Johan Tain

Equidistribution speed towards the Green current for endomorphisms of P k

Keywords: Green current, equidistribution speed AMS 2000 subject classication: 37F, 32H. 1

Equidistribution speed towards the Green current for endomorphisms of Pk

Introduction

Let f be a holomorphic endomorphism of algebraic degree d ≥ 2 on the complex projective space P k . The iterates f n = f • • • • • f dene a dynamical system on P k . It is well know that, if ω denotes the normalized Fubini-Study form on P k then the sequence d -n (f n ) * (ω) converges to a positive closed current T of bidegree (1, 1) called the Green current of f (see e.g. [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF]). It is a totally invariant current, whose support is the Julia set of f and that exhibits interesting dynamical properties. In particular, for a generic hypersurface H of degree s, the sequence d -n (f n ) * [H] converges to sT [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF]. Here, [H] denotes the current of integration on H and the convergence is in the sense of currents. In fact, if we denote by T p the self-intersection T ∧ • • • ∧ T, Dinh and Sibony proposed the following conjecture on equidistribution. Conjecture 1.1. Let f be a holomorphic endomorphism of P k of algebraic degree d ≥ 2 and T its Green current. If H is an analytic set of pure codimension p and of degree s which is generic in the Zariski sense, then the sequence d -pn (f n ) * [H] converges to sT p exponentially fast.

The aim of the paper is to prove the conjecture for p = 1. It is a direct consequence of the following more precise result on currents. Indeed, we only have to apply the theorem to S := s -1 [H] for hypersurfaces H which does not contain any element of A λ .

Theorem 1.2. Let f, T be as above and let 1 < λ < d. There exists a nite family A λ of periodic irreducible analytic sets such that if S is a positive closed (1, 1)-current of mass 1, whose dynamical potential u veries u L 1 (X) ≤ C for all X in A λ , then the sequence S n := d -n (f n ) * (S) converges exponentially fast to T. More precisely, for every 0 < β ≤ 2 and φ ∈ C β (P k ) we get

| S n -T, φ | ≤ A φ C β λ d nβ/2 , (1.1)
where A > 0 depends on the constants C and β but is independent of S, φ and n.

Here, the space L 1 (X) is with respect to the volume form ω dim(X) on X and C β (P k ) denotes the space of (k -1, k -1)-forms whose coecients are of class C β , equipped with the norm induced by a xed atlas. The dynamical potential of S is the unique quasi-plurisubharmonic function u such that S = dd c u + T and max P k u = 0. Note that A λ will be explicitly constructed. Theorem 1.2 still holds if we replace A λ by an analytic subset, e.g. a nite set, which intersects all components of A λ .

Remark 1.3. We can prove a similar result for a regular polynomial automorphism f : C k → C k outside its indeterminacy set, see [START_REF] Tain | Dynamique des endomorphismes holomorphes de l'espace projectif[END_REF].

Equidistribution problem without speed was considered in dimension 1 by Brolin [START_REF] Brolin | Invariant sets under iteration of rational functions[END_REF] for polynomials and by Lyubich [START_REF] Lyubich | Entropy properties of rational endomorphisms of the Riemann sphere[END_REF] and Freire, Lopes and Mañé [START_REF] Freire | An invariant measure for rational maps[END_REF] for rational maps. They proved that for every point a in P 1 , with maybe two exceptions, the preimages of a by f n converge towards the equilibrium measure, which is the counterpart of the Green current in dimension 1. In this context, a result in the direction of Theorem 1.2 was obtained in [START_REF] Drasin | Equidistribution and Nevanlinna theory[END_REF] (see also references therein).

In higher dimension, for p = k, simple convergence in Conjecture 1.1 was established by Fornaess and Sibony [START_REF] Erik | Complex dynamics in higher dimensions[END_REF], Briend and Duval [START_REF] Briend | Deux caractérisations de la mesure d'équilibre d'un endomorphisme de P k (C)[END_REF], Dinh and Sibony [START_REF] Dinh | Dynamique des applications d'allure polynomiale[END_REF]. Recently in [START_REF] Dinh | Equidistribution speed for endomorphisms of projective spaces[END_REF], Dinh and Sibony give exponential speed of convergence, which completes Conjecture 1.1 for p = k. The equidistribution of hypersurfaces was proved by Fornaess and Sibony for generic maps [START_REF] Erik | Complex dynamics in higher dimension[END_REF] and by Favre and Jonsson in dimension 2 [START_REF] Favre | Brolin's theorem for curves in two complex dimensions[END_REF]. The convergence for general endomorphisms and Zariski generic hypersurfaces was obtained by Dinh and Sibony in [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF]. These papers state convergence but without speed. In other codimensions, the problem is much more delicate. However, the conjecture was solved for generic maps in [START_REF] Dinh | Super-potentials of positive closed currents, intersection theory and dynamics[END_REF], using the theory of super-potentials.

We partially follow the strategy developed in [START_REF] Erik | Complex dynamics in higher dimension[END_REF], [START_REF] Favre | Brolin's theorem for curves in two complex dimensions[END_REF] and [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF], which is based on pluripotential theory together with volume estimates, i.e. a lower bound to the contraction of volume by f. These estimates are available outside some exceptional sets which are treated using hypothesis on the map f or on the current S.

The exceptional set A λ will be dened in Section 5. It is in general a union of periodic analytic sets possibly singular. In our proof of Theorem 1.2, it is necessary to obtain the convergence of the trace of S n to these analytic sets. So, we have to prove an analog of Theorem 1.2 where P k is replaced with an invariant analytic set. The geometry of the analytic set near singularities is the source of important technical diculties. We will collect in Section 2 and Section 3 several versions of Lojasiewicz's inequality which will allow us to work with singular analytic sets and also to obtain good estimates on the size of a ball under the action of f n . Such estimates are crucial in order to obtain the convergence outside exceptional sets.

Theorem 1.2 can be reformulated as an L 1 estimate of the dynamical potential u n of S n (see Theorem 6.1). The problem is equivalent to a size control of the sublevel set

K n = {u n ≤ -(λ/d) n }. Since T is totally invariant, we get that u n = d -n u • f n and f n (K n ) = {u ≤ -λ n }.
The above estimate on the size of ball can be applied provided that K n be not concentrated near the exceptional sets. The last property will be obtained using several generalizations of exponential Hörmander's estimate for plurisubharmonic functions that will be stated in Section 4. A key point in our approach is that, by reducing the domain of integration, we obtain uniform exponential estimates for non-compact families of quasi-plurisubharmonic functions.

We close this introduction by setting some notations and conventions. The symbols and mean inequalities up to constants which only depend on f or on the ambient space. To desingularize an analytic subset of P k , we always use a nite sequence of blow-ups of P k . Unless otherwise specied, the distances that we consider are naturally induced by embedding or smooth metrics for compact manifolds. For K > 0 and 0 < α ≤ 1, we say that a function u : X → C is (K, α)-Hölder continuous if for all x and y in X, we have |u(x) -u(y)| ≤ Kdist(x, y) α . We denote by B the unit ball of C k and for r > 0, by B r the ball centered at the origin with radius r. In P k , we denote by B(x, r) the ball of center x and of radius r. And, for X ⊂ P k an analytic subset, we denote by B X (x, r) the connected component of B(x, r)∩X which contains x. We call it the ball of center x and of radius r in X. It may have more than one irreducible component. Finally, for a subset Z ⊂ X, we denote by Z X,t or simply Z t , the tubular t-neighborhood of Z in X, i.e. the union of B X (z, t) for all z in Z. A function on X is called (strongly) holomorphic if it has locally a holomorphic extension to a neighborhood of the ambient space.

2

Lo jasiewicz's inequality and consequences

One of the main technical diculties of our approach is related to singularities of analytic sets that we will handle using blow-ups along smooth varieties.

In this section, we study the behavior of metric properties under blow-ups.

It will allow us to establish volume and exponential estimates onto singular analytic sets. We will frequently use the following Lojasiewicz inequalities. We refer to [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF] for further details.

Theorem 2.1. Let U be an open subset of C k and let h, g be subanalytic functions in U. If h -1 (0) ⊂ g -1 (0), then for any compact subset K of U, there exists a constant N ≥ 1 such that, for all z in K, we have

|h(z)| |g(z)| N .
In this paper, we only use the notion of subanalytic function in the following case. Let U be an open subset of C k and A ⊂ U be an analytic set. Every compact of U has a neighborhood V ⊂ U such that the function x → dist(x, A) and analytic functions on U are subanalytic on V. Moreover, the composition or the sum of two such functions is still subanalytic on V.

In particular, we have the following property.

Corollary 2.2. Let U be an open subset of C k and let A, B be analytic subsets of U. Then for any compact subset K of U, there exists a constant N ≥ 1 such that, for all z in K, we have

dist(z, A) + dist(z, B) dist(z, A ∩ B) N .
We briey recall the construction of blow-up that we will use later. If U is an open subset of C k which contains 0, the blow-up U of U at 0 is the submanifold of U ×P k-1 dened by the equations z i w j = z j w i for 1 ≤ i, j ≤ k where (z 1 , . . . , z k ) are the coordinates of C k and [w 1 : • • • : w k ] are the homogeneous coordinates of P k-1 . The sets w i = 0 dene local charts on U where the canonical projection π : U → U, if we set for simplicity i = 1 and w 1 = 1, is given by

π(z 1 , w 2 , . . . , w k ) = (z 1 , z 1 w 2 , . . . , z 1 w k ). If V ⊂ C p is an open subset, the blow-up of U × V along {0} × V is dened by U × V. This is the local model of a blow-up.
Finally, if X is a complex manifold, the blow-up X of X along a submanifold Y is obtained by sticking copies of the above model and by using suitable atlas of X. The natural projection π : X → X denes a biholomorphism between X \ Y and X \ Y where the set Y := π -1 (Y ) is called the exceptional hypersurface. If A is an analytic subset of X not contained in Y, the strict transform of A is dened as the closure of π -1 (A \ Y ).

We have the following elementary lemma.

Lemma 2.3. Let π : U ×V → U ×V be as above and Y denote the exceptional hypersurface in

U × V. Assume that U × V is bounded in C k × C p . Then for all z, z ∈ U × V, we have dist(z, z ) dist( z, z )(dist( z, Y ) + dist( z , Y )),
where z = π( z) and z = π( z ).

Proof. Since π leaves invariant the second coordinate, considering the maximum norm on U × V, the general setting is reduced to the case of blow-up of a point. Hence we can take V = {0}. The lemma is obvious if z or z is equal to 0. Since π is a biholomorphism outside Y , we can assume that 0 < z , z < By triangle inequality, we have

|z 1 w i -z 1 w i | ≥ |z 1 w i -z 1 w i | -|z 1 w i -z 1 w i | ≥ |z 1 ||w i -w i | -|z 1 -z 1 |.
Hence, by symmetry in z 1 and z 1 we get

2|z 1 w i -z 1 w i | ≥ (|z 1 | + |z 1 |)|w i -w i | -2|z 1 -z 1 |.
Therefore, there is a constant a > 0 independent of z and z such that

|z 1 -z 1 | + k i=2 |z 1 w i -z 1 w i | ≥ a(|z 1 | + |z 1 |)(|z 1 -z 1 | + k i=2 |w i -w i |).
The left-hand side corresponds to dist(z, z ). We also have that dist( z, z )

|z 1 -z 1 | + k i=2 |w i -w i | and dist( z, Y ) + dist( z , Y ) |z 1 | + |z 1 |. The result follows.
A similar result holds for analytic sets. More precisely, consider an irreducible analytic subset X of P k of dimension l and a smooth variety Y contained in X. Let π : P k → P k be the blow-up along Y and π the restriction of π to the strict transform X of X. Denote by Y and Y the exceptional hypersurfaces in P k and in X respectively. Lemma 2.4. There exists N ≥ 1 such that for all z and z in X

dist(z, z ) dist( z, z )(dist( z, Y ) + dist( z , Y )) N ,
Proof. The previous lemma gives the inequality with N = 1 if we substitute Y by Y . By Corollary 2.2 applied to A = X and B = Y , there exists N ≥ 1 such that dist( x, Y ) dist( x, Y ) N for all x in X. The result follows.

Here is the rst estimate on contraction for blow-ups. Lemma 2.5. There exists a constant N ≥ 1 such that for all

0 < t ≤ 1/2, if dist( x, Y ) > t and r < t/2, then π(B b X ( x, r)) contains B X (π( x), t N r).
Moreover, if N is large enough then the image by π of a ball of radius 0 < r ≤ 1/2 contains a ball of radius r N in X.

Proof. Let y be a point in X such that dist( x, y) = r and set x = π( x), y = π( y). The assumption on r gives that dist( y, Y ) > t/2. Therefore, we deduce from Lemma 2.4 that dist(x, y) rt N .

The rst assertion follows since t ≤ 1/2 and π is a biholomorphism outside Y .

For a general ball B of radius r in X, we can reduce the ball in order to avoid Y and then apply the rst statement. More precisely, as dim( Y ) ≤ l-1 there is a constant c > 0 such that for all ρ > 0, Y is cover by cρ -2(l-1) balls of radius ρ. On the other hand, by a theorem of Lelong [START_REF] Lelong | Fonctions plurisousharmoniques et formes diérentielles positives[END_REF][START_REF] Demailly | Complex analytic and dierential geometry[END_REF], the volume of a ball of radius ρ in X varies between c -1 ρ 2l and c ρ 2l for some c > 0. Hence, the volume of Y ρ is of order ρ 2 . Take ρ = c r l with c > 0 small enough. By counting the volume, we see that B is not contained in Y ρ . Therefore, B \ Y contains a ball of radius ρ/3. We obtain the result using the rst assertion.

In the same spirit, we have the following lemma. Lemma 2.6. Let Z be a compact manifold, Z be an irreducible analytic subset of P k and π : Z → Z be a surjective holomorphic map. Let A be an irreducible analytic subset of Z and dene A := π -1 (A). There exists N ≥ 1 such that A t N is included in π( A t ) for all t > 0 small enough. Moreover, if A is the union on two analytic sets

A 1 , A 2 such that A 2 := π( A 2 ) is strictly contained in A then π( A 1,t ) contains A t N \ A 2,t 1/2 .
Proof. Since dist( z, A) = 0 if and only if dist(π( z), A) = 0, we can apply Theorem 2.1 to these functions which implies the existence of

N ≥ 1 such that dist( z, A) N dist(π( z), A).
Therefore, since π is surjective, π( A t ) contains A t N for t > 0 small enough. Now, let A = A 1 ∪ A 2 be as above. Since π is holomorphic, there exists

c > 0 such that π( A 2,t ) ⊂ A 2,ct . Therefore, π( A 1,t ) contains A t N \ A 2,t 1/2 for t > 0 suciently small.
In the sequel, we will constantly use desingularization of analytic sets. The following lemma allows us to conserve integral estimates. Lemma 2.7. Let Z and Z be irreducible analytic subsets of Kähler manifolds. Let π : Z → Z be a surjective proper holomorphic map. Then, for every compact L of Z there exists

q ≥ 1 such that if v is in L q loc (Z) then v := v • π is in L 1 ( L).
Moreover, there exists c > 0, depending on L, such that

v L 1 ( b L) ≤ c v L q (π( b L)) .
Proof. Using a desingularization, we can assume that Z is a smooth Kähler manifold with a Kähler form ω. Denote by ω, n, m a Kähler form on Z and the dimensions of Z and Z respectively. Generic bers of π are compact of dimension m -n and form a continuous family. It follows that the integral of ω m-n on that bers is a constant.

Consider λ = π * (ω n ) ∧ ω m-n on Z. The last observation implies that π * ( λ) = ω n up to a constant. Therefore, if v is in L q loc (Z) then v is in L q loc ( Z, λ
). Moreover, we can write λ = h ω m where h is a positive function. If there exists τ > 0 such that h -τ is integrable on L with respect to ω m , we obtain for p = 1 + τ and q its conjugate that

b L | v| ω m = b L | v|h -1 λ ≤ b L | v| q λ 1/q b L h -p λ 1/p π( b L) |v| q ω n 1/q b L h -τ ω m 1/p v L q (π( b L)) .
It remains to show the existence of τ. The set {h = 0} is contained in the complex analytic set A where the rank of π is not maximal. More precisely, if π has maximal rank at z, then we can linearize π in a neighborhood of z.

Therefore, λ and ω m are comparable in that neighborhood. Since L is compact, the problem is local. Let z 0 be in L. We can nd a small chart U at z 0 and a holomorphic function φ on U such that A ∩ U is contained in {φ = 0}. We can also replace ω and ω by standard Euclidean forms on U and on a neighborhood of π(U ). So, we can assume that h is analytic. By Lojasiewicz inequality, for every compact K of U, there exists N ≥ 1 such that h(z) |φ(z)| N for all z in K. On the other hand, exponential estimate (cf. [START_REF] Hörmander | An introduction to complex analysis in several variables[END_REF] and Section 4) applied to the plurisubharmonic function log |φ| says that φ -α is in L 1 (K, ω m ) for some α > 0. Therefore, h -α/N belongs to L 1 (K, ω m ). We obtain the desired property near z 0 by taking τ ≤ α/N.

Finally, the following results establish a relation between the regularity of functions on an analytic set and that of their lifts to a desingularization. Proposition 2.8. Let Z, Z, π be as in Lemma 2.7 and v be a function on Z. Assume that the lift of v to Z is (K, α)-Hölder continuous. Then, for every compact L of Z, there exist constants 0 < α ≤ 1 and a > 0, independent of v, such that v is (aK, α )-Hölder continuous on L.

Proof. Let ∆ be the diagonal of Z × Z. We still denote by π the map induced on the product Z × Z and we set ∆ = π -1 (∆). As in Lemma 2.6, by Lojasiewicz inequality, we have

dist( a, ∆) M dist(π( a), ∆), if π( a) ∈ L × L.
Therefore, if we set a = ( x, x ) and π( a) = (x, x ), then we can rewrite this inequality as

(dist( x, z) + dist( x , z )) M dist(x, x ), (2.1)
for some z, z ∈ Z such that π( z) = π( z ). Now, let v be as in the proposition and v = v • π denote its lift. Taking the same notation as above we have

|v(x) -v(x )| = | v( x) -v( x )| ≤ | v( x) -v( z)| + | v( z ) -v( x )|, since π( z) = π( z ) which implies that v( z) = v( z ). Therefore, the assumption on v implies that |v(x) -v(x )| ≤ K(dist( x, z) α + dist( z , x ) α ),
and nally (2.1) gives

|v(x) -v(x )| ≤ aKdist(x, x ) α/M ,
where a > 0 depends only on α, L and π.

Corollary 2.9. For every compact L of Z, there exists 0 < α ≤ 1 such that every continuous weakly holomorphic function on Z is α-Hölder continuous on L. Moreover, every uniformly bounded family of such functions is uniformly α-Hölder continuous on L.

Proof. Recall that a continuous function on Z is weakly holomorphic if it is holomorphic on the regular part of Z. The result is known if Z is smooth with α = 1. Therefore, in general, it is enough to apply Proposition 2.8 to a desingularization of Z.

In Proposition 3.4, we will need a similar result in a local setting but with a uniform control of the constants. It is the aim of the two following results. Proposition 2.10. Let Z, Z, π and L be as in Proposition 2.8. Let v be a function dened on a ball

B Z (y, r) ⊂ L with 0 < r ≤ 1/2. Assume that v = v•π is (K, α)-Hölder continuous. Then, there exist constants 0 < α ≤ 1, a > 0 and N ≥ 1 independent of v, y and r such that v is (aK, α )-Hölder continuous on B Z (y, r N ).
Proof. The proof is the same as that of Proposition 2.8 except that we have to check that if x and x are in B Z (y, r N ) with N ≥ 1 large enough, then v is well dened at the points z and z dened in (2.1).

Since π is holomorphic, we have dist(x, π( z)) dist( x, z). Therefore, by (2.1) we have

dist(x, π( z)) dist( x, z) + dist( x , z ) dist(x, x ) 1/M .
Hence, if N is large enough, then z and z belong to π -1 (B Z (y, r)). Then, the result follows as in Proposition 2.8. Corollary 2.11. For every compact L of Z, there are constants

0 < α ≤ 1, K > 0 and N ≥ 1 such that if v is a continuous weakly holomorphic function on B Z (y, r) ⊂ L with |v| ≤ 1 then v is (K/r, α)-Hölder continuous on B Z (y, r N ). Proof. Let π : Z → Z be a desingularization. Let z be in π -1 (B Z (y, r/2)). Since π is holomorphic, there is a > 0 such that B b Z ( z, r/a) is contained in π -1 (B Z (y, r))
. Therefore, by Cauchy's inequality, v is a/r-Lipschitz on π -1 (B Z (y, r/2)). Hence, the result follows from Proposition 2.10.

Volume estimate for endomorphisms

The multiplicities of an endomorphism f are strongly related to volume estimates which were used successfully to solve equidistribution problems. In what follows, we generalize Lojasiewicz type inequalities obtained in [START_REF] Erik | Complex dynamics in higher dimension[END_REF], [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF] and [START_REF] Dinh | Equidistribution speed for endomorphisms of projective spaces[END_REF] to analytic sets, possibly singular. The aim is to control the size of a ball under iterations of f in an invariant analytic set. Singularities, in particular the points where the analytic sets are not locally irreducible, lead to technical diculties.

In this section, X always denotes an irreducible analytic set of a smooth manifold. In order to avoid some problems related to the local connectedness of analytic sets, instead of the distance induced by an embedding of X, we consider the distance ρ dened by paths in X. Namely, if x, y ∈ X then ρ(x, y) is the length of the shortest path in X between x and y. These two distances on X are related by the following result (see e.g. [START_REF] Bierstone | Semianalytic and subanalytic sets[END_REF]). Theorem 3.1. Let K be a compact subset of X. There exists a constant r > 0 such that for all x, y ∈ K we have

dist(x, y) ≤ ρ(x, y) dist(x, y) r .
The rst step to state volume estimate is the following result. Proposition 3.2. Let Γ ⊂ B × B and X ⊂ B be two analytic subsets with X locally irreducible and such that the rst projection π : B × B → B denes a ramied covering of degree m from Γ to X. There exist constants a > 0 and b ≥ 1 such that if x, y ∈ X ∩ B 1/2 then we can write π -1 (x) ∩ Γ = {x 1 , . . . , x m } and π -1 (y) ∩ Γ = {y 1 , . . . , y m }, with dist(x, y) ≥ adist(x i , y i ) bm . Moreover, a increases with m but is independent of Γ and b depends only on X.

Proof. By Theorem 3.1, to establish the proposition, we can replace dist(x, y) by ρ(x, y) on X. For w ∈ X Reg , we can dene the j-th Weierstrass polynomial,

k + 1 ≤ j ≤ 2k, on t ∈ C P j (t, w) = z∈π -1 (w)∩Γ (t -z j ) = m l=0 a j,l (w)t l ,
where z = ((z 1 , . . . , z k ), (z k+1 , . . . , z 2k )) ∈ B × B. The coecients a j,l are holomorphic on X Reg and uniformly bounded by m! since Γ ⊂ B × B. As X is locally irreducible, they can be extended continuously to X (see e.g. [START_REF] Demailly | Complex analytic and dierential geometry[END_REF]). It gives a continuous extension of the polynomials P j to X with P j (z j , π(z)) = 0 if z is in Γ. Moreover, by Corollary 2.9 there exists α > 0 such that the coecients a j,l are uniformly α-Hölder continuous on X ∩ B 3/4 with respect to ρ.

We claim that there is a constant a > 0 such that if x, y ∈ X ∩ B 1/2 and x ∈ B with x := (x, x ) ∈ Γ then there is y ∈ π -1 (y) ∩ Γ with ρ(x, y) ≥ adist( x, y) bm , where b = 1/α. From this, the result follows exactly as in the end of the proof of [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF]Lemma 4.3].

It remains to prove the claim. Fix c > 0 large enough and let r = ρ(x, y). We can assume that r is non-zero and suciently small, otherwise the result is obvious. Since the covering has degree m, we can nd an integer 2 ≤ l ≤ 4km such that for all k + 1 ≤ j ≤ 2k, no root of the polynomial P j (t, x) satises

c(l -1)r 1/bm ≤ | x j -t| ≤ c(l + 1)r 1/bm .
It gives a security ring over x which does not intersect Γ. Using the regularity of P j , it can be extended to a neighborhood of x. More precisely, for θ ∈ R we dene ξ j = x j + cle iθ r 1/bm and G j,c,θ (w) = c -m+1 P j (ξ j , w).

The choice of l implies that

|G j,c,θ (x)| = c -m+1 |P j (ξ j , x)| ≥ c -m+1 z∈π -1 (x) |ξ j -z j | ≥ cr α .
Moreover, we deduce from the properties on the coecients a j,l that the functions G j,c,θ are uniformly α-Hölder continuous on X ∩ B 3/4 with respect to (j, c, θ). Hence, if c is large enough, they do not vanish on B := {z ∈ X | ρ(x, z) < 2r} which contains y.

It implies that P j (t, w) = 0 if w is in B and |t -x j | = lcr 1/bm . Therefore, if we denote by Σ the boundary of the polydisc of center x and of radius lcr 1/bm , we have Γ ∩ (B × Σ) = ∅. Hence, since B is connected and contains y, by continuity there is a point y in π -1 (y) ∩ Γ with | x j -y j | ≤ lcr 1/bm . This completes the proof of the claim. Remark 3.3. Our proof shows that if we assume that m ≤ m 0 for a xed m 0 , then a depends only on the Hölder constants (K, α) associated with a j,l . Furthermore, if α is xed then a is proportional to K. Note that without assumption on the constants a and b, Proposition 3.2 can be deduced from Theorem 2.1.

The control of multiplicities of the covering gives the following more precise result. Proposition 3.4. Let X, Γ and π be as above. Let Z ⊂ X be a proper analytic subset. Assume that the multiplicity at each point of π -1 (x) is at most equal to s if x ∈ X \ Z. Then, there exist constants a > 0, b ≥ 1 and N ≥ 1 such that for all 0 < t ≤ 1/2 and x, y ∈ X ∩ B 1/2 with dist(x, Z) > t and dist(y, Z) > t, we can write

π -1 (x) ∩ Γ = {x 1 , . . . , x m } and π -1 (y) ∩ Γ = {y 1 , . . . , y m }, with dist(x, y) ≥ at N dist(x i , y i ) bs .
Proof. Let t > 0 and x ∈ X ∩ B 1/2 with dist(x, Z) > t. We want to nd a neighborhood B ⊂ X of x such that each component of Γ ∩ π -1 (B) denes a ramied covering of degree at most equal to s over B.

We construct an analytic set Y associated to the multiplicities of Γ.

Namely, rst dene Y ⊂ Γ s+1 by {z ∈ Γ s+1 | π • τ i (z) = π • τ j (z), 1 ≤ i, j ≤ s + 1}, where τ i , 1 ≤ i ≤ s + 1, are the canonical projections of Γ s+1 onto Γ. For i = j we set A i,j = {z ∈ Y | τ i (z) = τ j (z)}. Then, Y is dened by Y \ ∪ i =j A i,j . The map π 1 = π • τ 1 : Y → X denes a ramied covering. If x is generic in X then a point z ∈ Y over x ∈ X represents a family of (s + 1) distinct points in π -1 (x) ∩ Γ.
If π denotes the second projection of B × B onto B, consider the map h : Y → C (s+1) 2 dene by

h(z) = (π • τ i (z) -π • τ j (z)) 1≤i,j≤s+1 .
By construction, h(z) = 0 means precisely that there is a point in Γ with multiplicity greater than s over π 1 (z). It implies that π

1 (h -1 (0)) ⊂ Z. Hence, Theorem 2.1 implies there is a constant M > 0 such that if π(z) ∈ X ∩ B 1/2 then h(z) dist(z, h -1 (0)) M dist(π 1 (z), Z) M . (3.1)
Let a, b > 0 be the constants as in Proposition 3.2. As in the proof of that proposition, we use the distance function ρ on X. Fix γ > 0 small enough and set

B = {z ∈ X | ρ(x, z) < γt M bm }. For x = (x, x ) in Γ, we can choose 2 ≤ l ≤ 8m such that π -1 (x) ∩ Γ do not intersect the ring r(l -2) ≤ x -w ≤ r(l + 2),
where r = (a -1 γ) 1/bm t M . Hence, if H denotes the ball of center x and of radius rl, we have dist(π -1 (x) ∩ Γ, B × ∂H) > r. Therefore, by Proposition 3.2 Γ ∩ B × ∂H = ∅. It assures that π is proper on Γ ∩ B × H and then denes a ramied covering.

Moreover, this covering has degree at most equal to s. Otherwise, according to the radius of H, we have

min z∈π -1 1 (x) h(z) ≤ (s + 1) 2 2rl,
which is in contradiction with (3.1) if γ is small enough, since dist(x, Z) > t and r = (a -1 γ) 1/bm t M . Now, we want to apply Proposition 3.2 to this covering. But, in order to control the constants, we have to reduce B. Indeed, according to Remark 3.3, the constants of that proposition depend only on the Hölder continuity of the coecients a j,l (and on the degree of the covering). These coecients are bounded continuous weakly holomorphic functions dened on B then by Corollary 2.11 with L = X ∩ B 3/4 , they are (Kt -M , α)-Hölder continuous on B(x, t M ) for some M ≥ 1 large enough, 0 < α ≤ 1 and K > 0 independent of x and t. Therefore, after a coordinates dilation by t -M at x, we can apply Proposition 3.2 with the same exponent b and the second constant proportional to some power of t. Finally, let y ∈ X. We can assume that ρ(x, y) ≤ t M /2, otherwise the proposition is obvious. Hence, the previous observation implies that π -1 (x) ∩ Γ ∩ B × H = {x 1 , . . . , x s } and π -1 (y) ∩ Γ ∩ B × H = {y 1 , . . . , y s }, with ρ(x, y) ≥ a t N dist(x i , y i ) bs where a > 0, b ≥ 1 and N ≥ 1 are independent of x and t. More precisely, we can write N = M + N , where the contribution in t M comes from the dilation and that in t N comes from the estimate on Hölder continuity. The construction can be applied to each component of Γ ∩ π -1 (B). This gives the result.

We now consider the dynamical context. Assume that X ⊂ P k is an irreducible analytic set of dimensions l which is invariant by f. Denote by g the restriction of f to X. For x in X, we dene the local multiplicity of g at x as the maximal number of points in g -1 (z) which are near x for z ∈ X close enough to g(x). The local multiplicity is smaller than the topological degree i.e. the number of points in g -1 (x) for x generic in X. In our case, the topological degree of g is equal to d l , see [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF].

There exists a nite covering (U i ) i∈I of X by open subsets of P k such that X ∩ U i can be decomposed into locally irreducible components. Hence, we can apply Proposition 3.4 to the graph of g over each component of X ∩ U i . It gives the following corollary. Corollary 3.5. Let η > 1 and Z ⊂ X be a proper analytic subset. Assume that the local multiplicity of g is less than η outside g -1 (Z). Then there are constants a > 0, b ≥ 1 and N ≥ 1 such that if 0 < t < 1 and x, y are two points outside Z t where X is locally irreducible, then we can write g -1 (x) = {x 1 , . . . , x d l } and g -1 (y) = {y 1 , . . . , y d l } with dist(x, y) ≥ at N dist(x j , y j ) bη .

From this, we obtain the following size estimate for image of balls which is crucial in the proof of our main result. Corollary 3.6. Let δ > 1 and E ⊂ X be a proper analytic subset. Denote by E the preimage of E by g. Assume that the local multiplicity is less than δ outside E. There exist constants 0 < A ≤ 1, b ≥ 1 and N ≥ 1 such that if 0 < t ≤ 1/2, r < t/2 and x ∈ X \ E t , then g(B X (x, r)) contains a ball of radius At N r bδ . Moreover, b depends only on X.

Proof. Fix t > 0 and r < t/2. As in the proof of Lemma 2.5 with Y = X Sing ∪ g -1 (X Sing ), possibly after replacing r by cr l for some c > 0, we can assume that B X (x, r) and g(B X (x, r)) are contained in X Reg . The local multiplicity of f on X is bounded on d l . So, there exists 2 ≤ i ≤ 4d l , such that the ring { r(i-1) 4d l +1 ≤ dist(x, x ) ≤ r(i+1) 4d l +1 } contains no preimage of g(x). Thus, if x ∈ ∂B X (x, r i 4d l +1 ), then dist(x , g -1 (g(x))) ≥ r 4d l + 1 .

Moreover, we can apply Corollary 3.5 with η = d l and Z = ∅. Hence, there exist a, b > 0 such that g(B X (x, r i 4d l +1 )) ⊂ X \ E a(t/2) bd l . Therefore, we can apply once again Corollary 3.5 with η = δ, Z = E and a(t/2) bd l instead of t. We get, for some constants a > 0 and

N 0 ≥ 1 dist(g(x ), g(x)) ≥ a a t 2 bd l N 0 r 4d l + 1 bδ ,
and, since g is an open mapping near x B X (g(x), At N r bδ ) ⊂ g(B X (x, r))

with A = a N 0 a 2 N 0 bd l (4d l +1) bδ and N = N 0 bd l . Remark 3.7. When X is smooth, the ball in g(B X (x, r)) can be chosen centered at g(x).

Psh functions and exponential estimates

We refer to [START_REF] Demailly | Complex analytic and dierential geometry[END_REF] for basics on currents and plurisubharmonic (psh for short) functions. Let T be a positive closed (1, 1)-current of mass 1 on P k with continuous local potentials. Let us recall briey the associated notions of psh and weakly psh (wpsh for short) modulo T functions introduced in [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF].

Let Y be an analytic space. A function v : Y → R ∪ {-∞} is wpsh if it is psh on Y Reg and for y in Y, we have v(y) = lim sup v(z) with z ∈ Y Reg and z → y. These functions coincide with psh functions if Y is smooth. On compact spaces, the notion is very restrictive. However, if X is an analytic subset of P k , we have the more exible notion of wpsh modulo T function on X. Locally, it is the dierence of a wpsh function on X and a potential of T. If X is smooth, we say that the function is psh modulo T.

Note that the restriction of a psh modulo T function to an analytic subset is either wpsh modulo T or equal to -∞ on an irreducible component. If u is wpsh modulo T on X then dd c (u[X]) + T ∧ [X] is a positive closed current supported on X. On the other hand, if S is a positive closed (1, 1)-current on P k with mass 1, there is a psh modulo T function u on P k , unique up to a constant, such that S = dd c u + T.

These notions bring good compactness properties which permit to obtain uniform estimates. We have the following statements established in [START_REF] Dinh | Equidistribution towards the Green current for holomorphic maps[END_REF]. Proposition 4.1. Let (u n ) be a sequence of wpsh modulo T functions on X, uniformly bounded from above. Then there is a subsequence (u n i ) satisfying one of the following properties:

• There is an irreducible component Y of X such that (u n i ) converges uniformly to -∞ on Y \ X Sing .
• (u n i ) converges in L p (X) to a wpsh modulo T function u for every

1 ≤ p < +∞.
In the last case, lim sup u n i ≤ u on X with equality almost everywhere.

It implies the following lemma.

Lemma 4.2. Let G be a family of psh modulo T functions on P k uniformly bounded from above. Assume that each irreducible component of X contains an analytic subset Y such that the restriction of

G to Y is bounded in L 1 (Y ).
Then, the restriction of G to X is bounded in L 1 (X).

A classical result of Hörmander [START_REF] Hörmander | An introduction to complex analysis in several variables[END_REF] gives a uniform bound to exp(-u) in L 1 (B 1/2 ) for u in a class of psh functions in the unit ball of C k . Similar estimates can be obtained for compact families of quasi-psh functions. From now, we assume that T has (K, α)-Hölder continuous local potentials, with 0 < α ≤ 1 and K > 0. In the rest of this section, we establish exponential estimates for psh modulo T functions in dierent situations. A key observation in our approach is that Hölder continuity allows us to work with non-compact families. In the sequel, we will apply these estimates to T, the Green current associated to f. They allow us to control the volume of some sublevel sets of potentials of currents near exceptional sets.

As a consequence of classical Hörmander's estimate, we have the following lemma which will be of constant use. Here, ν denotes the standard volume form on C k and T is seen as a xed current on the unit ball B of C k . We assume that it admits a potential g which is (K, α)-Hölder continuous on B.

Lemma 4.3. Let v be a psh modulo T function in B t with v ≤ 0 and v(0) > -∞. Let 0 < s < -v(0) -1 and t > 0 be such that Kt α ≤ s -1 . There is a constant c > 0 independent of v, s and t such that

B t/2 exp(- sv 2 )ν ≤ ct 2k . (4.1)
Proof. As v is psh modulo T , we have v = v -g with v psh. We set

v(z) = v (z) -g(0) -Kt α . Then v is psh in B t , v(0) = v(0) -Kt α ≥ -2s -1
and v ≤ v ≤ 0 because g(z) -g(0) ≤ Kt α on B t . By [START_REF] Hörmander | An introduction to complex analysis in several variables[END_REF]Theorem 4.4.5] there exists c > 0 such that

B 1/2 exp(- s v(tz) 2 )ν ≤ c,
thus, by a change of variables z → tz, we get

B t/2 exp(- sv 2 )ν ≤ B t/2 exp(- s v 2 )ν ≤ ct 2k .
For the rest of the section, X always denotes an irreducible analytic subset of P k of dimension l and v is a psh modulo T function in P k with v ≤ 0. In Section 6 we will extend the previous result to the neighborhood of X, where the condition at 0 is replaced by an integrability condition on X. For this purpose, we have to control the size of sublevel sets of v in X. This is the aim of the following global result. Lemma 4.4. For X as above, there exists q 0 ≥ 1 with the following property. For q > q 0 we set = 2lq 0 /qα and take M > 0 and s ≥ 1 such that s 1+ v L q (X) ≤ M. Then, there exist constants a, c > 0 independent of v, q and s such that

X exp(-asv)ω l ≤ c. (4.2)
If X is smooth, we can choose q 0 = 1.

Proof. First, assume that X is a compact smooth manifold with a volume form η. Since X has dimension l, for t > 0 we can cover it by balls (B i ) i∈I with B i := B X (x i , t) and such that |I| ≤ c t -2l for some c > 0. Let t = s -1/α . As above, in each ball B X (x i , 2t) we can write v = v i -g i , where g i is a local potential of T. Using local charts at x i , we can identify B X (x i , 2t) with B 2t in C l . We consider v i (z) = s(v i (tz) -g i (0)). These functions are psh in B 2 . We show that they belong to a compact family, independent of v and s. Using a change of variables z → tz and Hölder's inequality, we get

v i L 1 (B 2 ) ≤ B 2 s|v(tz)|ν + B 2 s|g i (tz) -g i (0)|ν ≤ st -2l v L q (X) |B 2 | 1/p t 2l/p + 2 α K|B 2 | = s 1+ v L q (X) |B 2 | 1/p + 2 α K|B 2 | ≤ M |B 2 | 1/p + 2 α K|B 2 | ≤ M ,
where p is the conjugate of q, |B 2 | is the volume of B 2 and M is a positive constant. The family

U = {u ∈ P SH(B 2 ) | u L 1 (B 2 ) ≤ M } is compact so there exists a constant a > 0 such that exp(-au) L 1 (B) is uniformly bounded for all u ∈ U . Therefore, for i ∈ I Bt exp(-as(v i (z) -g i (0)))ν t 2l .
Moreover, the Hölder continuity implies that -sv(z

) ≤ K -s(v i (z) -g i (x i )) in B i . Hence, since (B i ) i∈I is a covering of X we obtain X exp(-asv)η ≤ i∈I B i exp(-asv)η ≤ i∈I B i exp(a(K -s(v i (z) -g i (x i ))))η i∈I t 2l ≤ c .
This implies the lemma if X is smooth with q 0 = 1.

In the general case, we consider a desingularization π : X → X with a volume form η on X. The map π is surjective, then by Lemma 2.7, there exists q 0 ≥ 1 such that

v L q/q 0 ( b X,η) v L q (X,ω l ) .
Moreover, π * (T ) possesses α-Hölder local potentials and v ≤ 0 is psh modulo π * (T ). Therefore, this choice of q 0 allows us to apply the lemma on X and get b X exp(-as v)η ≤ c.

The result follows since

X exp(-asv)ω l = b X exp(-as v)π * (ω l ) ≤ h ∞ b X exp(-as v)η,
where we write π * (ω l ) = hη.

The following estimate is a consequence of Lemma 4.3 and is related to the geometry of sublevel sets of psh modulo T functions. In Section 6, it will establish the existence of balls where we can apply our volume estimates. Lemma 4.5. For s ≥ 2 set F s = {x ∈ X | v(x) ≤ -s -1 }. There are constants β, c > 0 independent of v and s such that if F s contains no ball of radius s -β then

X exp(- sv 2 )ω l ≤ c.
Proof. We rst consider the case where X is smooth. Let t = 4 -1 (Ks) -1/α .

As in the proof of the previous lemma, we cover X by balls (B i ) i∈I of radius t with |I| ≤ c t -2l , c > 0. Assume there is no ball of radius t in F s . Hence, for each i ∈ I there exists

x i in B i such that v(x i ) > -s -1 .
The balls B i of center x i and of radius 2t cover X.

Thus X exp(- sv 2 )ω l ≤ i∈I B i exp(- sv 2 )ω l .
But, s < -v(x i ) -1 and K(4t) α ≤ s -1 therefore we can apply Lemma 4.3 on each ball

B i exp(- sv 2 )ω l t 2l .
Hence, we get

X exp(- sv 2 )ω l i∈I t 2l ≤ c ,
which gives the result when X is smooth with β > 1/α such that s -β < t.

If X is singular, we consider a desingularization π : X → X. By Lemma 2.5, there exists N ≥ 1 such that the image of a ball of radius r under π contains a ball of radius r N . Hence, if β is large enough, the hypothesis on F s assures there is no ball of radius t in F s = π -1 (F s ). Then, we can apply the lemma to v = v • π which is psh modulo π * (T ). We get

X exp(- sv 2 )ω l = b X exp(- s v 2 )π * (ω l ) ≤ c,
for some c > 0, since π * (ω l ) is smooth.

Exceptional sets

Let f be an endomorphism of P k of algebraic degree d ≥ 2. The aim of this section is to construct two families A λ and B λ of analytic sets where the iterate sequence f n has important local multiplicities. Let X ⊂ P k be an irreducible invariant analytic set. Dene κ X,n (x), or simply κ n (x) if no confusion is possible, as the local multiplicity of f n |X at x. It is a submultiplicative cocycles, namely it is upper semi-continuous for the Zariski topology on X, min X κ n = 1 and for any m, n ≥ 0 and x ∈ X we have the following sub-multiplicative relation

κ n+m (x) ≤ κ m (f n (x))κ n (x).
The inequality may be strict when X is singular. Dene

κ -n (x) := max y∈(f |X ) -n (x)
κ n (y).

We recall the following theorem of Dinh [START_REF] Dinh | Analytic multiplicative cocycles over holomorphic dynamical systems[END_REF], see also [START_REF] Favre | Multiplicity of holomorphic functions[END_REF].

Theorem 5.1. The sequence of functions κ 1/n -n converges pointwise to a function κ -. Moreover, for every λ > 1, the level set E λ (X) = {κ -≥ λ} is a proper analytic subset of X which is invariant under f |X . In particular, κ - is upper semi-continuous in the Zariski sense.

For a generic endomorphism of P k , E λ (P k ) is empty. In this case, Theorem 1.2 is already know in all bidegrees [START_REF] Dinh | Super-potentials of positive closed currents, intersection theory and dynamics[END_REF]. In our proof, we will proceed by induction, proving the exponentially fast convergence on X if it is already established on each irreducible component of E λ (X). But, even if E λ (X) is invariant, its irreducible components are periodic and not invariant in general. Therefore, if X is only periodic, we dene E λ (X) in the same way, replacing f by f p and λ by λ p , where p is a period of X. By Theorem 5.1, this denition is independent of the choice of p.

Fix 1 < λ < d. Dene the family B λ of exceptional sets as follows. First, we set P k ∈ B λ . If X is in B λ , we add to B λ all irreducible components of E λ (X). This family is nite and since the functions κ -are upper semicontinuous in the Zariski sense, there exists 1 < δ < λ such that B λ = B δ , or equivalently E λ (X) = E δ (X) if X ∈ B λ . This will give us some exibility in order to obtain estimates using an induction process.

As all elements of B λ are periodic, they are invariant under some iterate f n 0 . Let us remark that it is sucient to prove Theorem 1.2 for an iterate of f. Hence, we can assume that n 0 = 1, replacing f and λ by f n 0 and λ n 0 . Dinh also proved that

κ n 1 < δ n 1 outside (f |X ) -n 1 (E λ (X)) for some n 1 ≥ 1.
Once again, we can assume that n 1 = 1.

The second family A λ , that takes place in Theorem 1.2, is dened as the set of minimal elements for the inclusion in B λ . This family is not empty and each element of B λ contains at least one element of A λ . Note that no element of A λ is contained in another one. These analytic sets play a special role in the next section, to start induction and to obtain compactness properties. When P k is an element of A λ , it is the only element in A λ and the exceptional set is empty. Otherwise, dene the exceptional set as the union of all the elements of A λ .

6

Equidistribution speed

This section is devoted to the proof of Theorem 1.2. Fix an endomorphism f of algebraic degree d ≥ 2 of P k , and denote by T its Green current. Recall that T is totally invariant i.e. d -1 f * (T ) = T, and has (K, α)-Hölder continuous local potentials for some 0 < α ≤ 1, K > 0.

Fix C > 0 and 1 < λ < d, and let A λ , B λ be as in Section 5. Dene F λ (C) as the family of psh modulo T functions v on P k such that max P k v = 0 and v L 1 (X) ≤ C for all X ∈ A λ . By construction of A λ , Lemma 4.2 implies that F λ (C) is compact for each C > 0. Moreover, if X is an element of B λ , the restriction of F λ (C) to X forms a family of wpsh modulo T functions on X which is relatively compact in L p (X) for every 1 ≤ p < +∞.

If S is a positive closed (1, 1)-current of mass 1, it is cohomologous to T. Hence, there exists a unique psh modulo T function u on P k such that S = dd c u + T and max P k u = 0. We call u the dynamical potential of S. As T is totally invariant, the dynamical potential of

S n = d -n (f n ) * (S) is u n = d -n u • f n .
Since S n -T is a continuous linear operator on C 0 (P k ) whose norm is bounded, by interpolation theory between Banach spaces we have

S n -T C β S n -T β/2
C 2 , uniformly in S and n, see [START_REF] Triebel | Interpolation theory, function spaces, dierential operators[END_REF]. Consequently, in order to prove Theorem 1.2 we can assume that β = 2.

Moreover, it is easy to see that dd c φ ∞ φ C 2 for φ in C 2 (P k ). Therefore,

| S n -T, φ | = | dd c u n , φ | = | u n , dd c φ | φ C 2 u n L 1 (P k ) .
Hence, Theorem 1.2 is a direct consequence of the following theorem applied to p = 1 and X = P k .

Theorem 6.1. For each 1 ≤ p < +∞ and X ∈ B λ there exists a constant A X,p such that for all u ∈ F λ (C) and n ≥ 0 we have

u n L p (X) ≤ A X,p λ d n ,
where

u n = d -n u • f n .
As in Section 5, we can assume that each element of B λ is invariant by f, and there is 1 < δ < λ satisfying the following properties for all X in B λ :

• E λ (X) = E δ (X), • κ X,1 < δ outside E λ (X) = (f |X ) -1 (E λ (X)).
Let X be an element B λ of dimension l and λ 1 > 0 with δ < λ 1 < λ. Assume that Theorem 6.1 is true on each irreducible component of E λ = E λ (X) for λ 1 and all p ≥ 1. To prove it on X, we consider the sublevel set K n = {x ∈ X | u n (x) ≤ -s n } for a suitable constant s n . Exponential estimates on E λ will prove that its image by f i , 0 ≤ i ≤ n, cannot be concentrated near E λ . Therefore, volume estimates will imply that f

n (K n ) = {x ∈ X | u(x) ≤ -d n s n } is large if Theorem 6.
1 is false on X. Hence, a good choice of s n , allowed by the gap between λ 1 and λ, will give a contradiction.

We rst x some constants. In Corollary 3.6 the constant b depends only on X. Then, by replacing f by f n and δ by δ n with bδ n < δ n < λ n 1 , we can assume that b = 1. Let 0 < A ≤ 1, N ≥ 1 be the other constants of Corollary 3.6. Fix λ 2 , λ 3 > 0 such that • δ < λ 1 < λ 2 < λ 3 < λ, and • q > q 0 large enough such that λ 1 /d < (λ 2 /d) 1+ where and q 0 are dened in Lemma 4.4.

Multiplicities of f |X are controlled outside E λ . By induction hypothesis, we have a control of u n on E λ . We want to extend it to E λ . Let E be an irreducible component of E λ . The restriction of f to each component of (f |X ) -1 (E) is surjective onto E. Therefore, we deduce from Lemma 2.7 that there exists q ≥ 1 such that

v • f L q ((f |X ) -1 (E)) v L qq (E) ,
for all psh modulo T function v on P k . Hence, by induction hypothesis, there is a constant M > 0 such that u n L q ( e E λ ) ≤ M (λ 1 /d) n for n ≥ 1. The next step is to obtain exponential estimates in a neighborhood of E λ . Lemma 6.2. There exist constants c, η ≥ 1 and n

0 ≥ 1 such that if n ≥ n 0 then for all u ∈ F λ (C) we have e E λ,tn exp(-(d/λ 2 ) n u n )ω l ≤ c, where t n = (λ 2 /d) nη .
Proof. Let E be an irreducible component of E λ of dimension i. According to the choice of q, we can nd λ 2 < λ 2 such that λ 1 /d < (λ 2 /d) 1+ . Hence

u n L q (E) (d/λ 2 ) (1+ )n ≤ M (λ 1 /d) n (d/λ 2 ) (1+ )n ≤ M.
and by Lemma 4.4 with s = (d/λ 2 ) n we have

E exp(-a (d/λ 2 ) n u n )ω i ≤ c , for some constants a , c > 0. Therefore, if we set ρ n = (λ 2 /d) n , the volume in E of F n = {x ∈ E | u n (x) ≤ -ρ n } is smaller than c exp(-a (λ 2 /λ 2 ) n ). In particular, F n contains no ball of radius ρ 2/α n for n large enough. If X is smooth then set t n = ρ 1/α n .
As in Lemma 4.5, for n large enough, we can nd a covering of E tn by balls with center in E and of radius 2t n on which Lemma 4.3 holds. Hence, we get

Et n exp(-au n /ρ n )ω l ≤ c,
for some a, c > 0. The same argument with λ 2 slightly smaller shows that we can choose a = 1. We conclude the proof by summing on all irreducible components of E λ .

When X is singular, we consider a desingularization π : X → X. In order to establish the estimate near E, we proceed inductively as follows. Assume that there exists a triplet (A, a, θ) with a > 0, θ ≥ 1 and an analytic set A ⊂ E such that

E t θ \A t 1/θ exp(-au n /ρ n )ω l
is uniformly bounded in n ≥ 0 for t ≤ ρ n . Then, using the properties of the elements of B λ and dynamical arguments, we claim that a similar estimate holds if we substitute (A, a, θ) by some (A , a , θ ) with dim(A ) < dim(A). It will give the result for η large enough after less than l steps since dim(E) < l.

More precisely, let V be an irreducible component of A with maximal dimension. We distinguish two cases, according to whether V is in B λ or not. In the rst case, we know that for all p ≥ 1, u n L p (V ) (λ 1 /d) n . We set V := π -1 (V ). We denote by V 1 the union of all components of V which are mapped onto V and by V 2 the union of the other components of V . Therefore, Lemma 2.7 implies that

u n L p ( b V 1 ) (λ 1 /d) n ,
for all p ≥ 1, where u n = u n • π. Hence, the smooth version of the lemma implies that b

V 1,ρ 1/α n exp(-a u n /ρ n )π * (ω l )
is uniformly bounded for a > 0 small enough. Moreover, by Lemma 2.6, there exists a constant θ ≥ 1 such that π( V 1,t ) contains V t θ \ V 2,t 1/2 , where V 2 = π( V 2 ). It gives the desired result near V, since dim(V 2 ) < dim(V ).

From now, we can assume that no irreducible component of A with maximal dimension belongs to B λ (in particular A = E). Let V denote the union of all irreducible components of A with maximal dimension. In particular, these components are not totally invariant for f |E , therefore there exist an analytic set Z ⊂ E containing no component of V and an integer m ≥ 1 such that f m (Z) = V. We set Z = Z ∩ A. )ω l , (

since (f m |X ) * (ω l ) ω l . Moreover, for a small enough the right-hand side in (6.1) is bounded uniformly on n and dim(V ) = dim(Z ) < dim(V ) since Z contains no component of V. This together with the estimate outside A prove the claim with A = V .

From now, we x p ≥ 1 and for u in F λ (C) denote N (u) = {n ≥ 1 | u n L p (X) ≥ (λ/d) n } and denote by N the union of N (u) for all u. Our goal is to prove that N is nite, which will imply Theorem 6.1. For this purpose, we have the following result. On the other hand, let β be the constant in Lemma 4.5. For n suciently large we have (d/λ 3 ) n ≥ 2. Hence, Lemma 4.5 with s = (d/λ 3 ) n imply that K n has to contain a ball of radius (λ 3 /d) nβ , otherwise the right-hand side of (6.2) would be bounded uniformly on n, which is impossible since λ 3 < λ.

We can now complete the proof of the main theorem.

End of the proof of Theorem 6.1. If B ⊂ X is a Borel set then |B| denotes its volume with respect to the measure ω l . As we have already seen, the volume of a ball of radius r in X is larger than c r 2l , 0 < c ≤ 1. Therefore, observe that if x is in E λ,tn/2 then | E λ,tn ∩ B X (x, r)| = |B X (x, r)| ≥ c (r/2) 2l for r < t n /2. From now, assume in order to obtain a contradiction that N is innite. Consider u ∈ F λ (C) and n ∈ N (u) large enough. Fix also β large enough. So, we have (λ 3 /d) βn < t n /4 and c exp(-(λ 3 /λ 2 ) n ) ≤ c (A δ (t n /2) N δ (λ 3 /d) βn /2) 2l , (

where c is dened in Lemma 6.2. Let r 0 = (λ 3 /d) βn , and for 1 ≤ i ≤ n let r i = A(t n /2) N r δ i-1 . We will prove by induction that for 0 ≤ i ≤ n, f i (K n ) = {x ∈ X | u n-i (x) ≤ -λ n 3 /d n-i } contains a ball B i of radius r i . Since β is large, Lemma 6.3 implies that the assertion is true for i = 0. Let 0 ≤ i ≤ n -1 and assume the property is true for i. We deduce from Lemma 6.2 that e E λ,t n-i exp(-(d/λ 2 ) n-i u n-i )ω l ≤ c, and in particular

| E λ,tn ∩ B i | ≤ | E λ,t n-i ∩ f i (K n )| < c exp(-(λ 3 /λ 2 ) n λ i 2 ),
since t n ≤ t n-i . This and (6.3) imply that

|B i | ≥ c r 2l i > 2 2l | E λ,tn ∩ B i |,
since r i ≥ (A δ (t n /2) N δ r 0 ) δ i and δ < λ 2 . Consequently, the center of B i is not in E λ,tn/2 and by Corollary 3.6, f (B i ) ⊂ f i+1 (K n ) contains a ball B i+1 of radius r i+1 = A(t n /2) N r δ i . Note that we already reduced the problem to the case where the constant b in Corollary 3.6 is equal to 1.

Therefore, for all n in N (u) suciently large, the volume of f n (K n ) = {x ∈ X | u(x) ≤ -λ n 3 } is greater than D nδ n , with 0 < D < 1 independent

Lemma 6 . 3 .

 63 There are constants n 1 ≥ 1 andβ ≥ 1 such that if n is in N (u) with n ≥ n 1 then K n = {x ∈ X | u n (x) ≤ -(λ 3 /d) n } contains a ball of radius (λ 3 /d) βn .Proof. Since x p exp(x) if x ≥ 0, we deduce from the assumption on u n L p (X) that(λ/λ 3 ) n X (-(d/λ 3 ) n u n ) p ω l 1/p X exp(-(d/λ 3 ) n u n /2)ω l

  1. Moreover, up to an isometry of C k , we can assume that max |z i | = |z 1 | and max |z i | = |z 1 |. Indeed, we can send z and z into the plane generated by the rst two coordinates and then use the rotation group of this plane. Therefore, in the chart w 1 = 1, we have z = (z 1 , w 2 , . . . , w k ), z = (z 1 , w 2 , . . . , w k ) with |w i |, |w i | ≤ 1.

  The assumption on A and θ implies that if t ≤ ρ n thenZ t θ \A t 1/θ exp(-au n /ρ n )ω l is bounded uniformly on n. By Corollary 2.2, Z t θ ∩ A t 1/θ is contained in Z t 1/θ for some θ > θ. So, exp(-au n /ρ n )ω lis bounded uniformly on n. Fix a constant B > 1 large enough. We deduce from Corollary 3.6 applied to P k that for all t > 0,f m (Z t ) contains V B -1 t d mk . Moreover, since f m is Lipschitz, f m (Z t ) is contained in V Bt , where V = f m (Z ). So, we have f m (Z t θ \ Z t 1/θ ) ⊃ V t θ \ V t 1/θ ,for t > 0 small enough and θ > θ large enough. It follows that

	Z t θ \Z	t 1/θ
	V t θ \V	Z |X ) Z t θ \Z t 1/θ exp(-a u n+m λ m 2 ρ n+m )(f m t θ \Z t 1/θ u n+m λ m 2 exp(-a ρ n+m

t 1/θ exp(-a u n /ρ n )ω l ≤ * (ω l )

of u and n. This contradicts the inequality δ < λ 3 . Indeed, since F λ (C) is bounded in L q (X), by Lemma 4.4 there exists a > 0 such that

Hence, N is nite and in particular bounded by some n 2 ≥ 1. We conclude using the fact that the restriction of

to X is a relatively compact family of wpsh modulo T functions and then bounded in L p (X). Therefore, we have

if n ≤ n 2 and thus for every n ≥ 0 by the denition of N .


Johan Taflin

Speed of convergence towards attracting sets for endomorphisms of P k

Speed of convergence towards attracting sets for endomorphisms of P k

Introduction

Let f be a holomorphic endomorphism of algebraic degree d ≥ 2 on the complex projective space P k . A compact subset A of P k is called an attracting set if it has a trapping neighborhood U i.e. f (U ) U and A = ∩ n≥0 f n (U ) where

f n := f • • • • • f, n times. It follows that A is invariant, f (A) = A.
Furthermore, if A contains a dense orbit then A is a trapped attractor. Typical examples of such objects are fractal and their underlying dynamics are hard to study. We refer to [START_REF] Milnor | On the concept of attractor[END_REF], [START_REF] Ruelle | Elements of differentiable dynamics and bifurcation theory[END_REF] for general discussions on attractors and to [START_REF] Fornaess | Attractors in P 2 . In Several complex variables[END_REF], [START_REF] Jonsson | A nonalgebraic attractor in P 2[END_REF], [START_REF] Fornaess | Dynamics of P 2 (examples)[END_REF], [START_REF] Bonifant | Elliptic curves as attractors in P 2 . I. Dynamics[END_REF], [START_REF] Taflin | Invariant elliptic curves as attractors in the projective plane[END_REF] and references therein for examples of different types of attractors in P 2 .

Attracting sets are stable under small perturbations. Indeed, if f has an attracting set A = ∩ n≥0 f n (U ) then any small perturbation f of f has an attracting set defined by A = ∩ n≥0 f n (U ). For example, when f restricted to C k defines a polynomial self-map then the hyperplane at infinity P k \ C k is an attracting set. In the same way, it is easy to create examples where the attracting set is a projective subspace of arbitrary dimension. In this paper, we consider a family of endomorphisms, stable under small perturbations, which contains these examples. It was introduced by Dinh in [START_REF] Dinh | Attracting current and equilibrium measure for attractors on P k[END_REF] and we briefly recall the context.

In the sequel, we always assume that f possesses an attracting set A which has a trapped neighborhood U satisfying the following properties. There exist an integer 1 ≤ p ≤ k -1 and two projective subspaces I and L of dimension p -1 and k -p respectively such that I ∩ U = ∅ and L ⊂ U. We do not assume that L and I are invariant. Since I ∩ L = ∅, for each x ∈ L there exists a unique projective subspace I(x) of dimension p which contains I and such that L∩I(x) = {x}. Furthermore, for each x ∈ L we ask that U ∩I(x) is strictly convex as a subset of I(x) \ I C p . All these assumptions are stable under small perturbations of f. The geometric assumption on U is slightly stronger than the one of Dinh, who only requires U ∩ I(x) to be star-shaped at x. We need convexity in order the solve the ∂-equation on U. Indeed, under our assumption U is a (p -1)-convex domain in the sense of [START_REF] Henkin | Andreotti-Grauert theory by integral formulas[END_REF].

If E is a subset of P k , let C q (E) denote the set of all positive closed currents of bidegree (q, q), supported in E and of mass 1. It is well known that for any integer 1 ≤ q ≤ k and any smooth form S in C q (P k ), the sequence d -qn (f n ) * (S) converges to a positive closed current T q of bidegree (q, q) called the Green current of order q of f. We refer to [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF] for a detailed exposition on these currents and their effectiveness in holomorphic dynamics.

When q = k, it gives the equilibrium measure of f, µ := T k . It is exponentially mixing and it is the unique measure of maximal entropy k log d on P k . Moreover, it is hyperbolic and all its Lyapunov exponents are larger or equal to (log d)/2. The dynamics outside the support of µ is not very well understood. The aim of this paper is to continue the investigation started in [START_REF] Dinh | Attracting current and equilibrium measure for attractors on P k[END_REF] on the attracting sets described above, which do not intersect supp(µ). Indeed, since I ∩ U = ∅, by regularization there exists a smooth form S ∈ C k-p+1 (Ω), where Ω :

= P k \ U . As f -1 (Ω) ⊂ Ω, it follows that supp(T k-p+1 ) ∩ U = ∅, and hence supp(T q ) ∩ U = ∅ if q ≥ k -p + 1.
The set C p (U ) is non-empty since it contains the current [L] of integration on L and its regularizations in U. In the situation described above, Dinh proved that if R is a continuous element of C p (U ) then its normalized push-forwards by f n , d -(k-p)n (f n ) * (R), converge to a current τ which is independent of the choice of R. Moreover, the current τ gives us information on the geometry of A and on the dynamics of f |A : it is woven, supported in A and invariant i.e. f * (τ ) = d k-p τ. Our main result is that, with a natural additional assumption on f |U , stable under small perturbations, we obtain an explicit exponential speed of the above convergence for any R in C p (U ).

Theorem 1.1. Let f and τ be as above and assume that ∧ k-p+1 Df (z) < 1 on U . There is a constant 0 < λ < 1 such that for each 0 < α ≤ 2 the following property holds. There exists C > 0 such that for any element R of C p (U ) and any (k -p, k -p)-form ϕ of class C α on P k we have

| d -(k-p)n (f n ) * (R) -τ, ϕ | ≤ Cλ nα/2 ϕ C α .
(1.1)

In particular, τ is the unique invariant current in C p (U ) and d

-(k-p)n (f n ) * (R) converge to τ uniformly on R ∈ C p (U ).
Recall that f induces a self-map Df on the tangent bundle T P k which also gives a self-map ∧ q Df on the exterior power ∧ q T P k , 1 ≤ q ≤ k. In the sequel, all the norms on C α , L r , etc. are with respect to the Fubini-Study metric on P k . It also gives a uniform norm which induces an operator norm for ∧ q Df.

In the same spirit as Theorem 1.1, we proved in [START_REF] Taflin | Equidistribution speed towards the Green current for endomorphisms of P k[END_REF] that for a generic current S in C 1 (P k ), the sequence d -n (f n ) * (S) converges to T exponentially fast. However, the contexts are quite different. Here, we consider currents of arbitrary bidegree which are in general much harder to handle. Moreover, in [START_REF] Taflin | Equidistribution speed towards the Green current for endomorphisms of P k[END_REF] we deeply use that T has Hölder continuous local potentials. In the present situation, we can expect that the attracting current τ is always more singular. The idea to prove Theorem 1.1 is to use Henkin-Leiterer's solution with estimates of the dd c -equation on U in order to study separately the harmonic and non-harmonic parts of the left hand side term of (1.1). When dd c ϕ = 0 on U, we use the "geometry" of C p (U ), introduced in [Din07] and [START_REF] Dinh | Geometry of currents, intersection theory and dynamics of horizontal-like maps[END_REF], and Harnack's inequality to obtain exponential estimates. In order to deal with the non-harmonic part, we use the assumption on ∧ k-p+1 Df . This assumption comes naturally in several basic examples and their perturbations.

In [START_REF] Dinh | Attracting current and equilibrium measure for attractors on P k[END_REF], Dinh also showed that the equilibrium measure associated to A, defined by ν := τ ∧ T k-p , is invariant, mixing and of maximal entropy (k -p) log d on A. Theorem 1.1 is a first step in order to obtain other ergodic and stochastic properties on ν as exponential mixing or central limit theorem. We postpone this question in a future work.

Under the same assumptions, we deduce from the work of de Thélin [dT08], see also [START_REF] Dupont | Large entropy measures for endomorphisms of CP(k)[END_REF], the following result on ν.

Theorem 1.2. If f is as in Theorem 1.1, then the measure ν is hyperbolic. More precisely, counting with multiplicity it has k -p Lyapunov exponents larger than or equal to (log d)/2 and p Lyapunov exponents strictly smaller than -(k -p)(log d)/2.

Structural discs of currents

In this section we recall the notion of structural varieties of currents. It was introduced by Dinh and Sibony in order to put a geometry on the space C p (U ) which is of infinite dimension, see [START_REF] Dinh | Geometry of currents, intersection theory and dynamics of horizontal-like maps[END_REF] and [START_REF] Dinh | Attracting current and equilibrium measure for attractors on P k[END_REF]. The definition of structural varieties is based on slicing theory and they can be seen as complex subvarieties inside C p (U ). In [START_REF] Dinh | Super-potentials of positive closed currents, intersection theory and dynamics[END_REF], the authors developed the notion of super-potential which involves more deeply this geometry.

Slicing theory can be seen as a generalization to currents of restriction of smooth forms to submanifolds. We will briefly explain it in our context and refer to [START_REF] Federer | Geometric measure theory[END_REF] for a more complete account.

Let U be an open subset of P k satisfying the geometric hypothesis as above. Let V be a complex manifold of dimension l. We denote by π U and

π V the canonical projections of U × V to U and V respectively. If R is a positive closed current of bidegree (p, p) on U × V with π U (supp(R)) U then, for all θ in V , the slice R, π V , θ is well defined. For any (k -p, k -p)- form φ on U × V we have R, π V , θ (φ) = lim →0 R ∧ π * V (ψ θ, ), φ ,
where ψ θ, is an appropriate approximation in V of the Dirac mass at θ. It is a (p + l, p + l)-current on U × V supported on U × {θ} which can be identified to a (p, p)-current on U. A family of currents

(R θ ) θ∈V in C p (U ) is a structural variety if there exists a positive closed current R in U × V such that R θ = R, π V , θ . When V is isomorphic to the unit disc of C, we call (R θ ) θ∈V a structural disc.
Recall that in our situation f (U ) U. Under the geometrical assumption on U, Dinh constructed in [Din07, p.233] a family of structural discs in C p (U ). He uses that for each x ∈ L the set I(x) ∩ U is star-shaped at x to obtain a property similar to star-sharpness for C p (U ).

More precisely, up to an automorphism, we can assume that

I = {x ∈ P k | x i = 0, 0 ≤ i ≤ k-p}, L = {x ∈ P k | x i = 0, k-p+1 ≤ i ≤ k}, where x = [x 0 : • • • : x k ] are the homogeneous coordinates of P k . For θ ∈ C, A θ (x) := [x 0 : • • • : x k-p : θx k-p+1 : • • • : θx k ] is an automorphism of P k except for θ = 0 where it is the projection of P k \ I on L. Let set U := f (U ). As I(x) ∩ U is star-shaped at x ∈ L, there exists a simply connected open neighborhood V ⊂ C of [0, 1] such that A θ (U ) U for all θ in V . If S is in C p (U )
then the family (S θ ) θ∈V with S θ := (A θ ) * (S) defined a structural disc. Indeed, if Λ : P k × V → P k is the meromorphic map defined by Λ(x, θ) = (A θ ) -1 (x) and S := Λ * S then S θ = S , π V , θ , see [START_REF] Dinh | Attracting current and equilibrium measure for attractors on P k[END_REF] for more details. For any S in C p (U ), we have that S 1 = S and S 0 = [L] which is independent of S. In other words, any current in C p (U ) is linked to [L] by a structural disc in C p (U ). Moreover, S θ depends continuously on θ and we have the following important property.

Lemma 2.1 ( [START_REF] Dinh | Attracting current and equilibrium measure for attractors on P k[END_REF]). Let S be in C p (U ) and (S θ ) θ∈V be the structural disc described above. If φ is a real continuous (k -p, k -p)-form with dd c φ = 0 on U then the function θ → S θ , φ is harmonic on V.

q-Convex set and d-bar equation

The concept of q-convexity generalizes both Stein and compact manifolds. Andreotti and Grauert [START_REF] Andreotti | Théorème de finitude pour la cohomologie des espaces complexes[END_REF] obtained vanishing or finiteness theorems for q-convex manifolds and, in [START_REF] Henkin | Andreotti-Grauert theory by integral formulas[END_REF], Henkin and Leiterer developed a similar theory using integral representations. In particular, they obtained solutions of the ∂-equation with explicit estimates, which play a key role in our proof. For this reason, we use the conventions of [START_REF] Henkin | Andreotti-Grauert theory by integral formulas[END_REF].

If 1 ≤ q ≤ k is an integer then a real C 2 function ρ on an open subset V ⊂ P k is called q-convex if, in any holomorphic local coordinates, the Hermitian form

L ρ (x)t = k i,j=1
∂ 2 ρ(x) ∂z i ∂z j t i t j has at least q strictly positive eigenvalues at any point x ∈ V. Let 0 ≤ q ≤ k -1. We say that an open subset D of P k is strictly q-convex if there exists a (q + 1)-convex function ρ in a neighborhood V of ∂D such that

D ∩ V = {x ∈ V | ρ(x) < 0}.
Moreover, if the same condition is satisfied with V a neighborhood of D then D is called completely strictly q-convex.

The strict q-convexity has the following important consequence, see [HL88, Theorem 11.2].

Theorem 3.1. Let D be a strictly q-convex open subset of P k with C 2 boundary. If φ is a continuous ∂-exact form of bidegree (r, s) in a neighborhood of D with 0 ≤ s ≤ k, k-q ≤ r ≤ k, then there exists a continuous (s, r-1)-form ψ on D such that ∂ψ = φ and

ψ ∞,D ≤ C φ ∞,D
for some C > 0 independent of φ.

Furthermore, Andreotti and Grauert proved the following vanishing theorem, see [START_REF] Andreotti | Théorème de finitude pour la cohomologie des espaces complexes[END_REF] and [HL88, Theorem 12.7].

Theorem 3.2. If D is a completely strictly q-convex open subset of P k with C 2 boundary then H s,r (D, C) = 0 for any 0 ≤ s ≤ k and k -q ≤ r ≤ k.

Henkin and Leiterer [HL88, Theorem 5.13] give the following criteria of q-convexity, which is closely related to our geometric assumption on U with q = p -1.

Theorem 3.3. Let D be an open subset of P k with C 2 boundary. If for each x ∈ ∂D there exists a complex submanifold Y ⊂ P k of dimension q + 1, transverse to ∂D and such that Y ∩ D is a strictly pseudoconvex domain in Y, then D is strictly q-convex.

This result applies to our trapping neighborhood U with q = p-1. Indeed, observe that, possibly by exchanging U by a slightly smaller open set which contains f (U ), we can assume that ∂U is smooth and the intersection of ∂U with I(x) is transverse for all x ∈ L. The projective space I(x) has dimension p = q + 1 and U ∩ I(x) is strictly convex in I(x) \ I C p , so in particular strictly pseudoconvex in I(x). Therefore, by Theorem 3.3, U is strictly (p -1)-convex. In the sequel, we always choose such an attracting neighborhood U.

Up to an automorphism of P k , I is defined in homogeneous coordinates by

I = {x ∈ P k | x i = 0, 0 ≤ i ≤ k -p}. The function η(x) = |x k-p+1 | 2 + • • • + |x k | 2 |x 0 | 2 + • • • + |x k-p | 2 ,
is a (q + 1)-convex exhausting function of P k \ I, i.e. P k \ I is completely qconvex. In general, strictly q-convex subsets of a completely q-convex domain are not completely strictly q-convex. However, in our case it is easy to construct from a q-convex function ρ such that

U ∩ V = {x ∈ V | ρ(x) < 0}
for some neighborhood V of ∂U, a q-convex defining function defined in a neighborhood of U . Indeed, it is enough to compose (η, ρ) with a good approximation of the maximum function (see [HL88, Definition 4.12]). It will give a (q + 1)-convex function since the positive eigenvalues of the complex Hessians of ρ and η are in the same directions. Therefore, U is completely strictly (p-1)-convex and we have the following solution for the dd c -equation in symmetric bidegrees.

Theorem 3.4. Let U be as above. If ϕ is a C 2 (r, r)-form in a neighborhood of U with k -p ≤ r ≤ k, then there exists a continuous (r, r)-form ψ on U such that dd c ψ = dd c ϕ and

ψ ∞,U ≤ C dd c ϕ ∞,U
for some C > 0 independent of ϕ.

Proof. The proof follows closely the proof of Theorem 2.7 in [START_REF] Dinh | Dynamics of horizontallike maps in higher dimension[END_REF].

Without loss of generality, we can assume that ϕ is real and therefore dd c ϕ is also real. First, we solve the equation dξ = dd c ϕ with estimates. Let W be a small neighborhood of U , with the same geometric property and such that ϕ is defined on W. The maps A θ defined in Section 2 give a homotopy A : [0, 1] × W → W, A(θ, x) = A θ (x), between A 1 = Id and the projection A 0 of W on L. Since L has dimension k -p, A * 0 vanish identically on (r +1, r +1)forms if r ≥ k -p. Therefore, by homotopy formula (see e.g [BT82, p38]), there exists a form ξ on W such that dξ = dd c ϕ and ξ ∞,U dd c ϕ ∞,U . Moreover, possibly by exchanging ξ by (ξ+ξ)/2, we can assume that ξ = Ξ+Ξ where Ξ is a (r, r + 1)-form. As dξ is a (r + 1, r + 1)-form, it follows that ∂Ξ = 0 and dξ = ∂Ξ + ∂Ξ. Therefore, by Theorem 3.2, Ξ is ∂-exact and by Theorem 3.1, there exists a continuous (r, r)-form Ψ such that ∂Ψ = Ξ and Ψ ∞,U Ξ ∞,U . Finally, if ψ = -iπ(Ψ -Ψ) we have

dd c ψ = ∂∂(Ψ -Ψ) = ∂Ξ + ∂Ξ = dd c ϕ, and ψ ∞,U Ξ ∞,U dd c ϕ ∞,U .

Attracting speed

For R in C p (U ), we denote by R n its normalized push-forward by f n , i.e.

R n := d -(k-p)n (f n ) * (R).
To obtain (1.1), the first observation is that the norm of R n -τ, seen as a linear form on the space of continuous test (k -p, k -p)forms, is bounded independently of n and R. Therefore, it is sufficient to establish (1.1) for α = 2 and then apply interpolation theory between Banach spaces, see e.g. [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF], in order to obtain the general case. Let denote by X the set of all real continuous

(k -p, k -p)-forms φ on U such that dd c φ = 0 and | R -τ, φ | ≤ 1 for all R ∈ C p (U ). Observe that, since f (U ) U, if φ is in X then f * (φ)
is defined on U where it is still a real continuous form with dd c (f * (φ)) = 0. The set X is a truncated convex cone and the first part of the proof of Theorem 1.1 is to show that d -(k-p) f * acts by contraction on it. This result is available without any assumption on ∧ k-p+1 Df . It is based on Lemma 2.1 and Harnack's inequality for harmonic functions.

Lemma 4.1. There exists a constant 0 < λ 1 < 1 such that for any R in C p (U ), φ in X and n in N we have

| R n -τ, φ | ≤ λ n 1 . Proof. If R is in C p (U ) and φ in X, R 1 := d -(k-p) f * (R) is in C p (U )
and we define the function h R,φ on V by h R,φ (θ) := R 1,θ -τ, φ , where θ → R 1,θ is the structural disc described in Section 2. The definition of X implies that |h R,φ | ≤ 1 on V, for all R ∈ C p (U ) and φ ∈ X. Moreover, since R 1 is in C p (U ), it follows from Lemma 2.1 that all these functions are harmonic on V. Now, observe that if we take R = τ then h τ,φ (1) = 0 for all φ ∈ X, since d p-k f * τ = τ. Hence, as |h τ,φ | ≤ 1 on V, Harnack's inequality says that there exists 0 ≤ a < 1 such that |h τ,φ (0)| ≤ a for all φ in X. On the other hand, R 1,0 is a current independent of R. So, for all R ∈ C p (U ) and φ ∈ X we have h R,φ (0) = h τ,φ (0) and therefore |h R,φ (0)| ≤ a. Once again, we deduce from Harnack's inequality there exists 0 < λ 1 < 1, independent of R and φ, such that |h R,φ (1)| ≤ λ 1 or equivalently

R 1 -τ, φ λ 1 = | R -τ, φ 1 | ≤ 1,
where φ 1 = d -(k-p) f * (φ/λ 1 ). Moreover, φ 1 is defined on U and dd c φ 1 = 0. It follows that φ 1 is in X. Using the same arguments with φ 1 instead of φ gives

that | R 1 -τ, φ 1 | ≤ λ 1 which can be rewrite | R 2 -τ, φ | ≤ λ 2 1 . Inductively, we obtain that | R n -τ, φ | ≤ λ n 1 .
Remark 4.2. The constant λ 1 is not directly related to f. Indeed, it only depends on V i.e. on the size of U and the distance between ∂U and ∂f (U ).

If h is the unique biholomorphism between V and the unit disc in C such that h(0) = 0 and h(1) = α ∈]0, 1[ then Harnack's inequality gives explicitly that we can take a = 2α/(1 + α) and λ 1 = 4α/(1 + α) 2 .

In order to prove Theorem 1.1, we use Theorem 3.4 together with the assumption on ∧ k-p+1 Df and Lemma 4.1.

If ∧ k-p+1 Df (z) < 1 on U then by continuity, there exists a constant 0

< λ 2 < 1 such that ∧ k-p+1 Df (z) < λ 2 on U. Hence, if ϕ is a (k-p, k-p)- form of class C 2 , we have for ϕ i := d -i(k-p) (f i ) * (ϕ) with i ∈ N dd c ϕ i ∞,U λ 2i 2 d i(k-p) ϕ C 2 .
Here, the symbol means inequality up to a constant which only depends on our conventions and on U. By Theorem 3.4 with r = k -p, there exists a continuous (k -p, k -p)-form ψ i on U such that

dd c ψ i = dd c ϕ i and ψ i ∞,U dd c ϕ i ∞,U λ 2i 2 d i(k-p) ϕ C 2 .
We can now complete the proof of our main result.

End of the proof of Theorem 1.1. Let R be in C p (U ) and ϕ be a (k-p, k-p)form of class C 2 . Without loss of generality, we can assume that ϕ is real. Let 0 ≤ i ≤ n be two arbitrary integers. We set l := n -i. If R n , ϕ i and ψ i are defined as above then we have

R n -τ, ϕ = R l -τ, ϕ i = R l -τ, ϕ i -ψ i + R l -τ, ψ i , since τ is invariant. On the one hand, R l -τ, ψ i ψ i ∞,U λ 2i 2 d i(k-p) ϕ C 2 , (4.1)
since R l and τ are supported on U. On the other hand, observe that there exists a constant M ≥ 1 independent of ϕ such that d

-(k-p) f * (ϕ) ∞ ≤ M ϕ ∞ . Therefore, ϕ i -ψ i ∞,U ≤ M i ϕ ∞ + ψ i ∞,U ≤ M i ϕ ∞ + C λ 2i 2 d i(k-p) ϕ C 2 M i ϕ C 2 , and in particular | S -τ, ϕ i -ψ i | M i ϕ C 2 ,
for any S in C p (U ). Moreover, ϕ i -ψ i is a real continuous (k -p, k -p)-form on U and dd c (ϕ i -ψ i ) = 0. Hence, (ϕ i -ψ i )/(CM i ϕ C 2 ) belongs to X where C > 0 is a constant depending only on U and on our conventions. It follows by Lemma 4.1 that | R l -τ, ϕ i -

ψ i | ≤ CM i ϕ C 2 λ l 1 . (4.2)
To summarize, equations (4.1) and (4.2) imply that there are constants 0 < λ 1 , λ 2 < 1, and M ≥ 1 such that

| R n -τ, ϕ | ϕ C 2 M i λ l 1 + λ 2i 2 d i(k-p) .
If q ∈ N is large enough then M λ q 1 < 1. Therefore, if we choose n (q + 1)i, we obtain l iq and

| R n -τ, ϕ | ϕ C 2 λ n ,
where λ := max(λ 2 2 d -(k-p) , M λ q 1 ) 1/(q+1) < 1. This estimate holds for arbitrary n in N and is uniform on ϕ R.

Remark 4.3. In Theorem 1.1, it is enough to assume that ∧ k-p+1 Df (z) < d (k-p)/2 on U . Moreover, it is easy using small perturbations of a suitable polynomial map to construct examples with ∧ k-p+1 Df (z) as small as we want on U .

Hyperbolicity of the equilibrium measure

In this section, we prove Theorem 1.2. Recall that the equilibrium measure associated to A is given by ν := τ ∧ T k-p . It has maximal entropy on A equal to (k -p) log d, [START_REF] Dinh | Attracting current and equilibrium measure for attractors on P k[END_REF]. On the other hand, we have the following powerful result, see [dT08] and [START_REF] Dupont | Large entropy measures for endomorphisms of CP(k)[END_REF].

Theorem 5.1. If the Lyapunov exponents of ν are ordered so that

χ 1 ≥ • • • ≥ χ a-1 > χ a ≥ • • • ≥ χ k , then h(ν) ≤ (a -1) log d + 2 k i=a χ + i , (5.1)
where h(ν) denotes the entropy of ν and χ + i := max(χ i , 0). Now, let 1 ≤ c ≤ k be such that

χ 1 ≥ • • • ≥ χ c > 0 ≥ χ c+1 ≥ • • • ≥ χ k .
If we take a = c + 1 in Theorem 5.1, we obtain h(ν) ≤ c log d. Since h(ν) = (k-p) log d, it follows that c ≥ (k-p). It means there are at least k-p strictly

positive Lyapunov exponents. Moreover, if we have equality, c = k -p, Theorem 5.1 applied to the smallest a such that χ a = χ c gives

Hence, χ c ≥ (log d)/2. Note that in this part we do not need the assumption on ∧ k-p+1 Df .

It remains to prove that the assumptions of Theorem 1.1 imply that c ≤ k -p and χ c+1 < -(k -p)(log d)/2. It is not hard to deduce form Oseledec theorem [START_REF] Oseledec | A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems[END_REF] that the sum of the q largest Lyapunov exponents verifies

for ν-almost all z. Moreover, we have

Therefore, it follows that

Therefore, c ≤ k -p and we have seen above that in this case c = k -p and χ c ≥ (log d)/2. Finally, we have

Remark 5.2. Theorem 5.1 with a = 1 implies the Ruelle inequality, i.e.