∈ P 1 , so that diam(P 1 ) = 1 and for any w, dd c z log[z, w] = δ w -ω FS .

For any ω FS -psh function g on P 1 , i.e. such that ω FS + dd c g =: ν g is a probability measure on P 1 , we define the g-kernel function Φ g by setting

Φ g (z, w) := log[z, w] -g(z) -g(w) (2)
on P 1 × P 1 . For a probability measure ν on P 1 , set U g,ν := P 1 Φ g (•, w)dν (w) on P 1 . Then dd c z U g,ν = ν -ν g , so in the particular case where ν = ν g , we deduce that U g,νg ≡ I g :=

P 1 ×P 1 Φ g (ν g × ν g ) on P 1 .
Pick now f ∈ Rat d . For all (non-degenerate homogeneous polynomial) lift F : C 2 → C 2 of f , there exists a Hölder continuous ω FS -psh function g F :

P 1 → R such that lim n→∞ log F n d n -log • = g F • π
uniformly on C 2 \ {0}, which is called the dynamical Green function of F on P 1 . Since F is unique up to multiplication by α ∈ C * and g α•F = g F + (log |α|)/(d -1) for any α ∈ C * , the positive measure

ω FS + dd c g F =: µ f
is independent of the choice of F , and is in fact the unique maximal entropy measure of f on P 1 . For later use, we point out the equality

I g F = - 1 d(d -1) log |Res(F )|,
which is (a reformulation of) DeMarco's formula [START_REF] Demarco | Dynamics of rational maps: a current on the bifurcation locus[END_REF]Theorem 1.5] on the homogeneous capacity of the filled-in Julia set of F in C 2 .

Definition 2.1. -The dynamical Green function g f of f on P 1 is the unique ω FS -psh function on P 1 such that µ g f = µ f on P 1 and that I g f = 0.

Remark. -In particular, U g f ,µ f ≡ I g f = 0 on P 1 . Moreover, g F = g f for some lift F of f , which is unique up to multiplication by a complex number of modulus one.

The dynatomic and multiplier polynomials

We refer to [Si2, §4.1] and to [START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF][START_REF] Bassanelli | Distribution of polynomials with cycles of a given multiplier[END_REF][START_REF] Milnor | Geometry and dynamics of quadratic rational maps[END_REF] (see also [START_REF] Favre | Distribution of postcritically finite polynomials[END_REF]§6]) for the details on the dynatomic and multiplier polynomials and the related topics. Pick any f ∈ Rat d . For every n ∈ N * , let -Fix(f n ) be the set of all fixed points of f n in P 1 , and -Fix * (f n ) the set of all periodic points of f in P 1 having exact period n.

The n-th dynatomic polynomial of a lift F of f is a homogeneous polynomial

Φ * n (F, (z 0 , z 1 )) := k|n F k (z 0 , z 1 ) ∧ (z 0 , z 1 ) µ(n/k)
in z 0 , z 1 of degree d n ; there is a (finite) sequence (P (n) j ) j∈{1,...,dn} in C 2 \ {0} such that we have a factorization Φ * n (F, (z 0 , z 1 )) = dn j=1 ((z 0 , z 1 ) ∧ P

(n) j ), and setting z (n) j := π(P (n) j ) ∈ P 1 for each j ∈ {1, . . . , d n }, the sequence (z

(n) j ) dn
j=1 is independent of the choice of (P (n) j ) j∈{1,...,dn} and that of F , up to permutation. We recall that the set {z (n) j : j ∈ {1, . . . , d n }} is the disjoint union of Fix * (f n ) and the set of all periodic points z of f having exact period m < n and dividing n and whose multiplier (f m ) (z) is a n/m-th primitive root of unity. In particular, (

f n ) (z) = 1 for every z ∈ z (n) j : j ∈ {1, . . . , d n } \ Fix * (f n ), and for every z ∈ Fix * (f n ), we have #{j ∈ {1, . . . , d n } : z (n) j = z} = 1 if (f n ) (z) = 1. For every n ∈ N * , the n-th multiplier polynomial of f is the polynomial p n (f, w) := dn j=1 (f n ) (z (n) j ) -w 1/n (3)
in w of degree d n /n, which is unique up to multiplication in n-th roots of unity.

Let Λ be a quasi-projective variety parametrizing an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 . Then for any n ∈ N * , the n-th multiplier polynomial p n : Λ×C -→ C of (f λ ) λ∈Λ defined by Beware also that, since the existence of a cycle of given period and multiplier is invariant under Möbius conjugacy, the n-th multiplier polynomial p n : Rat d ×C → C of Rat d also descends to a regular function p n : M d × C → C, enjoying the same properties.

p n (λ, w) := p n (f λ , w)

A parametric version of Przytycki lemma

For a C 1 map f : P 1 → P 1 , the chordal derivative f # of f is the non-negative real valued continuous function

f # (z) := lim y→z [f (z), f (y)]
[z, y] on P 1 . For any rational map f ∈ Rat d , we set

M (f ) := sup P 1 (f # ) 2 ∈]1, +∞[.
We shall use the following, which is a direct consequence of [GV2, Lemma 3.1] and of the fact that the spherical and the chordal distance are equivalent on P 1 .

Lemma 2.2. -There exists a universal constant 0 < κ < 1 such that for any holomorphic family (f λ ) λ∈Λ of degree d rational maps with a marked critical point c : Λ → P 1 which does not lie persistently in a parabolic basin of f λ and is not persistently periodic, the following holds: for any n ∈ N * and any

λ ∈ Λ, if f n λ (c(λ)) = c(λ), then -either [f n λ (c(λ)), c(λ)] ≥ κ • M (f λ ) -n , -or c(λ) lies in the immediate basin of an attracting periodic point z(λ) of f λ of period dividing n, [c(λ), J λ ] ≥ κM (f λ ) -n , and 2[f n λ (c(λ)), c(λ)] ≥ [z(λ), c(λ)].

A length-area estimate

The modulus of an annulus A conformally equivalent to

A = {z ∈ C ; r < |z| < R} with 0 < r < R < +∞ is defined by mod(A) = mod(A ) = 1 2π log R r .
We shall use the following classical estimate ( [START_REF] Briend | Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP k[END_REF]Appendix]).

Lemma 2.3 (Briend-Duval

). -There exists a universal constant τ > 0 such that for any quasi-projective variety Λ, any Kähler metric ω on Λ and any pair of relatively compact holomorphic disks D 1 D 2 in Λ, so that D 2 \ D 1 is an annulus, we have min(1,mod(D 2 \ D 1 )) .

(diam ω (D 1 )) 2 ≤ τ • Area ω (D 2 )

Quantitative approximation of the Lyapunov exponent

Our precise result here can be stated as follows. This result relies on the combination of the arguments used in [START_REF] Okuyama | Quantitative approximations of the Lyapunov exponent of a rational function over valued fields[END_REF] as developed in Lemma 3.4 below and of the lemma "à la Przytycki" proved in [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF]. The locally uniform speed of convergence obtained here is not as fast as the pointwise one obtained in [START_REF] Okuyama | Quantitative approximations of the Lyapunov exponent of a rational function over valued fields[END_REF]. This is due to our need to control the dependence of the constants on f ∈ Rat d in the right-hand side. Here we obtain a continuous dependence.

Theorem 3.1. -There exists A ≥ 1 depending only on d such that for any r ∈]0, 1], any f ∈ Rat d , and any n ∈ N * , we have

1 d n 2π 0 log p n (f, re iθ ) dθ 2π -L(f ) ≤ A (C([f ]) + | log r|) σ 2 (n) d n , where C([f ]) = inf log sup P 1 f # 1 + sup P 1 |g f 1 |
, where the infimum is taken over all f 1 ∈ [f ] and where g f 1 is the dynamical Green function of f 1 normalized as in §2.3.

Of course, as the left-hand side of the inequality is invariant under Möbius conjugacy, it is sufficient to prove that for any 0 < r ≤ 1, any n ∈ N * and any f ∈ Rat d , we have

1 d n 2π 0 log p n (f, re iθ ) dθ 2π -L(f ) ≤ A log sup P 1 f # + sup P 1 |g f | + | log r| σ 2 (n) d n
for some constant A which depends only on d. So we pick f ∈ Rat d . In the following, the sums over subsets in Crit(f ), Fix(f n ), or Fix * (f n ) take into account the multiplicities of their elements. For any n ∈ N * , the cardinality of Fix(f n ) and that of Fix * (f n ) are d n + 1 and d n , respectively, taking into account of the multiplicity of each element of them as a fixed point of f n .

A non-quantitative version of Theorem 3.1 can be shown using the equidistribution of repelling cycles towards µ f and Pesin theory (both arguments being non-quantitative [START_REF] Bassanelli | Distribution of polynomials with cycles of a given multiplier[END_REF]). We instead use formula (5) to relate the multiplier of a n-periodic point with the distance between its orbit and the critical set (Lemma 3.2). Summing over all n-periodic points, we show we can control the left-hand side in Theorem 3.1, using Lemma 3.3, with the difference between the logarithm of [f n (c), c] for all critical points c and the logarithm of the multipliers. We then use Lemmas 2.2 and 3.5 to control that difference and Fatou-Shishikura inequality to bound the cardinality of attracting periodic orbits.

Relating multipliers with the distance between cycles and critical points

Recall that, by [START_REF] Okuyama | Quantitative approximations of the Lyapunov exponent of a rational function over valued fields[END_REF]Lemma 2.4] , we have

log(f # ) = L(f ) + c∈Crit(f ) Φ g f (•, c) + 2(g f • f -g f ) on P 1 . (5)
This formula plays a key role in the proofs of Lemma 3.2 and 3.3. Lemma 3.2. -Assume that f has no super-attracting cycles. Then for any n ∈ N * and any z ∈ Fix(f n ), we have [START_REF] Okuyama | A characterization of polynomials among rational functions in non-archimedean and complex dynamics[END_REF]Lemma 3.4], for every a ∈ P 1 ,

1 n c∈Crit(f ) n-1 j=0 log[f j (z), c] -log |(f n ) (z)| ≤ B 1 (f ), where B 1 (f ) := L(f ) + 2(2d -2) sup P 1 |g f |. Proof. -By (5) applied to f n , we have log((f n ) # ) = L(f n ) + c∈Crit(f n ) Φ g f n (•, c) + 2(g f n • f n -g f n ) = n • L(f ) + c∈Crit(f ) n-1 j=0 P 1 Φ g f (•, w) ((f j ) * δ c )(w) + 2(g f • f n -g f ) on P 1 . By
P 1 Φ g f (•, w) (f * δ a )(w) = Φ g f (f (•), a) on P 1 . In particular, for every z ∈ Fix(f n ), since (f n ) # (z) = |(f n ) (z)|, we have 1 n log |(f n ) (z)| = L(f ) + c∈Crit(f ) 1 n n-1 j=0 Φ g f (f j (z), c),
which with the definition (2) of the g f -kernel function Φ g f completes the proof.

Reduction to the critical dynamics

For any n ∈ N * and any 0 < r ≤ 1, we set

L r n (f ) := 1 d n 2π 0 log |p n (f, re iθ )| dθ 2π .
If f has no super-attracting cycles, for any m, n ∈ N * with m|n and any r ∈]0, 1], we set

u m,n (f, r) := 1 d m + 1 c∈Crit(f ) log[f m (c), c] - z∈Fix(f m ) |(f n ) (z)|<r 1 m log |(f m ) (z)| = u m,m (f, r m/n ). Lemma 3.3. -If f has no super-attracting cycles, then for any r ∈]0, 1] and any n ∈ N * , n (f, r) := 1 n(d n + 1) z∈Fix(f n ) |(f n ) (z)|≥r log |(f n ) (z)| -L(f ) = u n,n (f, r) + n (f ), where (d n + 1)| n (f )| ≤ 2(2d -2) sup P 1 |g f |.
Proof. -Pick r ∈]0, 1] and n ∈ N * , and set

µ n := z∈Fix(f n ) δ z , taking into account the multiplicity of each z ∈ Fix(f n ). Since (f n ) # (z) = |(f n ) (z)| for any z ∈ Fix(f n ), integrating the equation (5) against µ n gives 1 n P 1 log |(f n ) | µ n -(d n + 1)L(f ) = P 1 log(f # ) µ n -(d n + 1)L(f ) = c∈Crit(f ) P 1 Φ g f (c, •) µ n .
This may be rewritten

(d n + 1) n (f, r) = 1 n {|(f n ) |≥r} log |(f n ) | µ n -(d n + 1)L(f ) = c∈Crit(f ) P 1 Φ g f (c, •) µ n - {|(f n ) |<r} 1 n log |(f n ) | µ n . Using again that (f n ) # (z) = |(f n ) (z)| for any z ∈ Fix(f n
) and, by [START_REF] Okuyama | A characterization of polynomials among rational functions in non-archimedean and complex dynamics[END_REF]Lemma 3.5],

P 1 Φ g f (a, •) µ n = Φ g f (f n (a), a)
for every a ∈ P 1 , the definition (2) of the g f -kernel function Φ g f completes the proof.

Lemma 3.4.

-If f has no super-attracting cycles, then for any n ∈ N * and any r ∈]0, 1],

L r n (f ) -L(f ) - 1 d n m|n µ n m (d m + 1)u m,n (f, r) ≤ B(f, r) σ 0 (n) d n , where B(f, r) := (2d -2)(2 sup P 1 |g f | + | log r|).
Proof. -Pick r ∈]0, 1] and n ∈ N * . By the definition of d n , we have

1 = 1 d n m|n µ n m (d m + 1).
For any m ∈ N * dividing n and any z ∈ Fix(f m ), we have (f n ) (z) = (f m ) (z) n/m by the chain rule, and have

z∈Fix(f m ) |(f n ) (z)|≥r log |(f m ) (z)| - z∈Fix(f m ) log max |(f m ) (z)|, r m/n ≤ m(2d -2)| log r|
since the number of attracting periodic points of f of period dividing m is at most (2d -2)m. Recalling the definition (3) of p n , we have

L r n (f ) = 1 nd n dn j=1 2π 0 log (f n ) (z (n) j ) -re iθ dθ 2π = 1 nd n z∈Fix * (f n ) log max |(f n ) (z)|, r = 1 nd n m|n µ n m z∈Fix(f m ) n m log max |(f m ) (z)|, r m/n = 1 d n m|n µ n m 1 m z∈Fix(f m ) log max |(f m ) (z)|, r m/n ,
where the third equality is by the Möbius inversion. Hence recalling the definition of σ 0 (n), by Lemma 3.3, we have

L r n (f ) -L(f ) - 1 d n m|n µ n m (d m + 1)u m,n (f, r) ≤ 1 d n m|n (d m +1) m (f, r m/n )-u m,m (f, r m/n ) + (2d -2)| log r| • σ 0 (n) d n ≤ B(f, r) σ 0 (n) d n ,
which completes the proof.

3.3. Proof of Theorem 3.1 using the parametric version of Przytycki's lemma Lemma 3.5. -For any n ∈ N * , any z ∈ Fix(f n ), and any c ∈ Crit(f ),

[f n (c), c] ≤ 2 • M (f ) n • [c, z]. Proof. -Let M 1 := sup P 1 f # > 1. It is clear that the map f is M 1 -Lipschitz in the chordal metric [•, •]. If f n (z) = z, we have [f n (c), c] ≤ [f n (c), z] + [c, z] ≤ (M n 1 + 1) • [c, z] and the conclusion follows since M 1 > 1.
Proof of Theorem 3.1. -As there is no persistent parabolic and super-attracting cycle in Rat d , the set X of all elements in Rat d having neither super-attracting nor parabolic cycles and no multiple critical points is the complement of a pluripolar subset in Rat d , so X is dense in Rat d . Pick f ∈ X, n ∈ N * , and r ∈]0, 1].

(i) For any m ∈ N * dividing n, recalling the definition of u m,n (f, r), we have

u m,n (f, r) = u m,n (f, 1) + 1 d m + 1 z∈Fix(f m ) r≤|(f n ) (z)|<1 1 m log |(f m ) (z)|,
and recalling that f has at most (2d -2)m attracting periodic points of period dividing m and that

|(f m ) (z)| = |(f n ) (z)| m/n for any z ∈ Fix(f m ) by the chain rule, we have z∈Fix(f m ) r≤|(f n ) (z)|<1 1 m log |(f m ) (z)| ≤ (2d -2)m n | log r| ≤ (2d -2)| log r|.
Hence, recalling the definition of σ 0 (n), we have

1 d n m|n µ n m (d m + 1) (u m,n (f, r) -u m,n (f, 1)) ≤ (2d -2)| log r| • σ 0 (n) d n .
(ii) For any m ∈ N * dividing n, we have

z∈Fix(f m ) |(f n ) (z)|<1 m-1 j=0 log[f j (z), c] = m • z∈Fix(f m ) |(f m ) (z)|<1 log[z, c].
Recalling the definition of u m,n (f, r) and applying Lemma 3.2 to each such

z ∈ Fix(f m ) that |(f n ) (z)| < 1, we have (d m + 1)u m,n (f, 1) - c∈Crit(f ) log[f m (c), c] - z∈Fix(f m ) |(f m ) (z)|<1 log[z, c] = z∈Fix(f m ) |(f n ) (z)|<r 1 m c∈Crit(f ) m-1 j=0 log[f j (z), c] -log |(f m ) (z)| ≤ (2d -2)m • B 1 (f ),
where the last inequality holds since f has at most (2d -2)m attracting periodic points of period dividing m. Hence recalling the definition of σ 1 (n), we have

1 d n m|n µ n m (d m + 1)u m,n (f, 1) - 1 d n m|n µ n m c∈Crit(f ) log[f m (c), c]- z∈Fix(f m ) |(f m ) (z)|<1 log[z, c] ≤ (2d-2)B 1 (f ) σ 1 (n) d n .
We finally reduced the proof of Theorem 3.1 to estimating

δ n (f ) = δ n (f, 1) := 1 d n m|n µ n m c∈Crit(f ) log[f m (c), c] - z∈Fix(f m ) |(f m ) (z)|<1 log[z, c] .
(iii) We claim that for any c ∈ Crit(f ) and any m ∈ N * dividing n,

log[f m (c), c] - z∈Fix(f m ) |(f m ) (z)|<1 log[z, c] ≤ 2(2d -2)m 2 log M (f ) -log κ 2 ,
where κ ∈ (0, 1) is the absolute constant appearing in Lemma 2.2; recall that sup z,w∈P

1 [z, w] ≤ 1. Assume first that κ • M (f ) -m ≤ [f m (c), c].
Then by Lemma 3.5, we deduce that for any

z ∈ Fix(f m ), κ • 2 -1 M (f ) -2m ≤ 2 -1 M (f ) -m [f m (c), c] ≤ [z, c]
, so that since f has at most (2d -2)m attracting periodic points of period dividing m, we have

-m log M (f ) + log κ ≤ log[f m (c), c] - z∈Fix(f m ) |(f m ) (z)|<1 log[z, c], and log[f m (c), c] - z∈Fix(f m ) |(f m ) (z)|<1 log[z, c] ≤(2d -2)m 2m log M (f ) -log κ 2 ≤2(2d -2)m 2 log M (f ) -log κ 2 . Assume next that κ • M (f ) -m > [f m (c), c]
. By Lemma 2.2 applied to the trivial family (f ) and its (constant) marked critical point c (recall that the constant κ given by Lemma 2.2 depends only on d), c belongs to the immediate basin of an attracting periodic point z 0 of f of period k dividing m, and we have

2[f m (c), c] ≥ [z 0 , c] and [c, J f ] ≥ κM (f ) -m . Hence we have -log 2 ≤ log[f m (c), c] -log[z 0 , c], so that -log 2 ≤ log[f m (c), c] - z∈Fix(f m ) |(f m ) (z)|<1 log[z, c].
Noting that any attracting z ∈ Fix(f m )\{z 0 } lies in a Fatou component of f which does not contain z 0 , we also have 1

≥ [z, c] ≥ [c, J f ] ≥ κM (f ) -m for every such z ∈ Fix(f m ) \ {z 0 } that |(f m )(z)| < 1. Moreover, by Lemma 3.5, we have [f m (c), c] ≤ M (f ) m [z 0 , c]. Hence, since f has at most (2d -2)m attracting periodic points of period dividing m, we have log[f m (c), c] - z∈Fix(f m ) |(f m ) (z)|<1 log[z, c] ≤ (2d -2)m • (m log M (f ) -log κ) ≤ (2d -2)m 2 (log M (f ) -log κ) .
Hence the claim holds.

Since f has exactly 2d -2 critical points taking into account their multiplicities, letting

C 2 := 2(2d -2) 2 max {1, | log(κ/2)|}, we have |δ n (f )| ≤ C 2 • (log M (f ) + 1) σ 2 (n) d n , by the definition of σ 2 (n). (iv) Recall that 1 2 log d ≤ L(f ) = P 1 log(f # ) µ f ≤ log(sup P 1 f # ) and that by definition of d n , d n = m|n µ n m (d m + 1) ≥ m|n µ n m d m ≥ 1 -d -1 d n .
Hence, all the above intermediate estimates yield

1 d n 2π 0 log p n (f, re iθ ) dθ 2π -L(f ) ≤ B 3 (f, r) σ 2 (n) d n (6) for any f ∈ X, where B 3 (f, r) = C 3 (sup P 1 |g f | + log(sup P 1 f # ) + | log r|)
for some constant C 3 > 0 depending only on d. Since both sides of (6) depend continuously on f ∈ Rat d and X is dense in Rat d , the above estimate (6) still holds for any f ∈ Rat d .

Application: degeneration of the Lyapunov exponent

Consider a holomorphic family (f t ) t∈D * of degree d > 1 rational maps parametrized by the punctured unit disk, and assume it extends to a meromorphic family over D, i.e.

f t ∈ O(D)[t -1 ](z).
Theorem 3.6. -There exists a non-negative α ∈ R such that, as t → 0,

L(f t ) = α • log |t| -1 + o(log |t| -1 ).
This is a special case of [F, Theorem C] and can also be obtained as the combination of [DeM3, Proposition 3.1] and [START_REF] Demarco | Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity[END_REF]Theorem 1.4]. We provide here a simple proof as an application of Theorem 3.1.

Proof. -We can write p n (f t , w) = t -Nn h n (t, w), where h n : D × C → C is holomorphic and N n ∈ N. We rely on the following key lemma.

Lemma 3.7. -There exist C 1 , C 2 > 0 such that for any t ∈ D * 1/2 , sup z∈P 1 max |g ft (z)|, log(f # t (z)) ≤ C 1 log |t| -1 + C 2 .
Once Lemma 3.7 is at our disposal, by Theorem 3.1, there is C > 0 such that for any n ∈ N * and any

t ∈ D * 1/2 , L(f t ) - 1 d n 2π 0 log |p n (f t , e iθ )| dθ 2π ≤ C C 1 log |t| -1 + C 2 • σ 2 (n) d n ,
so that dividing both sides by log |t| -1 and making t → 0, there is

C > 0 such that for all n ∈ N * N n d n -C σ 2 (n) d n ≤ lim inf t→0 L(f t ) log |t| -1 ≤ lim sup t→0 L(f t ) log |t| -1 ≤ N n d n + C σ 2 (n) d n .
Indeed, as p n (f t , w) = t -Nn h n (t, w), where h n is analytic, we get

1 log |t| -1 2π 0 log |p n (f t , e iθ )| dθ 2π = N n + 2π 0 log |h n (f t , e iθ )| log |t| -1 dθ 2π = N n + o(1) as t → 0. Making n → ∞, we get lim t→0 L(f t ) log |t| -1 = lim n→∞ N n d n =: α ≥ 0.
This concludes the proof.

Proof of Lemma 3.7. -There is a meromorphic family (F t ) t∈D of homogeneous polynomial endomorphisms of C 2 such that for every t ∈ D * , F t is a lift of f t and that the holomorphic function t → Res(F t ) on D vanishes only at t = 0. According to [START_REF] Demarco | Bifurcations, intersections, and heights[END_REF]Lemma 3.3] (or [START_REF] Favre | Degeneration of endomorphisms of the complex projective space in the hybrid space[END_REF]Proposition 4.4]), there exist constants C ≥ 1 and β > 0 such that for any p ∈ C 2 \ {0} and any t ∈ D * ,

1 C |t| β ≤ F t (p) p d ≤ C. (7)
For any t ∈ D * , set u t (z) := log( F t (p) / p d ) on P 1 , where p ∈ π -1 (z). The function u t on P 1 is well-defined by the homogeneity of F t . Recalling the definition of g Ft , we have

g Ft (z) = ∞ n=0 (u t • f n t (z)
)/d n+1 uniformly on P 1 , so that by (7), sup

z∈P 1 |g Ft (z)| ≤ 1 d -1 sup z∈P 1 |u t (z)| ≤ 1 d -1 β log |t| -1 + log C .
Recalling the definition of I g F t and the formula To conclude the proof, we use the same strategy for giving an upper bound for log sup z∈P 1 f # t (z). Recall the following formula

I g F t = -(log |Res(F t )|)/(d(d -1)),
f # t (z) = 1 d |det DF t (p)| p 2 F t (p) 2
on P 1 , where p ∈ π -1 (z) (see, e.g., [START_REF] Jonsson | Sums of Lyapunov exponents for some polynomial maps of C 2 . Ergodic Theory Dynam[END_REF]Theorem 4.3]) and in particular, by (7), we have

log(f # t (z)) ≤ log |det DF t (p)| d • p 2d-2 + 2 β log |t| -1 + log C . Now write as F t = (P t , Q t ), P t (z, w) = d j=0 a j (t)z j w d-j , Q t (z, w) = d j=0 b j (t)z j w d-j with a j (t) = t -γ ãj (t) and b j (t) = t -γ bj (t)
for some γ ∈ N * and some ã, b ∈ O(D). In particular, there exists a constant C ≥ 1 such that for any t ∈ D(0, 1/2) and any 0 ≤ j ≤ d, we have max{|ã

j (t)|, | bj (t)|} ≤ C , so that | det DF t (p)| = ∂P t ∂z (p) ∂Q t ∂w (p) - ∂P t ∂w (p) ∂Q t ∂z (p) ≤ 2d 2 |t| -2γ d j, =0 |ã j (t) b (t)| • p 2(d-1) ≤ 2d 4 • C 2 p 2(d-1) • |t| -2γ
for any p ∈ C 2 . This gives a constant C ≥ 1 so that log(f # t (z)) ≤ 2(β+γ) log |t| -1 +log C on P 1 , which completes the proof.

Equidistribution towards the bifurcation currents

Let Λ be a quasi-projective variety such that, either Λ ⊂ M d , or parametrizing an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 .

The proof of Theorem A

Pick any compact subset K in Λ, and set

C 1 (K) := sup λ∈K C([f λ ]) ≥ 1 2 log d, where C([f λ ]
) is given by Theorem 3.1. We remark that for every n ∈ N * and every ρ ∈]0, 1],

T 1 n (ρ) := 1 d n 2π 0 [Per n (ρe iθ )] dθ 2π = dd c 1 d n 2π 0 log |p n (λ, ρe iθ )| dθ 2π .
Pick any 1 ≤ p ≤ min{m, 2d-2}, any n = (n 1 , . . . , n p ) ∈ (N * ) p , and any ρ = (ρ 1 , . . . , ρ p ) ∈ ]0, 1] p . Assume first that p = 1, i.e. n = n ∈ N * and ρ = ρ ∈]0, 1], and pick any continuous DSH (m -1, m -1)-form Ψ on Λ supported in K. By definition, we can write dd c Ψ = T + -T -, where T ± are positive measures of finite masses on Λ. By the Stokes theorem and Theorem 3.1, we have

T 1 n (ρ) -T bif , Ψ = K 1 d n 2π 0 log |p n (λ, ρe iθ )| dθ 2π -L(λ) dd c Ψ ≤ K 1 d n 2π 0 log |p n (λ, ρe iθ )| dθ 2π -L(λ) (T + + T -) ≤ AC 1 (K) (1 + |log ρ|) σ 2 (n) d n ( T + + T -)
, which completes the proof of Theorem A in this case by the definition of Ψ * DSH . We now assume that 2 ≤ p ≤ min{m, 2d -2}. Setting S j = S j (n, ρ) := ( 1≤ <j T bif ) ∧ ( j<k≤p T n k (ρ k )) for any 1 ≤ j ≤ p, which is a positive closed current of bidegree (p -1, p -1) on Λ, we have

T p n (ρ) -T p bif = p j=1 S j ∧ (T n j (ρ j ) -T bif ). (8)
Pick any continuous DSH (m -p, m -p)-form Ψ on Λ supported in K, and write dd c Ψ = T + -T -where T ± are positive closed (m -p + 1, m -p + 1) currents of finite masses on Λ. Then by Stokes formula, we have

T p n (ρ) -T p bif , Ψ = p j=1 S j ∧ (T 1 n j (ρ j ) -T bif ), Ψ = p j=1 K 1 d n 2π 0 log |p n (λ, ρe iθ )| dθ 2π -L(λ) S j ∧ dd c Ψ.
Since the masses can be computed in cohomology, by (4), there is C 2 > 0 independent of K, Ψ and T ± , such that for every 1

≤ j ≤ p, Λ S j ∧ (T + + T -) ≤ C 2 ( T + + T -).
Then by Theorem 3.1, we have

K 1 d n 2π 0 log |p n (λ, ρe iθ )| dθ 2π -L(λ) S j ∧ dd c Ψ ≤ AC 1 (K) (1 + | log ρ j |) σ 2 (n j ) d n j Λ S j ∧ (T + + T -) ≤AC 1 (K)C 2 (1 + | log ρ j |) σ 2 (n j ) d n j ( T + + T -)
for any 1 ≤ j ≤ p, which completes the proof of Theorem A.

Remark. -As in [START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF], we deduce from Theorem A the density in the support of T p bif of parameters having p distinct neutral cycles. We can actually give a more precise statement: taking any sequence of p-tuple of integers (n k ) in (N * ) p such that min j n j,k → ∞, we have that the set of parameters λ such that f λ has p distinct neutral cycles of exact periods n 1,k , . . . , n p,k for some k ∈ N * are dense in the support of T p bif .

The proof of Corollary B

Pick 1 ≤ p ≤ min{m, 2d -2}. We recall some basics on PB measures. For each ρ > 0, let λ Sρ the Lebesgue probability measure on the circle S ρ . Let θ : R + → R + be a smooth function with compact support in ]0, 1[ such that 1 0 θ(x)dx = 1. We consider the smooth measure ν defined as

ν := p j=1 1 0 λ Sρ j θ(ρ j )dρ j .
We say that a probability measure ν on (P 1 ) p is PB (or has bounded potential) if there exists a constant C ≥ 0 such that

| ν -ν, ϕ | ≤ C ϕ * DSH
for all ϕ which is DSH on (P 1 ) p , and then let C ν ≥ 0 be the minimal C ≥ 0 satisfying the above inequality for every ϕ. For example, ν is PB on (P 1 ) p , and λ Sρ is PB on P 1 . We claim that the positive closed (p, p)-current

T p n (ν) := 1 d |n| (P 1 ) p p j=1 [Per n j (w j )]ν(w 1 , • • • , w p )
on Λ is well-defined for any PB measure ν on (P 1 ) p and any n = (n 1 , . . . , n p ) ∈ (N * ) p . Indeed, the set of all w = (w 1 , . . . , w p ) ∈ C p such that p i=1 Per n i (w i ) is not of pure codimension p in Λ is analytic. Hence for any w = (w 1 , . . . , w p ) ∈ C p except for a pluripolar subset and any n = (n 1 , . . . , n p ) ∈ (N * ) p , the current p i=1 [Per n i (w i )] on Λ is well defined. In particular, since PB measures on (P 1 ) p give no mass to pluripolar sets, the current T p n (ν) is also well defined. Observe that T p n (ν) give no mass to pluripolar sets (hence to analytic sets) since it has bounded potentials. Arguing as above shows that for any w = (w 1 , . . . , w p ) ∈ C p outside a pluripolar subset and any n = (n 1 , . . . , n p ) ∈ (N * ) p , the current p i=1 [Per n i (w i )] is well-defined. So for any PB measures on (P 1 ) p , T p n (ν) gives no mass to analytic sets. Here is another description of (d |n| ) -1 p i=1 [Per n i (w i )] and T p n (ν); let Γ n be the analytic set of dimension m in Λ × (P 1 ) p defined as

Γ n := {(λ, (z 1 , . . . , z p )) ∈ Λ × (P 1 ) p : z j ∈ Fix * (f n j λ ) for every 1 ≤ j ≤ p}. Let F n : Γ n → (P 1
) p be a holomorphic map defined by

F n (λ, z 1 , . . . , z p ) = ((f n 1 λ ) (z 1 ), . . . , (f np λ ) (z p ))
, and P : Γ n → Λ be the restriction to Γ n of the projection Λ × (P 1 ) p → Λ. Consider π : Γ n → Γ n a desingularization of Γ n . The map F n := F n • π is holomorphic and the map P := P • π is an analytic map. If ν is a smooth PB measure in (P 1 ) p , then:

(9) T p n (ν) = 1 j n j d n j P * F n | Γn * (ν) on Λ.
Indeed, observe that, when testing against a smooth form, there is always one term that is smooth when computing the pull-back and push-forward.

Theorem 4.1. -Let Λ be a quasi-projective variety such that, either Λ ⊂ M d , or parametrizing an algebraic family (f λ ) λ∈Λ of degree d rational maps on P 1 . Then for any compact subset K in Λ, there exists C(K) > 0 such that for any 1 ≤ p ≤ min{m, 2d -2}, any n = (n 1 , . . . , n p ) ∈ (N * ) p , any PB measure ν in (P 1 ) p , and any continuous DSH-form Ψ of bidegree (m -p, m -p) on Λ supported in K, we have

T p n (ν) -T p bif , Ψ ≤ C(K) • (1 + C ν ) max 1≤j≤p 1 n j Ψ DSH .
Proof.

-Pick 1 ≤ p ≤ min{m, 2d -2} and a PB measure ν on (P 1 ) p . Consider first the case where ν is smooth. Pick n = (n 1 , . . . , n p ) ∈ (N * ) p and a smooth DSH form Ψ of bidegree (m -p, m -p) on Λ with compact support in K. By Theorem A and our choice of ν, there is C(K) > 0 depending only on K such that

T p n (ν) -T p bif , Ψ ≤ C(K) max 1≤j≤p σ 2 (n j ) d n j Ψ DSH ,
and we will show that

T p n (ν) -T p n (ν), Ψ ≤ C(K)C ν max 1≤j≤p 1 n j Ψ DSH .
By the above description of T p n (ν) and the definition of PB measures, we have

| T p n (ν) -T p n (ν), Ψ | ≤ C ν 1 j n j d n j F n * P * (Ψ) * DSH .
As taking dd c commutes with taking pull-pack or push-forward, writing as dd c Ψ = T + -T -, where T ± are smooth (because Ψ is smooth) positive closed currents of bidegree (m -p + 1, m -p + 1) of finite masses on Λ, one simply has to estimate the mass ( F ) * ( P * (T ± )) . Computing those masses can be done in cohomology testing against i =j ω i for all 1 ≤ j ≤ p, where ω i is the Fubini Study form on the i-th factor of (P 1 ) p . Set S n,j := p =1 n d n -1 P * F n * i =j ω i . By duality, this computation is the same as controlling S n,j , T ± for any j. Finally, for any j, one has to control the mass S n,j . By symmetry, consider the case where j = p. Let n = (n 1 , . . . , n p-1 ) and consider the associated map F n . Now take a generic point (z 0 1 , . . . z 0 p-1 ) ∈ P 1 p-1 and consider the line

L := {z = (z 1 , . . . , z p ) ∈ P 1 p , ∀i ≤ p - 1, z i = z 0 i }. Then the degree of F n -1 (L) equals d np times the degree of F n -1 (z 0 1 , . . . z 0 p-1 )
). So pushing-forward, we see that

S n,p ≤ C 1 n p ,
for some constant C ≥ 0 that does not depend on n.

In particular, we deduce that

p j=1 (n j d n j ) -1 F n * ( P * (Ψ)) * DSH ≤ C • max 1≤j≤p 1 n j Ψ DSH ,
where C ≥ 0 is (another) constant that does not depends on n, which implies the wanted result for Ψ and ν smooth. By a regularization argument [START_REF] Dinh | Super-potentials of positive closed currents, intersection theory and dynamics[END_REF], the result follows for Ψ continuous, replacing C(K) by a constant given by a (small) larger neighborhood of K.

Finally, we extend the result to any PB measure ν using again an approximation of ν.

Corollary B follows from Theorem 4.1 using classical pluripotential techniques as in [START_REF] Dinh | Distribution des valeurs de transformations méromorphes et applications[END_REF] or [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF].

Remark. -The order O(max j (n -1 j )) as min j (n j ) → ∞ in the right-hand side is sharp. Indeed, for the quadratic polynomials family (z 2 + λ) λ∈C , it has been shown in [BG] that the sequence (2 -n+1 [Per(n, e 2n )]) n (recall 2 n ∼ 2 n as n → ∞) of measures on C converges to dd c max{g, 4 -2 log 2}, where g is the Green function of the Mandelbrot set. Since 4 -2 log 2 > 0, this measure is not proportional to µ bif . On the other hand, if ν n = λ S e 2n we have C νn = O(n) as n → ∞, where λ S e 2n is the probability Lebesgue measure on the circle of center 0 and radius e 2n in C, which is PB. So one cannot improve the order O(n -1 ) as n → ∞ in the right-hand side for this family; otherwise, 2 -n [Per(n, e 2n )] would tend to µ bif as n → ∞.

Transversality of periodic critical orbit relations

Infinitesimal deformations of rational maps

Pick f ∈ Rat d . The orbit O(f ) := {φ -1 • f • φ ∈ Rat d : φ ∈ PSL 2 (C)} of f under the conjugacy action of PSL 2 (C) on Rat d is a 3 dimensional complex analytic submanifold in Rat d .
A tangent vector to Rat d at f is an equivalence class of holomorphic maps φ : D → Rat d such that φ(0) = f under the relation φ ∼ ψ iff φ (0) = ψ (0). The vector space of all tangent vectors at f is denoted by T f Rat d . A tangent vector ζ ∈ T f Rat d can be identified to a section of the line bundle f * (T P 1 ), where T P 1 denotes the holomorphic tangent bundle on P 1 . More specifically, to any tangent vector ζ ∈ T f Rat d , we attach a rational vector field η ζ on P 1 whose poles are in Crit(f ) by letting

η ζ (z) := -D z f -1 • ζ(z) ∈ T z P 1 , z ∈ P 1 .
If f has only simple critical points, then η ζ also has only simple poles (see [BE] for more details).

If f is postcritically finite, i.e., the postcritical set

P(f ) := n∈N * f n (Crit(f ))
of f is a finite subset in P 1 , then we denote by T (P(f )) the vector field on P(f ), and a vector field τ ∈ T (P(f )) is said to be guided by

ζ ∈ T f Rat d if τ = f * τ + η ζ on P(f ) and τ • f = ζ on Crit(f ).
For the sequel, we will rely on the following crucial result (see [BE, FG]).

Proposition 5.1 (Buff-Epstein).

-If f ∈ Rat d is postcritically finite and neither is conjugate to z ±2 nor is a Lattès map, then a tangent vector ζ ∈ T f Rat d is tangent to O(f ) if and only if there is a vector field τ ∈ T (P(f )) guided by ζ.

A transversality of periodic critical orbit relations

Let f ∈ Rat d be postcritically finite and hyperbolic of disjoint type, and let c 1 , . . . , c 2d-2 be 2d -2 distinct critical points of f . For any 1 ≤ i ≤ 2d -2, there is p i ∈ N * such that c i ∈ Fix * (f p i ), and there is an open neighborhood U of f in Rat d small enough so that c 1 , . . . , c 2d-2 can be followed holomorphically on U , that is for any 1 ≤ i ≤ 2d -2, there is a holomorphic map c i : U → P 1 such that c i (f ) = c i and that c i (g) ∈ Crit(g) for every g ∈ U .

We can choose an atlas of P 1 such that there is an affine chart of P 1 containing c 1 (g), . . . , c 2d-2 (g) for every g ∈ U , and define a map V : U -→ C 2d-2 by V(g) := g p 1 (c 1 (g)) -c 1 (g), . . . , g p 2d-2 (c 2d-2 (g)) -c 2d-2 (g) , g ∈ U.

We will need the following.

Theorem 5.2. -Let f ∈ Rat d be postcritically finite and hyperbolic of disjoint type. If f is not conjugate to z ±2 , then the linear map

D f V : T f Rat d → T 0 C 2d-2 is surjective and ker(D f V) = T f O(f ).
Though this result seems folklore, we could not find it in the above form in the literature. We provide here a proof for the sake of completeness, which is very much inspired by [BE,[START_REF] Epstein | Transversality principles in holomorphic dynamics[END_REF] (see also [FG]).

The proof of Theorem 5.2

From now on, we write u := du t dt t=0

for any holomorphic map t → u t defined on a disk D.

Proof of Theorem 5.2. -Under our assumption, the postcritically finite map f is neither a Lattès map, nor conjugate to z ±2 . Let us pick ζ ∈ ker(D f V), and choose a holomorphic disc t → f t ∈ Rat d with f 0 = f and such that ḟ = ζ. We shall use Proposition 5.1 and build a vector field τ ∈ T (P(f )) which is guided by ζ. Then counting dimensions will complete the proof. For any n ∈ N and any 1 ≤ i ≤ 2d -2, we set c i (t) = c i (f t ),

v n,i (t) := f n t (c i (t)), c i := c i (0), and v n,i := v n,i (0). It is clear that for any n ≥ 0, we have

vn+1,i = ζ(v n,i ) + D v n,i f • vn,i . (10)
We shall deduce the following from this equation.

Lemma 5.3. -Fix 1 ≤ i ≤ 2d -2. For all n, m ∈ N * , if v n,i = v m,i , then vn,i = vm,i .
Taking this result for granted , we continue to define a vector field τ on P(f ) that is guided by ζ. For any x ∈ P(f ), we set τ (x) := vn,i for some 1 ≤ i ≤ 2d -2 and some n ∈ N such that x = v n,i . Since f is of disjoint type, the previous lemma shows that τ is well-defined at x. It remains to check that τ is guided by ζ. The equality τ (f (c i )) = ζ(c i ) follows from the definition of τ and (10). When x = v n,i is not a critical point, then multiplying (10) by

D x f -1 gives τ = f * τ + η ζ at x. When x = c i is a critical point, since
x is a simple critical point, we may choose coordinates z at c i and w at f

(c i ) such that w = f t (z) = z 2 + t(a + O(z)) + O(t 2
). Since we may follow the critical point for |t| small, we may also suppose that c i (t) = 0 for all t so that f t (z) = z 2 + t(a + O(z 2 )) + O(t 2 ). We thus obtain ζ(z) = (a + O(z)) ∂ ∂w , and

η ζ (z) = (-a 2z + O(z)) ∂ ∂z .
Observe that in our coordinates we have τ (c i ) = ċi = 0, and τ (P (c i )) = d dt t=0 f t (c i (t)) = a ∂ ∂z . We may thus extend τ locally at c i and P (c i ) holomorphically by setting τ (z) ≡ 0 and τ (w) ≡ a. It follows that

f * τ (z) + η ζ (z) -τ (z) = a f (z) ∂ ∂z + - a 2z + O(z) ∂ ∂z -0 = O(z) ∂ ∂z .
It follows that f * τ + η ζ = τ at any critical point, which concludes the proof.

Proof of Lemma 5.3. -To simplify notation we write v k , c, p instead of v k,i , c i , p i respectively. Recall that p is a multiple of the exact period of v 0 = c. For any l ≥ 1, iterating the assertion (10) and using the fact that D(f p ) is vanishing at all points of the cycle, so in particular D v (l-1)p f p = 0, and that

v k+p = v k for all k ≥ 0, give vlp = ζ(v lp-1 ) + D v lp-1 f • ζ(v lp-2 ) + • • • + D v (l-1)p+1 f p-1 • ζ(v (l-1)p ) + D v (l-1)p f p • v(l-1)p = ζ(v lp-1 ) + D v lp-1 f • ζ(v lp-2 ) + • • • + D v (l-1)p+1 f p-1 • ζ(v (l-1)p ) = ζ(v p-1 ) + D v p-1 f • ζ(v p-2 ) + • • • + D v 1 f p-1 • ζ(v 0 ) = vp Since ζ ∈ ker(D f V), we also have v0 -vp = ċ -vp = D f V • ζ = 0 , whence vlp = v0 for all l ≥ 1 . Again by (10) we get vlp+1 = ζ(v lq ) + D v lp f • v lp = ζ(v 0 ) + D v 0 f • v 0 = v1 .
An immediate induction on k ≥ 0 then proves vlp+k = vk for all l ≥ 0. This proves the lemma.

Application to the space Rat 0 d

Denote by Rat 0 d the space of degree d rational maps on P 1 fixing 0, 1 and ∞. To be more precise, let us parametrize Rat d by Proof.

f ([z : t]) = d i=0 a i z i t d-i : d i=0 b i z i t d-i , [z : t] ∈ P 1 , with [a d : • • • : a 0 : b d : • • • : b 0 ] ∈ P 2d+1 \ {Res = 0}.
-Let ζ ∈ T f O(f ) ∩ T f Rat 0 d .
Then there exists a holomorphic germ m t ∈ Aut(P 1 ) centered at m 0 = id such that f t = m -1 t • f • m t and ζ = ḟ . Moreover, since (f t ) t is tangent to Rat 0 d , we can assume there are fixed points of f t satisfying x t ≡ 0, y t ≡ 1 and

z t ≡ ∞. Writing m t (z) = (a t z + b t )/(c t z + d t ) with a t d t -c t b t = 1, we get x t = -b t /a t , y t = (b t -d t )/(c t -a t ) and z t = -d t /c t . As m 0 = id, we have a t = 1+αt+O(t 2 ), b t = βt+O(t 2 ), c t = γt + O(t 2 ) and d t = 1 + δt + O(t 2 ).
We thus get -βt

+ O(t 2 ) ≡ 0, -γt + O(t 2 ) ≡ 0, 1 + (α + δ)t + O(t 2 ) = 1 and 1 + (δ - β + γ -α)t + O(t 2 ) ≡ 1, whence α = β = γ = δ = 0. As a consequence, m t = id + O(t 2 ) and m -1 t = id + O(t 2 ). Finally, differentiating f t = m -1 t • f • m t with respect to t and evaluating at t = 0 gives ζ = Df • ṁ = Df • 0 = 0 . This proves T f O(f ) ∩ T f Rat 0 d = {0}.
As above, we pick f ∈ Rat 0 d which is hyperbolic and postcritically finite with simple critical points c 1 , . . . , c 2d-2 . We also assume that for 1 ≤ i ≤ 2d -2, there exists p i ≥ 1 such that c i ∈ Fix * (f p i ). Let U ⊂ Rat 0 d be a neighborhood of f in which c i can be followed holomorphically as a critical point c i (g) of g for all i. We can choose an atlas of P 1 such that there is an affine chart containing c 1 (g), . . . , c 2d-2 (g) for every g ∈ U . We let

V(g) := (g p 1 (c 1 (g)) -c 1 (g), . . . , g p 2d-2 (c 2d-2 (g)) -c 2d-2 (g)) , g ∈ U.
From Theorem 5.2 and Lemma 5.4, we directly get the following. A fixed marked degree d rational map on P 1 is a (d + 2)-tuple (f, x 1 , . . . , x d+1 ) where f ∈ Rat d , and (x 1 , . . . , x d+1 ) ∈ (P 1 ) d+1 is a (d + 1)-tuple of all the fixed points of f , taking into account their multiplicities. A totally marked degree d rational map on P 1 is a 3d-tuple (f, x 1 , . . . , x d+1 , c 1 , . . . , c 2d-2 ) where (f, x 1 , . . . , x d+1 ) is a fixed marked rational map and (c 1 , . . . , c 2d-2 ) ∈ (P 1 ) 2d-2 is a (2d -2)-tuples of all critical points of f , taking into account their multiplicities.

Let Rat 0,fm d be the space of all fixed marked degree d rational maps (f, x 1 , . . . , x d+1 ) such that x 1 = 0, x 2 = 1 and x 3 = ∞. It is clear that Rat 0,fm d is smooth and quasiprojective. Let also Rat 0,tm d be the space of all totally marked degree d rational maps (f, x 1 , . . . , x d+1 , c 1 , . . . , c 2d-2 ) such that x 1 = 0, x 2 = 1 and x 3 = ∞. It is clear that both Rat 0,fm d and Rat 0,tm d are smooth and quasi-projective of dimension 2d -2. In both spaces, the action by conjugation of PSL 2 (C) extends naturally, by respecting the marking. Note that, in both cases, two tuples which are distinct cannot be conjugated, since an element in PSL 2 (C) fixes three distinct points if and only if it is the identity. Moreover, the conjugacy class of any marked tuple (f, ) where f has no multiple fixed point admits a representative in with x 1 = 0, x 2 = 1 and x 3 = ∞.

We finally let M fm d be the quotient of this action on the space of fixed marked rational maps. The space M fm d is an irreducible quasi-projective variety of dimension 2d -2 and its singular points are contained in the subvariety of M fm d consisting of all classes [(f, x 1 , . . . , x d+1 )] such that #{x 1 , . . . , x d+1 } ≤ 2. In particular, M fm d is smooth at any class [(f, x 1 , . . . , x d+1 )] such that f is hyperbolic. By the above, it is clear that the natural projection Rat 0,fm

d \ Per 1 (1) → M fm d \ Per 1 (1)
is actually a bijection. We note that the same construction of L, µ bif , Per n (w), and T p n (ρ) works on all the spaces introduced above (and even more generally) exactly as in the case of M d .

Parameterizing hyperbolic components of

Rat 0,fm d of disjoint type
Let Ω be a hyperbolic component in Rat 0,fm d . If J f is connected for any (f, x 1 , . . . , x d+1 ) ∈ Ω, then Ω is simply connected and contains a center, which is by definition the unique point (f, x 1 , . . . , x d+1 ) ∈ Ω such that #P(f ) < ∞, by [START_REF] Milnor | Hyperbolic components[END_REF]Theorem 9.3].

For any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 and any hyperbolic component Ω in Rat 0,fm d of type n, J f is connected for any (f, x 1 , . . . , x d+1 ) ∈ Ω, since all Fatou components of f are then topological disks by [START_REF] Przytycki | Remarks on the simple connectedness of basins of sinks for iterations of rational maps[END_REF]Proposition p.231]. In particular Ω has a center. We will also use the following in the sequel. Lemma 6.1. -For any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 , any hyperbolic component Ω in Rat 0,fm d of type n is simply connected and the fixed points and critical points are marked throughout Ω. More precisely, there are a holomorphic maps x 1 , . . . , x d+1 , c 1 , . . . , c 2d-2 :

Ω → P 1 such that Fix(f λ ) = {x 1 (λ), . . . , x d+1 (λ)} and Crit(f λ ) = (c 1 (λ), . . . , c 2d-2 (λ)) for any λ ∈ Ω.
Proof. -We have already seen that Ω is simply connected. Let τ : Rat 0,tm d → Rat 0,fm d be the natural finite branched cover. For any component Ω of τ -1 (Ω), Ω is a hyperbolic component of disjoint type in Rat 0,tm d and τ | Ω : Ω → Ω is an unramified cover, so is a biholomorphism, since Ω is simply connected.

In particular, we have a holomorphic map

(τ | Ω) -1 : Ω λ → (f λ , x 1 (λ), . . . , x d+1 (λ), c 1 (λ), . . . , c 2d-2 (λ)) ∈ Ω
and the holomorphic maps x 1 , . . . , x d+1 , c 1 , . . . , c 2d-2 follow all the fixed points and critical points of f λ , respectively.

Pick any hyperbolic component Ω in Rat 0,fm d of type n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 , and let x 1 , . . . , x d+1 , c 1 , . . . , c 2d-2 : Ω → P 1 be the marking of all the fixed points and critical points of (f λ ) λ∈Ω given by Lemma 6.1. For any i ∈ {1, . . . 2d -2} and any λ ∈ Ω, let w i (λ) ∈ D be the multiplier of the attracting cycle C i (λ) of f λ of exact period n i and whose immediate attractive basin contains c i (λ). The multiplier map W Ω : Ω → D 2d-2 on Ω is defined by W Ω (λ) := (w 1 (λ), . . . , w 2d-2 (λ)), λ ∈ Ω.

Let λ Ω be the center in Ω. Noting also that #P(f λ ) < ∞ for any λ ∈ W -1 {0}, we have

W -1 {0} = {λ Ω }. Theorem 6.2. -The map W Ω : Ω → D 2d-2 is a biholomorphism.
Proof. -Write W for W Ω . First, we prove that W is surjective and finite. According to [START_REF] Grauert | of Grundlehren der Mathematischen Wissenschaften[END_REF]§3 page 179], this implies that W is a finite and possibly ramified covering. Next, we show that W is locally invertible at λ Ω . Since W -1 {0} = {λ Ω }, this implies W has degree 1, i.e., is a biholomorphism. Let us first prove that W is surjective. We proceed using the classical surgery argument: for any 0 < ε < 1, we construct a continuous map σ : D(0, 1 -ε) 2d-2 → Ω such that W • σ = id. We sketch the construction referring to [START_REF] Carleson | Complex dynamics. Universitext: Tracts in Mathematics[END_REF]Theorem VIII.2.1] or [Do] for detail.

Choose λ = (f, x 1 , . . . , x d+1 ) ∈ Ω and for any 1 ≤ i ≤ 2d -2, let U 1,i , . . . , U n i ,i be all the components of the immediate basin of the attracting cycle

C i (λ) such that c i (λ) ∈ U 1,i . Since c i (λ) is a simple critical point of f , U 1,i is simply connected and there exists a conformal map ϕ i : U 1,i → D such that ϕ i • f n i • ϕ -1 i (ξ) = ξ • ξ + w i 1 + wi ξ , |ξ| < 1,
where w i := w i (λ). Fix ∈ (0, 1). For any ρ = (ρ 1 , . . . , ρ 2d-2 ) ∈ D(0, 1 -ε) 2d-2 , we can define a continuous map fρ by setting fρ = f outside the union of all U j,i , and such that

ϕ i • f n i ρ • ϕ -1 i (ξ) = ξ • ξ+ρ i 1+ ρi ξ
on the open disk |ξ| < 1 -r containing the critical point of the Blashke product in the right hand side. Notice that ( fρ ) ρ is a continuous family of quasiregular maps of P 1 . We now solve the Beltrami equation for the unique Beltrami form which is 0 on the complement the U j,i 's and invariant under fρ : there is a continuous family of quasiconformal homeomorphism ψ ρ :

P 1 → P 1 such that f ρ := ψ ρ • fρ • ψ -1
ρ is a rational map and depends again continuously on ρ and that

ψ ρ (x 1 (λ)) = 0, ψ ρ (x 2 (λ)) = 1, and ψ ρ (x 3 (λ)) = ∞. Then the d + 2 tuple (ψ ρ • fρ • ψ -1 ρ , ψ ρ (x 1 (λ)), . . . , ψ ρ (x d+1 (λ))
) lies in Ω by the above continuous dependence and is mapped to ρ by W by the chain rule.

Let us show that W is finite, i.e., #W -1 (w) < ∞ for any w = (w 1 , . . . , w 2d-2 ) ∈ D 2d-2 . Suppose to the contrary that for some w ∈ D 2d-2 , #W -1 (w) = ∞. Then there is an infinite set contained in i Per n i (w i ) ∩ Ω. In particular, the quasi-projective subvariety Λ := i Per n i (w i ) has dimension > 0 and any λ ∈ Λ has 2d -2 distinct attracting cycles of respecting periods n 1 , . . . , n 2d-2 . The holomorphic family (f λ ) λ∈Λ thus has no bifurcations. Since J f λ = P 1 for some λ ∈ Λ, (f λ ) λ∈Λ is not a family of Lattès maps. Hence by [START_REF] Mcmullen | Families of rational maps and iterative root-finding algorithms[END_REF]Theorem 2.2], (f λ ) λ∈Λ is trivial. Since the natural projection Rat 0,fm d → M d has finite fibers, thi implies that the quasi-projective variety Λ is a finite set. This is a contradiction.

Let us finally see the local invertibility of W at λ Ω . Since p is a biholomorphism on Ω, the map W is locally invertible at λ Ω if and only if W is locally invertible at

a := (f (λ Ω ), x 1 (λ Ω ), . . . , x d+1 (λ Ω )) ∈ Rat 0,fm
Ω = [(f, x 1 , . . . , x d+1 )].
Let us remark that, since f has only simple fixed points, the restriction π Ω : Ω → Ω of π to Ω is an (unramified) cover. We can choose an atlas of P 1 such that there is an affine chart of P 1 containing {c 1 (g), . . . , c 2d-2 (g)} for every g ∈ Ω, and define

V : Ω → C 2d-2 by V (g) := (g n 1 (c 1 (g)) -c 1 (g), . . . , g n 2d-2 (c 2d-2 (g)) -c 2d-2 (g)) , g ∈ Ω.
According to Corollary 5.5, we have ker(

D f V ) = {0}. Beware that Ŵ := W • π -1 is a holomorphic map from an open neighborhood of f in Ω to C 2d-2 , so it is sufficient to prove that ker(D f Ŵ ) ⊂ ker(D f V ).
Let v ∈ T f Rat 0 d , and pick a holomorphic disk (f t ) t∈D in Rat 0 d such that f 0 = f and ḟ = v. For any t ∈ D and any 1

≤ i ≤ 2d -2, set w i (t) := w i (f t ), c i (t) := c i (f t ), Ŵ (t) = Ŵ (f t ) and V (t) = V (f t ).
For any t ∈ D and any 1 ≤ i ≤ 2d -2, let C i (t) be the attracting cycle of f t whose immediate attractive basin contains c i (t), so that there is a holomorphic function

z i on D such that z i (t) ∈ C i (t) for any t ∈ D (so w i (t) = (f n i t ) (z i (t))) and that z i (0) = c i (0). Then for any 1 ≤ i ≤ 2d -2, we find ẇi = d(f n i t ) dt t=0 (z i (0)) + (f n i ) (z i (0)) • żi = ∂(f n i t ) ∂t t=0 (c i (0)) + (f n i ) (c i (0)) • żi ,
and since (f n i t ) (c i (t)) = 0 for any t ∈ D, we also have

0 = ∂(f n i t ) ∂t t=0 (c i (0)) + (f n i ) (c i (0)) • ċi (0). Hence for any 1 ≤ i ≤ 2d -2, ẇi = (f n i ) (c i (0)) • ( żi -ċi ) ,
and we also note that (

f n i ) (c i (0)) = 0 since f is hyperbolic of disjoint type. If in addition v ∈ ker(D f Ŵ ), then for any 1 ≤ i ≤ 2d -2, ẇi = 0 (and by definition z i (0) = c i (0)) hence we have z i (t) -c i (t) = O(t 2 ). For any 1 ≤ i ≤ 2d -2, the i-th component of V (t) is f n i t (c i (t)) -c i (t) =f n i t (z i (t)) + (f n i t ) (z i (t))(c i (t) -z i (t)) + O((c i (t) -z i (t)) 2 ) -c i (t) =(1 -w i (t))(z i (t) -c i (t)) + O(t 4 ) = O(t 2 ) as t → 0, so that v ∈ ker(D f V ).
(T 2d-2,fm n (ρ))(Ω fm ) = #Stab(n, ρ) d |n| .
2. the measure T 2d-2 n (ρ) has full mass on the union of all hyperbolic components in M d of type n, and for any such a component Ω,

(T 2d-2 n (ρ))(Ω) = #Stab(n, ρ) d |n| .
Proof. -Consider the case Rat 0,fm

d first. Pick ρ ∈]0, 1[ 2d-2
, and observe that

T 2d-2,fm n (ρ) = 1 (2π) 2d-2 d |n| [0,2π] 2d-2 ,∀i =j, θ i =θ j 2d-2 j=1 [Per n j (ρ j e iθ j )]dθ 1 • • • dθ 2d-2 ,
as finite measures on Rat 0,fm d , since we only remove a set of Lebesgue measure zero in [0, 2π] 2d-2 . Hence T 2d-2,fm n (ρ)-almost every point has 2d -2 distinct attracting cycles. For the second part, let Ω be a hyperbolic component in Rat 0,fm d of type n = (n 1 , . . . , n 2d-2 ). By Theorem 6.2, we know that the multiplier map W = (W 1 , . . . , W 2d-2 ) : Ω → D 2d-2 is a biholomorphism. In particular, the intersection 2d-2 j=0 Per n j (w i ) is smooth and transverse in Ω for all w ∈ D 2d-2 . This implies ( 11)

d |n| T 2d-2,fm n (ρ) = σ∈Stab(n,ρ) 2d-2 i=1 dd c log max{|W σ(i) |, ρ σ(i) }
on Ω, which has mass #Stab(n, ρ) on Ω. This concludes the proof for the case of Rat 0,fm of p to Rat 0,fm d \ Per 1 (1) is a finite unbranched cover of M d \ Per 1 (1). Indeed, p can only branch at parameters (f, x 1 , . . . , x d+1 ) where at least two of x 1 , . . . , x d+1 coincide, and those parameters are contained in Per 1 (1).

Let Ω ⊂ M d be a hyperbolic component of type n and let Ω be a connected component of p -1 (Ω). Then the restriction p| Ω : Ω → Ω is an unbranched cover. Since multipliers do not depend on the marking of critical points, the multiplier map

W Ω : Ω → D 2d-2 descends to a biholomorphism Ω → D 2d-2 . We now observe that T 2d-2,fm n (ρ) = (p) * (T 2d-2 n (ρ)
) and the conclusion follows as above.

Fix any ρ ∈]0, 1[ 2d-2 and any n ∈ (N * ) 2d-2 . Note that by construction,

T 2d-2,fm n (ρ) = (p) * (T 2d-2 n (ρ)) and µ fm bif = (p) * (µ bif ).
Since Rat 0,fm d is an affine variety, we can assume Rat 0,fm d ⊂ C N for some N . Consider the function log + |Z|, defined on C N , and let ϕ : Rat 0,fm d → R be its restriction to Rat 0,fm d . The function ϕ is psh, continuous, non-negative and dd c ϕ has finite mass in Rat 0,fm d . We have the lemma: Lemma 6.5. -There exist constants C 1 , C 2 > 0 that depend only on d such that, for any compact subset K of Rat 0,fm d , if C(K) is the constant in Theorem A, then we have the following inequality:

C(K) ≤ C 1 • ϕ ∞,K + C 2 .
Proof. -We follow closely the proof of [DeM3, Proposition 3.1] (see also [START_REF] Favre | Degeneration of endomorphisms of the complex projective space in the hybrid space[END_REF]Proposition 4.4]) and adapt it to the present situation. Since H 1 (Rat d , R) = 0, by [START_REF] Bassanelli | Bifurcation currents in holomorphic dynamics on P k[END_REF]Lemma 4.9], there exists a family of non-degenerate homogeneous polynomial lifts to C 2 of the family Rat d . We thus may choose a family F of non-degenerate homogeneous polynomial lifts to C 2 of the family Rat 0,fm d . Set V := Rat 0,fm d . We may regard this family F as a homogeneous non-degenerate polynomial maps with coefficients in the ring C

[V ]. Note that Res(F ) ∈ C[V ] and, in particular, log | Res(F )| ≤ αϕ(f ) + β for some constants α, β ≥ 0 independent of f ∈ V .
We now want to prove that there exists m ≥ 1 and C > 0 such that for any f ∈ Rat 0,fm d and any (x, y) ∈ C 2 \ {0},

1 C e -mϕ(f ) ≤ F (x, y) 2 (x, y) 2d ≤ Ce mϕ(f ) .
We work with the maximum norm (x, y) = max{|x|, |y|} on C 2 . The upper bound follows easily from the fact that F 1 , F 2 ∈ C[V ][x, y] and the triangle inequality. By the homogeneity of F 1 and F 2 , it is sufficient to verify the lower bound whenever (x, y) = 1. By the item (c) of [START_REF] Joseph | The arithmetic of dynamical systems[END_REF]Proposition 2.13], there exists homogeneous polynomials

g 1 , g 2 , h 1 , h 2 ∈ C[V ][x, y] d-1 such that g 1 (x, y)F 1 (x, y) + g 2 (x, y)F 2 (x, y) = Res(F )x 2d-1 and (12) h 1 (x, y)F 1 (x, y) + h 2 (x, y)F 2 (x, y) = Res(F )y 2d-1 . (13) Again, since g 1 , g 2 , h 1 , h 2 ∈ C[V ][x, y], there are constants A, B ≥ 0 independent of f ∈ V such that max {|g 1 (x, y)|, |g 2 (x, y)|, |h 1 (x, y)|, |h 2 (x, y)|} ≤ Ae Bϕ(f ) if (x, y) ≤ 1. When x = 1, equation (12) gives |Res(F )| ≤ 4 max {|g 1 (x, y)|, |g 2 (x, y)|} • F (x, y) ≤ 4A Bϕ(f ) F (x, y) .
We proceed similarly with equation ( 13) when y = 1 and the conclusion follows.

Following exactly the proof of Lemma 3.7 gives C 1 , C 2 ≥ 0 such that max sup

z∈P 1 log f # (z), sup z∈P 1 |g f (z)| ≤ C 1 ϕ(f ) + C 2 for any f ∈ Rat 0,fm d .
Recall that we picked n ∈ (N * ) 2d-2 . Let ε > 0, and set ρ = (1/2, . . . , 1/2), so in particular that Stab(n, ρ) = Stab(n). Take R > 0 large enough so that supp(T 2d-2,fm n (ρ)) is contained in the intersection B(0, R) between V and the open ball in C N of radius R and centered at 0. Observe that this is possible since there are at most finitely many of type n and for a hyperbolic component Ω of type n, W -1 ω (D 2d-2 1/2 ) ⊂ Ω is relatively compact in V (for d = 2, this is known to be true for the whole component Ω [START_REF] Lawrence | Bounded hyperbolic components of quadratic rational maps[END_REF]).

For any A > 0, we pick the following test function

Ψ A := 1 A min{max (ϕ, A) -2A, 0} on V.
Then, Ψ A is continuous and DSH on V and dd c Ψ A = T + A -T - A for some positive closed currents of finite masses, where T ± A ≤ C /A and for some C > 0 depending neither on A nor on T ± A (e.g. [START_REF] Dinh | Super-potentials of positive closed currents, intersection theory and dynamics[END_REF]Lemma 2.2.6 ]). Then observe that Ψ A is equal to -1 in B(0, e A ), and 0 outside B(0, e 2A ). Applying Theorem A with the control of Lemma 6.5 implies:

T 2d-2,fm n (ρ), Ψ A -µ fm bif , Ψ A ≤ (1 + log 2) C 1 ϕ(e 2A ) + C 2 max 1≤j≤2d-2 σ 2 (n j ) d n j C A .
Taking A = log R so that ϕ(e 2A ) = 2A, there is a constant C d > 0 depending only on d, we have

T 2d-2,fm n (ρ) -µ fm bif , -Ψ A ≤ C d max j (σ 2 (n j )/d n j ). As R → ∞, we have µ fm bif , -Ψ A → µ fm bif and in turn T 2d-2,fm n (ρ) -µ fm bif ≤ C d max 1≤j≤2d-2 σ 2 (n j ) d n j .
Let us go back to M d . Since the measures T 2d-2 n (ρ) and µ bif (resp. T 2d-2,fm n (ρ) and µ fm bif ) give no mass to algebraic subvarieties of M d (resp. of Rat 0,fm d ), we have Kiwi and Rees. Take m > n ≥ 2, they computed, in the critically marked moduli space M cm 2 , the number n IV (n, m) of all hyperbolic components Ω in M cm 2 of type (n, m) such that any [(f, c 1 , c 2 )] ∈ Ω has two distinct attracting cycles of respective exact periods j, k with j|n and k|m, respectively and their immediate attractive basins contain c 1 , c 2 , respectively. They prove

T 2d-2 n (ρ) -µ bif = 1 deg(p) T 2d-2,fm
n IV (n, m) = 5 3 2 n-3 + 1 12 - 1 4 n q=2 φ(q)ν q (n) 2 q -1 2 m + ε 1 (n, m),
where m) and |ν q (n) -2 n /(2(2 q -1))| ≤ 1/2. Their computation in particular yields that for any m > n ≥ 2,

|ε 1 (n, m)| ≤ 2 n + 2 2gcd(n,
n IV (n, m) = 1 3 - 1 8 n q=1 φ(q) (2 q -1) 2 2 n+m + ε 2 (n, m),
where |ε 2 (n, m)| ≤ C • 2 m for some C ≥ 1 independent of n, m. We now note that the natural projection π : M cm 2 → M 2 is of degree 2 and is unramified over any [f ] ∈ M 2 of disjoint type, and for any

[(f, c 1 , c 2 )] ∈ M cm 2 , π -1 {[f ]} = {[f, c, c ], [f, c , c]}. In particular, n IV (n, n + 1) = j|n, k| (n+1)
N (j, k).

Since we have N (j, k) ≤ C • 2 j+k by Bézout's theorem and

d n • d n+1 = 2 2n+1 + O(2 n ) as n → ∞, the above gives N (n, n + 1) d n • d n+1 = n IV (n, n + 1) 2 2n+1 + o(1) = 1 3 - 1 8 +∞ q=1 φ(q) (2 q -1) 2 + o(1), as n → ∞.
The conclusion follows from Theorem C, since #Stab(n, n + 1) = 1.

Weak genericity of postcritically finite hyperbolic rational maps

The moduli space M d of degree d rational maps is known to be an irreducible affine variety of dimension 2d -2 which is defined over Q (see [START_REF] Joseph | The space of rational maps on P 1[END_REF][START_REF] Milnor | Geometry and dynamics of quadratic rational maps[END_REF]); and all non-flexible Lattès postcritically finite degree d rational maps are known to be defined over Q (see e.g. [START_REF] Joseph | The arithmetic of dynamical systems[END_REF]). These properties are, for the moduli space of critically marked degree d polynomials, the starting point of the work [FG]. The idea developed there is to apply Yuan's equidistribution theorem [Y] to get the equidistribution of pcf maps towards the bifurcation measure.

The use of this equidistribution result requires: (1) defining an adelic semi-positive metric on an ample line bundle L → Md which associated height function h satisfies h([f ]) = 0 for all non-Lattès pcf map f , and which induced Monge-Ampère measure is proportional to µ bif . ( 2) showing that any sequence (X k ) of Galois invariant finite sets of postcritically finite parameters is

weakly generic in M d in that Card(X k ∩ C) = o (Card(X k )) as Card(X k ) → ∞ for any proper affine subvariety C in M d defined over Q. This is stronger than the Zariski density of k X k in M d .
Contrary to the case of polynomials, item (1) seems very difficult to establish and could even be wrong as stated. Here we focus on item (2).

For any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 , we set

X n := {[f ] ∈ M d : f has 2d -2 periodic critical points c 1 , . . . , c 2d-2
of respective exact periods n 1 , . . . , n 2d-2 }, so that C n ⊂ X n . A consequence of our counting of hyperbolic components is that any sequences of sets of centers of hyperbolic components of disjoint type is weakly generic.

Theorem F. -For any sequence (n(k)) k of (2d -2)-tuples n(k) = (n 1,k , . . . , n 2d-2,k ) in (N * ) 2d-2 satisfying min j (n j (k)) → ∞ as k → ∞, the sequence (X n(k) ) k is Galois-invariant and weakly generic in M d .

Remark. -This result in particular implies that k

X n(k) is Zariski dense in M d , which refines [DeM3, Theorem A].
For proving this weak genericity property, we prove a stronger result in the moduli space M cm d of critically marked degree d rational maps on P 1 , i.e., the orbit space of PSL 2 (C) in the space Rat cm d of critically marked degree d rational maps (f, c 1 , . . . , c 2d-2 ), where f ∈ Rat d and (c 1 , . . . , c 2d-2 ) is a (2d -2)-tuple of all critical points of f , counted with multiplicity. This is also an irreducible affine variety of dimension 2d -2 which is defined over Q and the natural finite branched cover p : M cm d → M d is of degree (2d -2)! and also defined over Q.

For any n ∈ N * and any 1 ≤ j ≤ 2d -2, let Per j (n) be the analytic hypersurface

Per j (n) := {[(f, c 1 , . . . , c 2d-2 )] ∈ M cm d : Φ * n (F, C j ) = 0}
in M cm d , where F and C j are lifts of f and c j , respectively (see Section 2.4 for the definition of Φ * n ); the degree of the hypersurface Per j (n) is bounded from above by Cd n for some constant C ≥ 1 depending only on d (see e.g. [START_REF] Joseph | The arithmetic of dynamical systems[END_REF] for more details), and since M cm d is quasi-projective, Per j (n) is actually an algebraic hypersurface of M cm d . For any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 , set Y n := 2d-2 j=1 Per j (n j ) ⊂ M cm d . We prove here the following as an application of our counting result. Theorem 6.6. -For any proper algebraic subvariety V in M cm d , there exists a constant C > 0 such that for all n ∈ (N * ) 2d-2 , we have

Card Y n ∩ V /Card Y n ≤ C • d -(min j n j )/2 .
For the proof, we follow the strategy of [START_REF] Favre | Distribution of postcritically finite polynomials[END_REF]Theorem 5.3] and we rely on the following, which is just an adaptation of [START_REF] Favre | Distribution of postcritically finite polynomials[END_REF]Lemma 5.4]. Lemma 6.7. -Let V be any irreducible algebraic subvariety of dimension q in M cm d and let p be a smooth point in M cm d . Assume that V is also smooth at p. Pick hypersurfaces H 1 , . . . , H 2d-2 intersecting transversely at p. Then there is I ⊂ {1, . . . , 2d -2} of cardinality 2d -2 -q such that p is an isolated point of V ∩ j∈I H j .

Proof of Theorem 6.6. -The case dim V = 0 is an immediate consequence of Theorem C, since V is a finite set in that case. We thus assume q := dim V ∈ {1, . . . , 2d -3}.

Pick

n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 . Let Z n be the subset in Y n consisting of all [(f, c 1 , . . . , c 2d-2 )] ∈ M d such that the orbits of c 1 , . . . , c 2d-2 of f are also disjoint. We claim that there is a constant C > 0 depending only on d such that Card Y n \ Z n ≤ C d |n|-(min i n i )/2 ; for, since Y n \ Z n consists of all [f ] ∈ M cm
d such that f has a super-attracting cycles of exact period n i and containing at least two distinct critical points for some i, we have

Y n \ Z n ⊂ 2d-2 i=1 j =i n i =n j [n i /2] k=0 {f k (c i ) = c j } ∩ : =i Per (n ) .
Since also deg({f k (c i ) = c j }) ≤ Cd k for some C > 0 independent of i, j and k, by Bézout's Theorem, there exists constant C 0 , C depending only on d such that

Card(Y n \ Z n ) ≤ 2d-2 i=1 j =i n i =n j [n i /2] k=0 Cd k : =i Cd n ≤ 2d-2 i=1 j =i n i =n j C 0 d |n|-n i /2 ≤ C d |n|-(min i n i )/2 .
Set N = N q := 2d -2 -q ∈ N * and recall that N = dim V . Let V reg be the regular locus of V . We also claim that there is a constant C > 0 depending only on d such that

Card V reg ∩ Z n ≤ C deg(V ) I d N j=1 n i j ,
where here and below the sums I range over all N -tuples I = (i 1 , . . . , i N ) of distinct indices in {1, . . . , 2d -2}; indeed, for any such choice I, we set Y I := N j=1 Per i j (n i j ), and let F I be the set of all isolated points of V ∩ Y I . By Bézout's Theorem, we have ( 14)

Card(F I ) ≤ deg(V ) N j=1 deg(Per i j (n i j )) = C 2 deg(V )d N j=1 n i j
V reg ∩ Z n ≤ C 5 I d -j / ∈I n i j Card(Y n ) and Card Y n \ Z n ≤ C 5 d -(min i n i )/2 Card(Y n ),
where C 5 > 0 depends only on V , d and q. Since V sing := V \ V reg is an algebraic subset in M cm d of codimension 2d -2 -q + 1, the proof is complete by a finite induction.

Proof of Theorem F. -Pick any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 , and for any permuta-

tion σ ∈ S 2d-2 , set σ(n) := (n σ(1) , . . . , n σ(2d-2) ). Let us first remark that p -1 (X n ) = σ∈S 2d-2 Y σ(n) .
The Galois-invariance of X n follows from the Galois-invariance of Y σ(n) for any σ ∈ S 2d-2 . Similarly, for any irreducible subvariety Z ⊂ M d defined over Q, we can apply Theorem 6.6 to any irreducible component of the algebraic subset

V = p -1 (Z) in M cm d .
The fact that p is a finite branched cover together with the assumption min j (n j (k)) → +∞ as k → ∞ completes the proof.

Distribution of hyperbolic maps with given multipliers in M d

This section is devoted to the proof of Theorem E. Pick any n = (n 1 , . . . , n 2d-2 ) ∈ (N * ) 2d-2 and any w = (w 1 , . . . , w 2d-2 ) ∈ D 2d-2 .

(1)). Set C fm n,w := π -1 fm (C n,w ), µ fm n,w := #Stab(n, w) d |n| C fm n,w δ [(f,x 1 ,...,x d+1 )] ,
and µ fm n := µ fm n,(0,...,0) . Then µ fm n,w = (π fm ) * (µ n,w ) for all w and µ fm n = (π fm ) * (µ n ). In particular, for any DSH and continuous function Ψ on M d with compact support,

µ n,w -µ bif , Ψ = 1 (d + 1)! µ n,w -µ bif , (π fm ) * (π fm ) * Ψ = 1 (d + 1)! µ fm n,w -µ fm bif , (π fm ) * Ψ .
Hence, it is sufficient to prove the desired estimates in the fixed marked moduli space M fm d . So, pick any compact set K in M fm d , and any either

C 1 or C 2 function Ψ on M fm d with support in K. Set ρ = (ρ 1 , . . . , ρ 2d-2 ) := (max(|w 1 |, 1/2), . . . , max(|w 2d-2 |, 1/2)) ∈ ([1/2, 1[) 2d-2 , so that ρ j ∈ [|w j |, 1[ for any 1 ≤ j ≤ 2d -2.
By Theorem A and the upper bound of the DSH-norm by the C 2 -norm, there is C(K) > 0 depending only on K such that

T 2d-2,fm n (ρ) -µ fm bif , Ψ ≤ C(K) max 1≤j≤2d-2 σ 2 (n j ) d n j Ψ C 2 if Ψ is C 2
and then, by interpolation between Banach spaces, that

T 2d-2,fm n (ρ) -µ fm bif , Ψ ≤ C(K) max 1≤j≤2d-2 σ 2 (n j ) d n j 1/2 Ψ C 1 if Ψ is C 1 .
Whence the proof of Theorem E reduces to showing

µ fm n -T 2d-2,fm n (ρ), Ψ ≤ C max 1≤j≤2d-2 1 d n j • Ψ C 2 if Ψ is C 2 , and (15)
µ fm n,w -T 2d-2,fm n (ρ), Ψ ≤ C max 1≤j≤2d-2 -1 d n j log ρ j 1/2 Ψ C 1 if Ψ is C 1 , ( 16
)
where C > 0 depends only on d.

A reduction to work on algebraic curves

Observe that the measure µ fm n,w -T 2d-2,fm n (ρ) on M fm d has its support contained in the union of all hyperbolic components of type n. Let Ω be such a component, and W = (W 1 , . . . , W 2d-2 ) : Ω → D 2d-2 the multiplier map on Ω. Letting λ r be the normalized Lebesgue measure on ∂D r , by ( 11) we have on Ω,

d |n| (T 2d-2,fm n (ρ) -µ fm n,w ) =#Stab(n, w)W * (λ ρ 1 ⊗ • • • ⊗ λ ρ 2d-2 ) -#Stab(n, w) 2d-2 i=1 [Per n i (w i )] = σ∈Stab(n,w) W * σ(1) (λ ρ 1 ) ∧ • • • ∧ W * σ(2d-2) (λ ρ 2d-2 ) -W * σ(1) (δ w 1 ) ∧ • • • ∧ W * σ(2d-2) (δ w 2d-2 ) = σ∈Stab(n,w) 2d-2 j=1 S σ,j ,
where for any σ ∈ Stab(n) and any 1 ≤ j ≤ 2d -2, the measure S σ,j is defined as

S σ,j := 1≤i<j W * σ(i) (λ ρ i ) ∧ W * σ(j) (λ ρ j ) -W * σ(j) (δ w j ) ∧ k>j W * σ(k) (δ w k ) = Sj-1 1≤i<j W * σ(i) (δ u i )∧ W * σ(j) (λ ρ j )-W * σ(j) (δ w j ) ∧ k>j W * σ(k) (δ w k )dλ Sj-1 (u 1 , . . . , u j-1 ) on Ω, setting λ Sj-1 := λ ρ 1 ⊗ • • • ⊗ λ ρ j-1 on Sj-1 := ∂D ρ 1 × • • • × ∂D ρ j-1 if j > 1 and S σ,1 := W * σ(1) (λ ρ 1 ) -W * σ(1) (δ w 1 ) ∧ k>1 W * σ(k) (δ w k ).
Recall that a wedge-product over the empty set is equal to 1 and that an intersection over the empty set is the whole space.

For any σ ∈ Stab(n, w), any 1 ≤ j ≤ 2d -2, and any u ∈ Sj-1 (if j > 1), let Λ σ,j (u) or Λ σ,1 be the set of all [(f, x 1 , . . . , x d+1 )] ∈ M fm d having a cycle of exact period n σ(i) and multiplier u i ∈ D ρ i for any 1 ≤ i < j and a cycle of exact period n σ(k) and multiplier w k for any k > j. Hence

Λ σ,j (u) ⊂ 1≤i<j Per n i (u σ(i) ) ∩ k>j Per n k (w σ(k) )
is an algebraic curve and, by Bézout's theorem, its area is

≤ C • d |n|-n j for some constant C depending only on d. Set W Λ σ,j (u) := p j • (W| Ω∩Λ σ,j (u) ) or W Λ σ,1 := p 1 • (W| Ω∩Λ σ,1
), where p j : D 2d-2 → D is the projection on the j-th coordinate. Then the measure

1≤i<j W * σ(i) (δ u i ) ∧ W * σ(j) (λ ρ j ) -W * σ(j) (δ w j ) ∧ k>j W * σ(k) (δ w k ) is equal to W * Λ σ,j (u) (λ ρ j -δ w j ). ( 17
)
on Ω ∩ Λ σ,j (u) if j > 1, and the measure S σ,1 is itself equal to W Λ σ,1 (λ ρ 1 -δ w 1 ) on Ω ∩ Λ σ,1 . 7.3. Proof of (16): the case of arbitrary multipliers in D 2d-2 Assume that Ψ is C 1 and test (17) against Ψ. For any σ ∈ Stab(n, w), any 1 ≤ j ≤ 2d -2, and any u := (u i ) i<j ∈ Sj-1 (if j > 1). We continue to fix Ω as in Subsection (7.2) and let O := W -1 Λ σ,j (u) (w j ). Then Λ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ w j ) = Λ σ,j (u)∩Ω (Ψ -Ψ(O)) • W * Λ σ,j (u) (λ ρ j ), so that by the mean value inequality:

Λ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ w j ) ≤ Λ σ,j (u)∩Ω |Ψ -Ψ(O)| • W * Λ σ,j (u) (λ ρ j ) ≤C • Ψ C 1 diam(W -1 Λ σ,j (u) (D(0, ρ j ))),
where the diameter is computed with respect to the distance induced by β and the constant C > 0 only depends on the choice of the C 1 -norm. By the length-area estimate (Lemma 2.3), we have diam(W -1 Λ σ,j (u) (D(0, ρ j ))) 2 ≤ τ • Area(Ω ∩ Λ σ,j (u)) min{1, 1 2π log(1/ρ j )} = τ • Area(Ω ∩ Λ σ,j (u))

| log ρ j |/(2π) since W -1 Λ σ,j (u) (D(0, ρ j )) W -1 Λ σ,j (u) (D(0, 1)) = Ω ∩ Λ σ,j (u) in Λ σ,j (u) are holomorphic disks and ρ j ≥ 1/2. Using Cauchy-Schwarz inequality gives

Ω Λ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ ρ j ) ≤C • Ψ C 1 • Ω τ • Area(Ω ∩ Λ σ,j (u)) | log ρ j |/(2π) 1/2 ≤C Ω τ | log ρ j |/(2π) 1/2 Ω Area(Ω ∩ Λ σ,j (u)) 1/2 Ψ C 1 ≤C τ | log ρ j |/(2π) 1/2
N fm (n) 1/2 (Area(Λ σ,j (u))) 1/2 Ψ C 1 , so that recalling that N fm (n) = N M fm d (n) ≤ C 1 d |n| by Bézout's theorem and Area(Λ σ,j (u)) ≤ C 2 d |n|-n j , where C 1 , C 2 > 0 depend only on d, we have

Ω Λ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ ρ j ) ≤ C 3 d -n j τ | log ρ j |/(2π) 1/2 d |n| Ψ C 1 ,
where C 3 > 0 depends only on d. Similarly,

Ω Λ σ,1 ∩Ω Ψ • W * Λ σ,1 (λ ρ 1 -δ ρ 1 ) ≤ C 3 d -n 1 τ | log ρ 1 |/(2π) 1/2 d |n| Ψ C 1 .
Since the right-hand sides are independent of u and σ, recalling (17), we have

µ fm n,ρ -T 2d-2,fm n (ρ), Ψ ≤ C 4 max 1≤j≤2d-2 1 d n j | log ρ j | 1/2 Ψ C 1 ,
where C 4 > 0 depends only on d (and actually not on K). Hence (16) holds.

Proof of (15): the center of components

Assume that Ψ is C 2 and test (17) against Ψ. Pick any σ ∈ Stab(n), any 1 ≤ j ≤ 2d -2, and any (u i ) i<j ∈ Sj-1 (if j > 1). We continue to fix Ω as in Subsection (7.2) for a while and let O := W -1 Λ σ,j (u) (0, . . . , 0) be the center of the disk Ω ∩ Λ σ,j (u). Then we have

≤ C • Ψ C 2 • diam(W -1 Λ σ,j (u) (D(0, ρ j ))) 2 ,
where the diameter is again computed with respect to the distance induced by β and the constant C > 0 depends only on the choice of the C 2 -norm. Again by the length-area estimate (Lemma 2.3), we have

Λ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ 0 ) ≤ C • Ψ C 2 • τ • Area(Ω ∩ Λ σ,j (u)) | log(1/2)|/(2π)
since W -1 Λ σ,j (u) (D(0, ρ j )) W -1 Λ σ,j (u) (D(0, 1)) = Ω ∩ Λ σ,j (u) in Λ σ,j (u) are holomorphic disks (here, ρ j ≡ 1/2 by definition). Hence

Λ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ 0 ) ≤ C • Ψ C 2 • Area(Ω ∩ Λ σ,j (u)),
where C > 0 depends only on d, so recalling that Ω Area(Ω∩Λ σ,j (u)) ≤ Area(Λ σ,j (u)) ≤ C 1 d |n|-n j , where C 1 > 0 depends only on d, Ω Λ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ 0 ) ≤C C 1 • Ψ C 2 d |n|-n j , and similarly,

Ω Λ σ,1 ∩Ω Ψ • W * Λ σ,1 (λ ρ 1 -δ 0 ) ≤C C 1 • Ψ C 2 d |n|-n 1 .
Since the right-hand sides are independent of u and σ, recalling (17), we have

µ fm n -T 2d-2,fm n (ρ), Ψ ≤ C Ψ C 2 max 1≤j≤2d-2 d -n j ,
where C > 0 depends only on d. Hence (15) holds.

Remark. -One cannot hope the (even qualitative) convergence µ n → µ bif for bounded DSH observables; indeed, consider the DSH function φ A := min{0, max{log |Z|, -A}/A} on C 2d-2 for some A > 0, which is identically equal to 0 outside the ball B(0, exp(-A)) and equal to -1 at 0. Furthermore, it is DSH and its DSH norm can be taken arbitrarily small for A 1. By a change of coordinates, one can then construct a DSH function in M d which is equal to -1 at the center of a given hyperbolic component and 0 outside that component, with arbitrarily small DSH norm. Summing this constructions over sufficiently many hyperbolic components, we can construct an observable ψ A which is bounded and DSH with µ n , ψ A → µ bif , ψ A . Nevertheless, it would be interesting to find a space of test functions independent of the choice of coordinates for which a similar statement as Theorem E holds. The following (see [START_REF] Favre | Distribution of postcritically finite polynomials[END_REF]Theorem 6.8]) will also be useful in the sequel.

Theorem 8.2. -The map W : H → D d-1 is a biholomorphism.

Counting hyperbolic components of disjoint type

As in the case of rational maps, we denote by N P (n) the number of hyperbolic components of type n = (n 0 , . . . , n d-2 ) in the family (P c,a ) (c,a)∈C d-1 . When n j = n for all j = and n j ≥ 1 for all j, we have N P (n) = (d -1)! • d |n| . This result is an immediate consequence of Theorem 8.2. Indeed, all such components contain one postcritically finite parameter, counted with multiplicity, and all of them are contained in C d . The result follows from Bézout's Theorem and the fact that deg(Per j (n j )) = d n j .

Our aim here is to give a good generalization of the above statement, including the case when n j = n for all j, . The first observation is that any hyperbolic component H in C d-1 of type n is contained in the compact set C d . We rely on the following lemma, which is an immediate adaptation of Lemma 6.4 (hence we omit the proof). Remark. -This result is coherent with the above remark concerning the case n j = n for all j = , since in that case, Stab(n) = {id}. The above statement can also be interpreted as follows; the number of postcritically finite parameters for which all critical points are periodic with prescribed exact periods n 0 , . . . , n d-2 ≥ 2 and at least 2 critical

  Notations. -Let µ : N * → {-1, 0, 1} be the Möbius function. Define the sequence (d n ) in N * by d n := m|n µ n m (d m + 1) ∈ N * , or equivalently d n + 1 = m|n d m for any n ∈ N * , so that d n = d n + O(d n/2 ) as n → ∞.

  is holomorphic, and since Λ is a quasi-projective variety, this p n : Λ × C → C is actually a regular function with deg w (p n (λ, w)) = d n /n for all λ ∈ Λ and with deg λ (p n (λ, w)) ≤ Cd n for all w ∈ C, where C > 0 depends only on the family (f λ ) λ∈Λ , see e.g. [BB3, §2.2]. For any n ∈ N * and any w ∈ C, we set Per n (w) := {λ ∈ Λ ; p n (λ, w) = 0} and denote by [Per n (w)] the current of integration defined by the zeros of p n (•, w) on Λ. Remark that for all w ∈ C and all n ∈ N * , since deg λ (p n (λ, w)) ≤ Cd n , we have (4) 1 d n [Per n (w)] ≤ C.

  The space Rat 0 d is then determined by the equations b d = 0, a 0 = 0, i a i = j b j and is thus clearly a smooth subvariety of Rat d of pure dimension 2d -2. Lemma 5.4. -The complex submanifolds Rat 0 d and O(f ) in Rat d intersect transversely at any f ∈ Rat 0 d .

  Corollary 5.5. -Pick any postcritically finite and hyperbolic f ∈ Rat 0 d of disjoint type. The map V f : U → C 2d-2 is a local biholomorphism at f . 6. Counting the centers of hyperbolic components of disjoint type in M d 6.1. The marked spaces Rat 0

d.

  The conclusion follows from Lemma 6.3 below by the inverse function theorem. Lemma 6.3. -The linear map D a W is invertible. Proof. -Let π : Rat 0,fm d → Rat 0 d ⊂ Rat d be the natural branched cover, and set Ω := π(Ω), which is the hyperbolic component in Rat 0 d containing f with λ

6. 3 .

 3 Counting hyperbolic components : the mass of µ bif in M d We now prove Theorem C and Corollary D. To avoid confusions, for any ρ and any n, denote T 2d-2 n (ρ) and µ bif on Rat 0,fm d by T 2d-2,fm n (ρ) and µ fm bif , respectively. Observe first the following. Lemma 6.4. -Fix any ρ ∈]0, 1[ 2d-2 and any n ∈ (N * ) 2d-2 . Then 1. the measure T 2d-2,fm n (ρ) has full mass on the union of all hyperbolic components in Rat 0,fm d of type n, and for any such a component Ω fm ,

\

  be the natural finite branched cover of degree (d + 1)!. Observe that the restriction p := p| Rat 0,fm d Per 1 (1)

  = #Stab(n)N (n)/d |n| (by Lemma 6.4) completes the proof of Theorem C. Proof of Corollary D. -Let us begin with describing [KR, Theorem 1.1] by

\

  7.1. A reduction to work on M fm d Let π fm : M fm d → M d be the natural quotient map. It is a finite branched cover of degree deg(π fm ) = (d + 1)! and recall the definition of C n,w from the introduction. Again, for any ρ and any n, denote T 2d-2 n (ρ) and µ bif on M fm d by T 2d-2,fm n (ρ) and µ fm bif , respectively (though we already used that notation on Rat 0,fm d , this is not an issue since the projection Rat 0,fm d Per 1 (1) → M fm d \ Per 1 (1) is a bijection and none of the considered objects give mass to Per 1

Λ

  σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ 0 ) = Λ σ,j (u)∩Ω (Ψ -Ψ(O)) • W * Λ σ,j (u) (λ ρ j ) = Λ σ,j (u)∩Ω (Ψ(z) -(D O Ψ) (z -O) -Ψ(O)) ((W Λ σ,j (u) ) * (λ ρ j ))(z),the latter equality holding by the mean value theorem for harmonic functions, that is:Λ σ,j (u)∩Ω (D O Ψ) (z -O) • (W Λ σ,j (u) ) * (λ ρ j )(z) = (D O Ψ) (O -O) = 0.By the mean value inequality, we haveΛ σ,j (u)∩Ω Ψ • W * Λ σ,j (u) (λ ρ j -δ 0 ) ≤ Λ σ,j (u)∩Ω |Ψ(z) -(D O Ψ) (z -O) -Ψ(O)| • (W Λ σ,j (u) ) * (λ ρ j )(z)

  w i (c, a) ∈ D be the multiplier of the attracting cycle that has exact period n i . In this way we get a holomorphic map W = W H : H → D d-1 defined by W(c, a) := (w 0 (c, a), . . . , w d-2 (c, a)), (c, a) ∈ H.

  Lemma 8.3. -For any ρ ∈]0, 1[ d-1 and any n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 with min j n j ≥ 2, the measure T d-1 n (ρ) has full mass on the union of all hyperbolic components Ω ⊂ C d such that for all (c, a) ∈ Ω, P c,a has d -1 distinct attracting cycles in C of respective exact periods n 0 , . . . , n d-2 . Furthermore, it gives mass #Stab(n)/d |n| to each of those components.Here is the precise statement.Theorem 8.4. -There exists a constant C ≥ 1 depending only on d, such that for any n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 with min j n j ≥ 2, we have0 ≤ 1 -#Stab(n) • N P (n) (d -1)! • d |n| ≤ C max 0≤j≤d-2 σ 2 (n j ) d n j .Proof. -Pick any n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 with min j n j ≥ 2. Set ρ := (1/2, . . . , 1/2), and pick a smooth cut-off function Ψ on C d-1 such that Ψ = 1 on C d . Applying Theorem A yieldsT d-1 n (ρ), Ψ -(dd c L) d-1 , Ψ ≤ C Ψ * DSH max 0≤j≤d-2 σ 2 (n j ) d n j ,where C > 0 only depends on supp(Ψ) and d. As seen in the previous Subsection, we have (dd c L) d-1 , Ψ = (d -1)!, and by Lemma 8.3 and supp(T dis complete also by N P (n) ≤ d |n| /#Stab(n).

  for some constant C 2 > 0 depending only on d. Since Card V reg ∩ Z n ≤ I Card(F I ) by Lemma 6.7, the claim holds. According to Theorem C, there is a constant C 4 > 0 depending only on d such that Card(Y n ) ≥ Card(Z n ) ≥ C 4 d |n| provided min j n j is large enough.Hence, the above two claims imply Card

8. Distribution of hyperbolic maps in P cm d 8.1. A good parametrization of P cm d We refer to [START_REF] Dujardin | Distribution of rational maps with a preperiodic critical point[END_REF]§5] and [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF]§2] for the material of this section. Recall that the critically marked moduli space P cm d of degree d polynomials is the space of affine conjugacy classes of degree d polynomials with d -1 marked critical points in C. We define a finite branched cover of C d-1 → P cm d as follows. For c = (c 1 , . . . , c d-2 ) ∈ C d-2 and a ∈ C, let

where σ k (c) is the monic elementary degree k symmetric polynomial in the c i 's. This family is known to be a finite branched cover of P cm d . Remark also that the (finite) critical points of P c,a are exactly c 0 , c 1 , . . . , c d-2 , taking into account their multiplicity, where we set c 0 := 0, and that they depend algebraically on (c, a) ∈ C d-1 . From now on, we work on the parameter space C d-1 of the family (P c,a ) (c,a)∈C d-1 rather than P cm d itself, without loss of generality.

The dynamical Green function of P c,a is the continuous psh function g c,a :

, where the convergence is locally uniform in (c, a, z) ∈ C d . For any 0 ≤ j ≤ d -2, the function g j (c, a) := g c,a (c j ) is psh and continuous on C d-1 and, setting T j := dd c g j , we have dd c L = j T j and T j ∧ T j = 0.

In this family it is now classical to define the bifurcation measure on C d-1 as a probability measure

Then supp µ bif is compact and coincides with the Shilov boundary of the connectedness locus

d n = k|n D k by Möbius inversion, and

For any n ∈ N * , the n-th dynatomic polynomial of P c,a is defined as

and for any 0 ≤ j ≤ d -1 and any n ∈ N * , we set

Moreover, the following holds (see [START_REF] Favre | Distribution of postcritically finite polynomials[END_REF]Theorem 6.1]).

Theorem 8.1. -For any n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 satisfying min j n j > 1 and any (c, a) ∈ d-1 j=0 Per j (n j ) such that P c,a has only simple critical points in C, the (d -1) hypersurfaces Per j (n j ) are smooth and intersect transversely at (c, a).

Pick any n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 . We say a hyperbolic component H in C d-1 (or the family (P c,a ) (c,a)∈C d-1 ) to be of (disjoint) type n if for every (c, a) ∈ H, P c,a admits d-1 distinct attracting periodic orbits of respective exact periods n 0 , . . . , n d-2 in C. Then all critical points of P c,a in C for (c, a) ∈ H are simple. For each 0 ≤ i ≤ d -2, we let points lie in the same super-attracting cycle, counted with multiplicity of intersection of the Per τ (j) (n j ) for all τ ∈ S d-1 , is bounded from above by C max j≤d-2 (σ 2 (n j )/d n j ) • d |n| . This is a much better estimate than the one we can obtain by naive arguments. Indeed, without taking the multiplicity into account we can naively get a bound from above of the form Cd |n|-min j n j /2 max j≤d-2 n j , see e.g. the proof of the upper bound on Card(Y n \ Z n ) in the proof of Theorem 6.6.

An immediate application of this Theorem is the following: Corollary 8.5. -For any integer n ≥ 2, we have

where C ≥ 1 is given by Theorem 8.4. In particular,

Proof. -In the present case, we have #Stab(n, . . .

Since µ bif is a probability measure, the result follows from Theorem 8.4 above.

Remark. -In fact, we have proved that, counted with multiplicity, the number of intersection points of the Per j (n) for which at least two critical points lie in the same periodic orbit is bounded from above by a constant times σ 2 (n)d (d-2)n .

Distribution of polynomials with (d -1) attracting cycles

Pick n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 with min j n j ≥ 2 and w := (w 0 , . . . , w d-2 ) ∈ D d-1 . As in the case of rational maps, we let C n,w be the (finite) set of parameters (c, a) ∈ C d-1 such that P c,a has d -1 distinct attracting cycles in C of respective exact periods n 0 , . . . , n d-2 and multipliers w 0 , . . . , w d-2 . We also let

The only modification from the case of rational maps is the multiplication by 1/(d -1)!.

From the normalization µ bif = (dd c L) d-1 /(d-1)!, we see easily that this factor should also appear in the definition of ν n,w . An argument similar to that in the proof of Theorem E gives the following.

Theorem 8.6. -There exists a constant C > 0 depending only on d such that

) and any n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 with min j n j ≥ 2,

2. for any Ψ ∈ C 1 c (C d-1 ), any w = (w 0 , . . . , w d-2 ) ∈ D d-1 and any n = (n 0 , . . . , n d-2 ) ∈ (N * ) d-1 with min j n j ≥ 2,

Remark. -The key difference with the case of the moduli space M d of degree d rational maps is the existence of a universal constant C > 0. This is a consequence of the fact that C n,w ∪ supp(µ bif ) ⊂ C d , which is compact in C d-1 , for all n and all w. This compactness property implies the existence of a universal constant C 1 > 0 in the conclusion of Theorem A in the family (P c,a ) (c,a)∈C d-1 .

We now come to our last result in the spirit of Theorem B of [START_REF] Gauthier | Distribution of postcritically finite polynomials II: Speed of convergence[END_REF]: for any n ∈ N * , we want to prove the measure equidistributed on parameters (c, a) ∈ C d-1 satisfying c j ∈ Fix * (P n c,a ) for any 0 ≤ j ≤ d -2 converges towards the bifurcation measure, with an exponential speed of convergence.

Corollary 8.7. -There exists a constant C > 0 depending only on d such that for any integer n ≥ 2 and any Ψ ∈ C 2 c (C d-1 ), we have

Proof. -For any integer n ≥ 2, we have d-2 j=0 [Per j (n)] = (d n ) d-1 ν n,(0,...,0) , so that we can directly apply Theorem 8.6.