Julie Déserti

JONQUIÈRES MAPS AND SL(2; C)-COCYCLES by

Keywords:

. For "(α, β) well chosen" of modulus 1 the centraliser of f α,β is trivial, the topological entropy of f α,β is 0, there exist two domains of linearisation : in the first one the closure of the orbit of a point is a torus, in the other one the closure of the orbit of a point is the union of two circles. On P 1 C × P 1 C any f α,β can be viewed as a cocyle ; using recent results about SL(2; C)-cocycles ([1]) we determine the LYAPUNOV exponent of the cocyle associated to f α,β .

Introduction

In this article we deal with a family of birational maps ( f α,β ) given by f α,β : P 2 where α, β denote two complex numbers with modulus 1, case where we know almost nothing about the dynamics. Let us consider the set Ω of pairs of complex numbers of modulus 1 that satisfy diophantine condition. The family ( f α,β ) satisfies the following properties ( [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF]):

• for (α, β) ∈ Ω the centraliser of f α,β , that is the set of birational maps of P 2 C that commutes with f α,β , is isomorphic to Z;

• the topological entropy of f α,β is 0;

• rotation domains of ranks 1 and 2 coexist: there is a domain of linearisation where the orbit of a generic point under f α,β is a torus, and there is an other domain of linearisation where the orbit of a generic point under f 2 α,β is a circle.

We can also see f α,β on P 1 C × P 1 C since all the computations of [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF] have been done in an affine chart they may all be carried on P 1 C × P 1 C ; the sets P 1 C × S 1 ρ , where S 1 ρ = {y ∈ C | |y| = ρ}, are invariant.

Let us define A α,ρ n : S 1 ρ → M(2; C) given in terms of A α,ρ (y) = α y 1 1 by

A α,ρ n (•) = A α,ρ (β n •)A α,ρ (β n-1 •) . . . A α,ρ (β •)A α,ρ (•).
To compute f n α,β (x, y) is equivalent to compute A α,ρ n (y) as soon as f k α,β (x, y) = (-1, α) for any 1 ≤ k ≤ n.

Using [START_REF] Avila | Global theory of one-frequency Schrödinger operators[END_REF] we are able to determine the LYAPUNOV exponent of the cocycle (A α,ρ , β):

Theorem A. -The LYAPUNOV exponent of (A α,ρ , β) is
• positive as soon as ρ > 1;

• zero as soon as ρ ≤ 1.

More precisely f α,β is semi conjugate to αx+y 2 x+1 , β 1/2 y and the LYAPUNOV exponent of the cocycle B α,ρ , β 1/2 , where

B α,ρ (y) = α y 2 1 1 ,
is equal to max(0, ln ρ).

In the next section we introduce the family ( f α,β ) and its properties ( §1). Then we deal with the recent works of AVILA on SL(2; C)-cocyles. In the last section we give the proof of Theorem A (see §2). Let us explain the sketch of it. We associate to B α,ρ , β 1/2 a cocycle B α,ρ , β 1/2 that belongs to SL(2; C). We first determine lim ρ→0 L B α,ρ , β 1/2 , and then

lim ρ→+∞ L B α,ρ , β 1/2
where L(C, γ) denotes the LYAPUNOV exponent of the SL(2; C)-cocyle (C, γ). In both cases, we get 0. Using [1, Theorem 5] we obtain that L B α,ρ , β 1/2 vanishes everywhere; it allows us to determine L A α,ρ , β since

L B α,ρ (y), β 1/2 = L B α,ρ (y), β 1/2 + max(0, ln ρ),
and since A α,ρ , β and β 1/2 , B α,ρ are conjugate.

Acknowledgment. -I would like to thank Artur AVILA for very helpful discussions, Dominique CERVEAU for his constant support, and Serge CANTAT for his remarks. Thanks also to the referee whose comments help me to improve the text.

Some properties of the family ( f α,β )

A rational map φ from P 2 C into itself is a map of the form (x : y : z) φ 0 (x, y, z) :

φ 1 (x, y, z) : φ 2 (x, y, z) ,
where the φ i 's are some homogeneous polynomials of the same degree without common factor; φ is birational if it admits an inverse of the same type. We will denote by Bir(P 2 C ) the group of birational maps of P 2 C , also called the CREMONA group. The degree of φ, denoted deg φ, is the degree of the φ i 's. The degree is not a birational invariant: deg ψφψ -1 = deg φ for generic birational maps φ and ψ. The first dynamical degree of φ given by

λ(φ) = lim n→+∞ deg φ n 1/n ,
is a birational invariant; it is strongly related to the topological entropy h top (φ) of φ (see [START_REF] Gromov | On the entropy of holomorphic maps[END_REF][START_REF] Yomdin | Volume growth and entropy[END_REF])

h top (φ) ≤ log λ(φ) (1.1)
Any birational map φ admits a resolution

S π 2 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ π 1 ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ P 2 C φ / / ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 C ,
where π 1 , π 2 : S → P 2 C are sequences of blow-ups (see [START_REF] Beauville | Complex algebraic surfaces[END_REF] for example). The resolution is minimal if and only if no (-1)-curve of S is contracted by both π 1 and π 2 . The base-points of φ are the points blown-up in π 1 , which can be points of P 2 C or infinitely near points. We denote by b(φ) the number of such points, which is also equal to the difference of the ranks of Pic(S) and Pic(P 2 C ), and thus equals to b(φ -1 ). The dynamical number of base-points of φ introduced in [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF] is by definition

µ(φ) = lim n→+∞ b(φ n ) n ;
it is a real positive number that satisfies µ(φ n ) = |n µ(φ)| for any n ∈ Z, µ(ψφψ -1 ) = µ(φ), and allows us to give a characterization of birational maps conjugate to automorphisms:

Theorem 1.1 ([8]

). -Let S be a smooth projective surface; the birational map φ ∈ Bir(S) is conjugate to an automorphism of a smooth projective surface if and only if µ(φ) = 0.

The behavior of φ ∈ Bir(P 2 C ) is strongly related to the behavior of deg φ n n∈N (see [START_REF] Gizatullin | Rational G-surfaces[END_REF][START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF][START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]); up to birational conjugacy exactly one of the following holds:

1. the sequence deg φ n n∈N is bounded and either φ is of finite order, or φ is an automorphism of P 2 C ; 2. there exists an integer k such that

lim n→+∞ deg φ n n = k 2 µ(φ)
2 and φ is not an automorphism;

there exists an integer

k ≥ 3 such that lim n→+∞ deg φ n n 2 = k 2 κ(φ)
9 where κ(φ) ∈ Q is a birational invariant, and φ is an automorphism; 4. the sequence deg φ n n∈N grows exponentially (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] for more precise dynamical properties). In the first three cases λ(φ) = 1, in the last one λ(φ) > 1. In case 2. (resp. 3.) the map φ preserves a unique fibration which is rational (resp. elliptic).

In case 1. (resp. 2., resp. 3, resp. 4) we say that φ is elliptic (resp. a JONQUIÈRES twist, resp. an HALPHEN twist, resp. hyperbolic).

Let us give some examples. Let

φ(x, y) = a(y)x + b(y) c(y)x + d(y) , αy + β γy + δ
be an element of the JONQUIÈRES group PGL(2; C(y)) ⋊ PGL(2; C); either φ is elliptic (for instance φ : (x : y : z) (yz : xz : xy)), or φ is a JONQUIÈRES twist (for example φ : (x : y : z) (xz : xy : z 2 ) for which the unique invariant fibration is y/z = constant). The map φ : Clearly elliptic birational maps have a poor dynamical behavior contrary to hyperbolic ones. The study of automorphisms of positive entropy is strongly related with birational maps of P 2 C : Theorem 1. 2 ([9]). -Let S be a compact complex surface that carries an automorphism φ of positive topological entropy.

P 2 C P 2 C (x : y : z) (2y + z)(y + z) : x(2y -z) : 2z(y + z)
• Either the KODAIRA dimension of S is zero and φ is conjugate to an automorphism on the unique minimal model of S that necessarily is a torus, or a K3 surface or an ENRIQUES surface; • or the surface S is a non-minimal rational one, isomorphic to P 2

C blown up at n points, n ≥ 10, and φ is conjugate to a birational map of P 2 C . This yields many examples of hyperbolic birational maps for which we can establish a lot of dynamical properties ([18, 4, 5, 6, 7, 14, 13]).

Another way to measure chaos is to look at the size of centralisers. Let us give two examples. The polynomial automorphisms of C 2 having rich dynamics are HÉNON maps; furthermore a polynomial automorphism of C 2 is a HÉNON one if and only if its centraliser is countable. Let us now consider rational maps on S 1 ; if the centraliser of such maps is not trivial (1) , then the JULIA set is "special". The centraliser of an elliptic birational map of infinite order is uncountable ( [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]). The centralisers of HALPHEN twists are described in [START_REF] Gizatullin | Rational G-surfaces[END_REF]. The centraliser of an hyperbolic map is countable ( [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). In [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF] we end the story by studying centralisers of JONQUIÈRES twists. If the 1. The centraliser of a map φ is trivial if it coincides with the iterates of φ.

fibration is fiberwise invariant, then the centraliser is uncountable ; but if it isn't, then generically the centraliser is isomorphic to Z. We don't know a lot about dynamics of these maps, in this article we will thus focus on a family of such maps. We consider the JONQUIÈRES maps

f α,β : P 2 C P 2 C (x : y : z) (αx + y)z : βy(x + z) : z(x + z)
where α, β denote two complex numbers with modulus 1. The base-points of f α,β are

(1 : 0 : 0), (0 : 1 : 0), (-1 : α : 1).

Any f α,β preserves a rational fibration (the fibration y = constant in the affine chart z = 1). Each element of the family ( f α,β ) has first dynamical degree 1 hence topological entropy zero (1.1); more precisely one has ([8, Example 4.3])

µ( f α,β ) = 1 2
so f α,β is not conjugate to an automorphism (Theorem 1.1). The centralizer of f α,β is isomorphic to Z (see [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF]Theorem 1.6]). The idea of the proof is the following: the point p = (1 : α : 1) is blown-up onto a fiber of the fibration y = constant. Let ψ be an element of

Cent( f α,β ) = g ∈ Bir(P 2 C ) | g • f α,β = f α,β • g ;
since ψ blows down a finite number of curves there exists a positive integer k (chosen minimal) such that f k α,β (p) is not blown down by ψ. Replacing ψ by ψ = ψ f k-1 α,β one gets that ψ(p) is an indeterminacy point of f α,β . In other words ψ permutes the indeterminacy points of f α,β . A more precise study allows us to establish that p is fixed by ψ. The pair (α, β) being in Ω, the closure of the negative orbit of p under the action of f α,β is ZARISKI dense; since ψ fixes any element of the orbit of p one obtains ψ = id.

Let us recall that if ψ is an automorphism on a compact complex manifold M, the FATOU set F (ψ) of ψ is the set of points that have a neighborhood V such that f n We can also see f α,β on P 1 C × P 1 C and that is what we will do in the sequel since all the computations of [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF] have been done in an affine chart they may all be carried on P 1 C × P 1 C ; the sets P 1 C × S 1 ρ are invariant. In [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF] we show that there are two rotation domains for f 2 α,β , one of rank 1, and the other one of rank 2 (2) ; in the first case we give here a more precise statement than in [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF]:

|V | n ∈ N is a normal family. Set G(U) = φ : U → U | φ = lim n j →+∞ ψ n j ; we say that U is a rotation domain if G(U) is a subgroup of Aut(U). An equivalent definition is the following: a component U of F (ψ) which is invariant by ψ is a rotation domain if ψ |U is conjugate to a linear rotation. If U is a rotation domain, G(U) is a compact LIE group
Theorem 1.3. -Assume that (α, β) belongs to Ω.
There exists a strictly positive real number r such that f α,β is conjugate to (αx, βy) on P 1

C × D(0, r) where D(0, r) denotes the disk centered at the origin with radius r.

There exists a strictly positive real number r such that

f 2 α,β is conjugate to x β , z β 2 on P 1 C × D(0, r).
Remark 1.4. -The point (α -1, 0) is also a fixed point of f α,β where the behavior of f α,β is the same as near (0, 0).

Proof. -The first assertion is proved in [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF].

Let us consider the map ψ(x, z) = a(z)x+b(z) c(z)x+1 , z . The equation

ψ -1 f 2 α,β ψ = x β , z β 2 yields β a β -2 z c(z) + β a β -2 z a(z) -c β -2 z a(z) + α a β -2 z a(z) +z α 2 a β -2 z c(z) -α c β -2 z c(z) -c β -2 z c(z) -c β -2 z a(z) = 0, (1.2)
β a β -2 z -β a(z) + z α 2 a β -2 z -αβ c(z) -β c(z) -β a(z) -α c β -2 z -c β -2 z +β(α + β) a(z)b β -2 z + (α + β) b(z)a β -2 z + β 2 b β -2 z c(z) -b(z)c β -2 z +z α 2 βb β -2 z c(z) -b(z)c β -2 z = 0 (1.3)
and

(α + 1) z + b(z) -β b β -2 z -α 2 zb β -2 z + zb(z) -(α + β) b β -2 z b(z) = 0 (1.4)
Let us set

a(z) = ∑ i≥0 a i z i , b(z) = ∑ i≥0 b i z i , c(z) = ∑ i≥0 c i z i .
We easily get a 0 = 1 -β, b 0 = 0 and c 0 = α + β.

Relation (1.4) implies that b 1 = β(1 + α) 1 -β & β b ν 1 -β 1-2ν + F i (b i | i < ν) = 0 ∀ ν > 1, (1.3)
yields

a ν β 1-2ν -β + b ν (α + β)a 0 1 + β 1-2ν + c 0 β 2-2ν -1 + G i (a i , b i , c i | i < ν) = 0 and (1.2) to c ν a 0 β -β -2ν + a ν (α + β)a 0 1 + β -2ν + c 0 β 1-2ν -1 + H i (a i , b i , c i | i < ν) = 0
2. There already exists an example of automorphism of positive entropy with rotation domains of rank 1 and 2 (see [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF]), but f α,β is not conjugate to an automorphism on a rational surface.

where the F i 's, G i 's and H i 's denote universal polynomials; this allows to compute b ν , a ν and c ν . Thus we get a formal conjugacy of f 2 α,β to its linear part. Since this linear part satisfies a Rüssmann condition (see [START_REF] Rüssmann | Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition[END_REF]Theorem 2.1] condition (2)), according to [19, Theorem 2.1] any formal linearizing map conjugating f 2 α,β to its linear part is convergent on a polydisc.

About SL(2; C)-cocycles

A (one-frequency, analytic) quasiperiodic SL(2; C)-cocycle is a pair (A, β), where β ∈ R and A : S 1 1 → SL(2; C) is analytic, and defines a linear skew product acting on C 2 × S 1 1 by (x, y) → (A(y) • x, βy).

The iterates of the cocyle are given by (A n , nβ) where A n is given by

A n (y) = A β n-1 y . . . A(y) n ≥ 1, A 0 (y) = id, A -n (y) = A n (β -n y) -1 . The LYAPUNOV exponent L(A, β) of a quasiperiodic SL(2; C)-cocycle (A, β) is given by lim n→+∞ 1 n S 1 1 ln ||A n (y)|| dy.
A quasiperiodic SL(2; C)-cocycle (A, β) is uniformly hyperbolic if there exist analytic functions u, s : S 1 1 → P 2 C , called the unstable and stable directions, and n ≥ 1 such that for any y ∈ S UH is open, and A → L(A, β) is analytic over UH (regularity properties of the LYAPUNOV exponent are consequence of the regularity of the unstable and stable directions which depend smoothly on both variables).

Definition. -Let (A, β) be a quasiperiodic SL(2; C)-cocycle.

If L(A, β) > 0 but (A, β) ∈ UH , then (A, β) is nonuniformly hyperbolic. If A ∈ C ω SL(2; C), S 1
1 admits a holomorphic extension to |Im y| < δ then for |ε| < δ we can define A ε ∈ C ω SL(2; C), S 1 1 by A ε (y) = A(y + iε). The LYAPUNOV exponent L(A ε , β) is a convex function of ε. We can thus introduce the following notion. The acceleration of a quasiperiodic SL(2; C)-cocyle (A, β) is given by

ω(A, β) = lim ε→0 + 1 2πε L(A ε , β) -L(A, β) . Remark 2.1.
-The convexity of the LYAPUNOV exponent in function of ε implies that the acceleration is decreasing.

Since the LYAPUNOV exponent is a convex and continuous function the acceleration is an upper semi-continuous function in R \ Q ×C ω SL(2; C), S 1 1 . The acceleration is quantized:

Theorem 2.2 ([1]). -If (A, β) is a SL(2; C)-cocycle with β ∈ R Q, then ω(A, β) is always an integer.
A direct consequence is the following:

Corollary 2.3. -The function ε → L(A ε , β) is a piecewise affine function of ε.
It is thus natural to introduce the notion of regularity. A cocycle

(A, β) ∈ C ω SL(2; C), S 1 1 × R \ Q is regular if L(A ε , β) is affine for ε in a neighborhood of 0. In other words (A, β) is regular if the equality L(A ε , β) -L(A, β) = 2πεω(A, β)
holds for all ε small, and not only for the positive ones. Regularity is equivalent to the acceleration being locally constant near (A, β).

It is an open condition in C ω SL(2; C), S 1 1 × R \ Q. The following statement gives a characterization of the dynamics of regular cocycles with positive LYAPUNOV exponent:

Theorem 2.4 ([1]

). -Let (A, β) be a SL(2; C)-cocycle with β ∈ R Q. Assume that L(A, β) > 0; then (A, β) is regular if and only if (A, β) is UH .

One striking consequence is the following:

Corollary 2.5 ([1]

). -For any (A, β) in C ω SL(2; C), S 1 1 × R \ Q there exists ε 0 such that • L(A ε , β) = 0 (and ω(A, β) = 0) for every 0 < ε < ε 0 ,

• or (A ε , β) ∈ UH for every 0 < ε < ε 0 .

Remark 2.6. -Let us mention that there is a link between SL(2; C)-cocycles and SCHRÖDINGER operators (see [START_REF] Avila | Global theory of one-frequency Schrödinger operators[END_REF] for more details).

Proof of Theorem A

Suppose that ρ = 1, and let us consider the cocycle (B α,ρ , β 1/2 ) where B α,ρ (y) = α y 2 1 1 .

Since αx + y

x + 1 , βy (x, y 2 ) = (x, y 2 ) αx + y 2

x + 1 , β 1/2 y the cocycles (A α,ρ , β) and (B α,ρ , β 1/2 ) have the same behavior. Using two different arguments of monodromy (one for ρ < 1, and the other one for ρ > 1) we see that there is a continuous determination for the square root of det B α,ρ (y) = αy 2 . Let us set B α,ρ (y) = 1 αy 2 B α,ρ (y) ∈ SL(2; C) that is thus defined on two different domains of analyticity.

According to Theorem 1.3 one has L B α,ρ , β 1/2 = 0 when ρ is close to both 0 and ∞.

Assume that L B α,ρ , β 1/2 is non constant. When B α,ρ is holomorphic, so in particular when ρ < 1 and ρ > 1, the acceleration is decreasing (Remark 2.1); furthermore the acceleration is positive for ρ < 1 and negative for ρ > 1 (because L is continuous). Theorem 2.2 thus implies ω B α,1 + , β 1/2 -ω B α,1 -, β 1/2 ≤ -2.

By definition of B α,ρ we have L B α,ρ (y), β 1/2 = L B α,ρ (y), β 1/2 -S 1 ρ ln αy 2 dy = L B α,ρ (y), β 1/2max(0, ln ρ).

Even though B α,ρ (y), β 1/2 is not a SL(2; C)-cocycle, the LYAPUNOV exponent is still a convex function of log ρ (see for example [START_REF] Avila | Complex one-frequency cocycles[END_REF]). The jump of ω(B α,ρ (y), β 1/2 ) is thus ≥ 0, and the jump for the second term of the right member is -1. Therefore the jump of L B α,ρ (y), β 1/2 is ≥ -1: contradiction.

C P 2 C

 2 (x : y : z) (αx + y)z : βy(x + z) : z(x + z)

  is an HALPHEN twist ([15, Proposition 9.5]). HÉNON automorphisms give by homogeneization examples of hyperbolic maps.

  , and the action of G(U) on U is analytic real. Since G(U) is a compact, infinite, abelian LIE group, the connected component of the identity of G(U) is a torus of dimension 0 ≤ d ≤ dim C M. The integer d is the rank of the rotation domain. The rank coincides with the dimension of the closure of a generic orbit of a point in U.

1 1 ,

 1 A(y) • u(y) = u(βy) A(y) • s(y) = s(βy), and for any unit vector x ∈ s(y) (resp. x ∈ u(y)) we have ||A n (y) • x|| < 1 (resp. ||A n (y) • x|| > 1). The unstable and stable directions are uniquely characterized by those properties, and clearly u(y) = s(y) for any y ∈ S 1 1 . If (A, β) is uniformly hyperbolic, then L(A, β) > 0. Let us denote by UH ⊂ C ω SL(2; C), S 1 1 the set of A such that (A, β) is uniformly hyperbolic. Uniform hyperbolicity is a stable property:


Julie Déserti

SOME PROPERTIES OF THE GROUP OF BIRATIONAL MAPS GENERATED BY THE AUTOMORPHISMS OF P n C AND THE STANDARD INVOLUTION by

Keywords:

We give some properties of the subgroup G n (C) of the group of birational self-maps of P n C generated by the standard involution and the group of automorphisms of P n C . We prove that there is no nontrivial finite-dimensional linear representation of G n (C). We also establish that G n (C) is perfect, and that G n (C) equipped with the Zariski topology is simple. Furthermore if ϕ is an automorphism of Bir(P n C ), then up to birational conjugacy, and up to the action of a field automorphism ϕ |G n (C) is trivial.

Introduction

The group Bir(P 2 C ) of birational self-maps of P 2 C , also called the Cremona group of rank 2, has been the object of a lot of studies. For finite subgroups let us mention for example [START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF][START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF][START_REF] Bogomolov | On stable conjugacy of finite subgroups of the plane Cremona group[END_REF]; other subgroups have been dealt with ( [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF][START_REF] Déserti | Sur les sous-groupes nilpotents du groupe de Cremona[END_REF]), and some group properties have been established ( [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF][START_REF] Déserti | Le groupe de Cremona est hopfien[END_REF][START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF][START_REF] Cerveau | Transformations birationnelles de petit degré, volume 19 of Cours Spécialisés[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Cantat | Normal subgroups in the Cremona group[END_REF][START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF][START_REF] Blanc | Simple relations in the Cremona group[END_REF][START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF]). One can also find a lot of properties between algebraic geometry and dynamics ([26, 13, 7]). The Cremona group in higher dimension is far less well known; let us mention some references about finite subgroups ( [START_REF] Prokhorov | Simple finite subgroups of the Cremona group of rank 3[END_REF][START_REF] Yu | On birational involutions of P 3[END_REF][START_REF] Prokhorov | p-elementary subgroups of the Cremona group of rank 3[END_REF][START_REF] Popov | Jordan groups and automorphism groups of algebraic varieties[END_REF][START_REF] Mundet I Riera | Finite group actions on homology spheres and manifolds with nonzero euler characteristic[END_REF]), about algebraic subgroups of maximal rank ( [START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF][START_REF] Vinberg | Algebraic transformation groups of maximal rank[END_REF][START_REF] Umemura | Sur les sous-groupes algébriques primitifs du groupe de Cremona à trois variables[END_REF][START_REF] Umemura | On the maximal connected algebraic subgroups of the Cremona group[END_REF][START_REF] Umemura | Maximal algebraic subgroups of the Cremona group of three variables. Imprimitive algebraic subgroups of exceptional type[END_REF]), about other subgroups ( [START_REF] Popov | Some subgroups of the Cremona groups[END_REF][START_REF] Popov | Tori in the Cremona groups[END_REF]), about (abstract) homomorphisms from PGL(r + 1; C) to the group Bir(M) where M denotes a complex projective variety ( [START_REF] Cantat | Morphisms between Cremona groups and a characterization of rational varieties[END_REF]), and about maps of small bidegree ( [START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF][START_REF] Pan | Transformations birationnelles quadratiques de l'espace projectif complexe à trois dimensions[END_REF][START_REF] Pan | Sur les transformations de Cremona de bidegré (3, 3)[END_REF][START_REF] Hudson | Cremona Transformations in Plane and Space[END_REF][START_REF] Déserti | On cubic birational maps of P 3 C[END_REF]).

In this article we consider the subgroup of birational self-maps of P n C introduced by Coble in [START_REF] Coble | Point sets and allied Cremona groups[END_REF] G n (C) = σ n , Aut(P n C ) where σ n denotes the involution (z 0 : z 1 : . . . :

z n )    n ∏ i=0 i =0 z i : n ∏ i=0 i =1 z i : . . . : n ∏ i=0 i =n z i    .
The author is supported by ANR Grant "BirPol" ANR-11-JS01-004-01.

Hudson also deals with this group ( [START_REF] Hudson | Cremona Transformations in Plane and Space[END_REF]): "For a general space transformation, there is nothing to answer either to a plane characteristic or Noether theorem. There is however a group of transformations, called punctual because each is determined by a set of points, which are defined to satisfy an analogue of Noether theorem, and possess characteristics, and for which we can set up parallels to a good deal of the plane theory."

Note that the maps of G 3 (C) are in fact not so "punctual" ( [8, §8]). It follows from Noether theorem ( [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF][START_REF] Šafarevič | Algebraic surfaces[END_REF]) that G 2 (C) coincides with Bir(P 2 C ); it is not the case in higher dimension where G n (C) is a strict subgroup of Bir(P n C ) (see [START_REF] Hudson | Cremona Transformations in Plane and Space[END_REF][START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF]). However the following theorems show that G n (C) shares good properties with G 2 (C) = Bir(P 2 C ). In [START_REF] Cerveau | Transformations birationnelles de petit degré, volume 19 of Cours Spécialisés[END_REF] we proved that for any integer n ≥ 2 the group Bir(P n k ), where k denotes an algebraically closed field, is not linear; we obtain a similar statement for G n (k), n ≥ 2:

Theorem A.
-If k is an algebraically closed field, there is no nontrivial finite-dimensional linear representation of G n (k) over any field.

The group G n (C) contains some "big" subgroups:

Proposition B. -
-The group of polynomial automorphisms of C n generated by the affine automorphisms and the Jonquières ones is a subgroup of G n (C).

-If g 0 , g 1 , . . ., g k are some generic automorphisms of

P n C , then g 0 σ n , g 1 σ n , . . . , g k σ n ⊂ G n (C) is a free subgroup of G n (C).
Remark 1.1. -For the meaning of "generic" see the proof of Proposition 4.3.

In [START_REF] Cerveau | Transformations birationnelles de petit degré, volume 19 of Cours Spécialisés[END_REF] we establish that

G 2 (C) = Bir(P 2 C ) is perfect, i.e. [G 2 (C), G 2 (C)] = G 2 (C)
; the same holds for any n:

Theorem C. -If k is an algebraically closed field, G n (k) is perfect.
In [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF] we determine the automorphisms group of G 2 (C) = Bir(P 2 C ); in higher dimensions we have a similar description. Before giving a precise result, let us introduce some notation: the group of the field automorphisms acts on Bir(P n C ): if f is an element of Bir(P n C ), and κ is a field automorphism we denote by κ f the element obtained by letting κ acting on f . Theorem D. -Let ϕ be an automorphism of Bir(P n C ). There exist κ an automorphism of the field C, and ψ a birational map of

P n C such that ϕ( f ) = κ (ψ f ψ -1 ) ∀ f ∈ G n (C).
The question "is the Cremona group simple ?" is a very old one; Cantat and Lamy recently gave a negative answer in dimension 2 (see [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF]). One can consider the same question when G 2 (k) is equipped with the Zariski topology (k denotes here an algebraically closed field) ; Blanc looked at it, and obtained a positive answer ( [START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF]). What about G n (k) ? Proposition E. -If k is an algebraically closed field, the group G n (k), equipped with the Zariski topology, is simple.

Organisation of the article. -We first recall a result of Pan about the set of group generators of Bir(P n C ), n ≥ 3 (see §2); we then note that as soon as n ≥ 3, there are birational maps of degree n = deg σ n that do not belong to G n (C). In §3 we prove Theorem A, and in §4 Proposition B. Let us remark that the fact that the group of tame automorphisms is contained in G n (C) implies that G n (C) contains maps of any degree, it was not obvious a priori. In §5 we study the normal subgroup in G n (C) generated by σ n (resp. by an automorphism of P n C ); it allows us to establish Theorem C. We finish §5 with the proofs of Theorem D, and Proposition E.

Acknowledgments. -I would like to thank D. Cerveau for his helpful and continuous listening. Thanks to the referee that helps me to improve the exposition. Thanks to I. Dolgachev for pointing out me that Coble introduced the group G n (C) in [START_REF] Coble | Point sets and allied Cremona groups[END_REF], and to J. Blanc, J. Diller, F. Han, M. Jonsson, J.-L. Lin for their remarks and comments.

About the set of group generators of

Bir(P n C ), n ≥ 3 2.1. Some definitions. -A polynomial automorphism ϕ of C n is a map C n → C n of the type (z 0 , z 1 , . . . , z n-1 ) → ϕ 0 (z 0 , z 1 , . . . , z n-1 ), ϕ 1 (z 0 , z 1 , . . . , z n-1 ), . . . , ϕ n-1 (z 0 , z 1 , . . . , z n-1 ) , with ϕ i ∈ C[z 0 , z 1 , . . . , z n-1 ], that is bijective; we denote ϕ by ϕ = (ϕ 0 , ϕ 1 , . . . , ϕ n-1 ). A rational self-map φ : P n C P n
C is given by (z 0 : z 1 : . . .

: z n ) φ 0 (z 0 , z 1 , . . . , z n ) : φ 1 (z 0 , z 1 , . . . , z n ) : . . . : φ n (z 0 , z 1 , . . . , z n )
where the φ i are homogeneous polynomials of the same positive degree, and without common factor of positive degree. Let us denote by C[z 0 , z 1 , . . . , z n ] d the set of homogeneous polynomials in z 0 , z 1 , . . ., z n of degree d. The degree of φ is by definition the degree of the φ i . A birational self-map of P n C is a rational self-map that admits a rational inverse. The set of polynomial automorphisms of C n (resp. birational self-maps of P n C ) form a group denoted Aut(C n ) (resp. Bir(P n C )).

2.2.

A result of Pan. -Let us recall a construction of Pan ( [START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF]) which, given a birational self-map of P n C , allows one to construct a birational self-map of

P n+1 C . Let P ∈ C[z 0 , z 1 , . . . , z n ] d , Q ∈ C[z 0 , z 1 , . . . , z n ] ℓ , and let R 0 , R 1 , . . ., R n-1 ∈ C[z 0 , z 1 , . . . , z n-1 ] d-ℓ be some homogeneous poly- nomials. Denote by Ψ P,Q,R : P n C P n C and Ψ : P n-1 C P n-1
C the rational maps defined by

Ψ P,Q,R = QR 0 : QR 1 : . . . : QR n-1 : P & Ψ R = R 0 : R 1 : . . . : R n-1 . Lemma 2.1 ([35]). -Let d, ℓ be some integers such that d ≥ ℓ+1 ≥ 2. Take Q in C[z 0 , z 1 , . . . , z n ] ℓ , and P in C[z 0 , z 1 , . . . , z n ] d without common factors. Let R 1 , . . ., R n be some elements of C[z 0 , z 1 , . . . , z n-1 ] d-ℓ . Assume that P = z n P d-1 + P d Q = z n Q ℓ-1 + Q ℓ with P d-1 , P d , Q ℓ-1 , Q ℓ ∈ C[z 0 , z 1 , . . . , z n-1 ] of degree d -1, resp. d, resp. ℓ -1, resp. ℓ and such that (P d-1 , Q ℓ-1 ) = (0, 0). The map Ψ P,Q,R is birational if and only if Ψ R is birational.
Let us give the motivation of this construction:

Theorem 2.2 ([29, 35]

). -Any set of group generators of Bir(P n C ), n ≥ 3, contains uncountably many non-linear maps.

We will give an idea of the proof of this statement.

Lemma 2.3 ([35]

). -Let n ≥ 3. Let S be an hypersurface of

P n C of degree ℓ ≥ 1 having a point p of multiplicity ≥ ℓ -1.
Then there exists a birational self-map of P n C of degree d ≥ ℓ + 1 that blows down S onto a point.

Proof. -One can assume without loss of generality that p = (0 : 0 : . . . : 0 : 1). Denote by q ′ = 0 the equation of S, and take a generic plane passing through p given by the equation h = 0. Finally Proof of Theorem 2.2. -Let us consider the family of hypersurfaces given by q(z 1 , z 2 , z 3 ) = 0 where q = 0 defines a smooth curve C q of degree ℓ on {z 0 = z 4 = z 5 = . . . = z n = 0}. Let us note that q = 0 is birationally equivalent to P n-2

choose P = z n P d-1 + P d such that • P d-1 = 0; • pgcd (P, hq ′ ) = 1. Now set Q = h d-ℓ-1 q ′ , R i = z i ,
C × C q . Furthermore q = 0 and q ′ = 0 are birationally equivalent if and only if C q and C q ′ are isomorphic. Note that for ℓ = 2 the set of isomorphism classes of smooth cubics is a 1-parameter family, and that according to Lemma 2.3 for any C q there exists a birational self-map of P n C that blows down C q onto a point. Hence any set of group generators of Bir(P n C ), n ≥ 3, has to contain uncountably many non-linear maps. One can take d = ℓ + 1 in Lemma 2.3. In particular Corollary 2.4. -As soon as n ≥ 3, there are birational maps of degree n = deg σ n that do not belong to G n (C).

Remark 2.5. -The maps Ψ P,Q,R that are birational form a subgroup of Bir(P n C ) denoted by J 0 (1; P n C ), and studied in [START_REF] Pan | Cremona maps[END_REF] : in particular J 0 (1; P 3 C ) inherits the property of Theorem 2.2.

2.3.

A first remark. -Let φ be a birational map of

P 3 C . A regular resolution of φ is a morphism π : Z → P 3 C which is a sequence of blow-ups π = π 1 • . . . • π r along smooth irreducible centers, such that -φ • π : Z → P 3 C is a birational morphism, -and each center B i of the blow-up π i : Z i → Z i-1 is contained in the base locus of the induced map Z i-1 P 3
C . It follows from Hironaka that such a resolution always exists. If B is a smooth irreducible center of a blow-up in a smooth projective complex variety of dimension 3, then B is either a point, or a smooth curve. We define the genus of B as follows: it is 0 if B is a point, the usual genus otherwise. Frumkin defines the genus of φ to be the maximum of the genera of the centers of the blow-ups in the resolution of φ (see [START_REF] Frumkin | A filtration in the three-dimensional Cremona group[END_REF]), and shows that this definition does not depend on the choice of the regular resolution. In [START_REF] Lamy | On the genus of birational maps between 3-folds[END_REF] an other definition of the genus of a birational map is given. Let us recall that if E is an irreducible divisor contracted by a birational map between smooth projective complex varieties of dimension 3, then E is birational to P 1 C × C , where C denotes a smooth curve ( [START_REF] Lamy | On the genus of birational maps between 3-folds[END_REF]). The genus of a birational map φ of P 3

C is the maximum of the genera of the irreducible divisors in P 3

C contracted by φ. Lamy proves that these two definitions of genus agree ( [START_REF] Lamy | On the genus of birational maps between 3-folds[END_REF]). Let φ be in Bir(P 3 C ), and let H be an irreducible hypersurface of P 3 C . We say that H is φexceptional if φ is not injective on any open subset of H (or equivalently if there is an open subset of H which is mapped into a subset of codimension ≥ 2 by φ). Let φ 1 , . . ., φ k be in Bir(P 3 C ), and

let φ = φ k • . . . • φ 1 . Let H be an irreducible hypersurface of P 3 C . If H is φ-exceptional, then there exists 1 ≤ i ≤ k and a φ i -exceptional hypersurface H i such that -φ i-1 • . . . • φ 1 realizes a birational isomorphism from H to H i ; -φ i contracts H i .
In particular one has the following statement.

Proposition 2.6. -The group G 3 (C) is contained in the subgroup of birational self-maps of P 3 C of genus 0.

Non-linearity of G n (C)

If V is a finite dimensional vector space over C there is no faithful linear representation Bir(P n C ) → GL(V ) (see [ The image by ρ of these maps satisfy Birkhoff Lemma so p ≤ n: contradiction. In any dimension we have the same property: G n (C) is not linear, i.e. if V is a finite dimensional vector space over C there is no faithful linear representation G n (C) → GL(V ). Actually G n (C) satisfies a more precise property due to Cornulier in dimension 2 (see [START_REF] Cornulier | Nonlinearity of some subgroups of the planar Cremona group[END_REF]):

Proposition 3.1.
-The group G n (C) has no non-trivial finite dimensional representation.

Lemma 3.2. -The map ς = z 0 z n-1 : z 1 z n-1 : . . . : z n-2 z n-1 : z n-1 z n : z 2 n belongs to G n (C). Proof. -We have ς = a 1 σ n a 2 σ n a 3 where a 1 = z 2 -z 1 : z 3 -z 1 : . . . : z n -z 1 : z 1 : z 1 -z 0 , a 2 = z n-1 + z n : z n : z 0 : z 1 : . . . : z n-2 , a 3 = z 0 + z n : z 1 + z n : . . . : z n-2 + z n : z n-1 -z n : z n .
Proof of Proposition 3.1. -We adapt the proof of [START_REF] Cornulier | Nonlinearity of some subgroups of the planar Cremona group[END_REF].

Let us now work in the affine chart

z n = 1. By Lemma 3.2 in G n (C) there is a natu- ral copy of H = (C * ) n ⋊ Z; indeed ς = z 0 z n-1 , z 1 z n-1 , . . . , z n-2 z n-1 , z n-1 ≃ Z acts on (α 0 z 0 , α 1 z 1 , . . . , α n-1 z n-1 ) | α i ∈ C * ≃ (C *
) n and H is the group of maps

(α 0 z 0 z k n-1 , α 1 z 1 z k n-1 , . . . , α n-2 z n-2 z k n-1 , α n-1 z n-1 ) | α i ∈ C * , k ∈ Z .
Consider any linear representation ρ : H → GL(k; C). If p is prime, and if ξ p is a primitive p-root of unity, set g p = (ξ p z 0 , ξ p z 1 , . . . , ξ p z n-1 ), h p = (ξ p z 0 , ξ p z 1 , . . . , ξ p z n-2 , z n-1 ).

Then h p = [ς, g p ] commutes with both φ and g p . By [2, Lemma 1] if ρ(g p ) = 1, then k ≥ p.

Picking p to be greater than k, this shows that if we have an arbitrary representation f : G n (C) → GL(k; C), the restriction f |PGL(n+1;C) is not faithful. Since PGL(n + 1; C) is simple, this implies that f is trivial on PGL(n + 1; C). We conclude by using the fact that the two involutions -id and σ n are conjugate via the map ψ given by

z 0 + 1 z 0 -1 , z 1 + 1 z 1 -1 , . . . , z n-1 + 1 z n-1 -1
and ψ = a 1 σ n a 2 where a 1 and a 2 denote the two following automorphisms of

P n C a 1 = z 0 + 1, z 1 + 1, . . . , z n-1 + 1 , a 2 = z 0 -1 2 , z 1 -1 2 , . . . , z n-1 -1 2 . Remark 3.3. -Proposition 3.1 is also true for G n (k)
where k is an algebraically closed field.

Subgroups of G n (C)

4.1. The tame automorphisms. -The automorphisms of C n written in the form (φ 0 , φ 1 , . . . , φ n-1 ) where

φ i = φ i (z i , z i+1 , . . . , z n-1 )
depends only on z i , z i+1 , . . ., z n-1 form the Jonquières subgroup J n ⊂ Aut(C n ). A polynomial automorphism (φ 0 , φ 1 , . . . , φ n-1 ) where all the φ i are linear is an affine transformation. Denote by Aff n the group of affine transformations; Aff n is the semi-direct product of GL(n; C) with the commutative unipotent subgroup of translations. We have the following inclusions

GL(n; C) ⊂ Aff n ⊂ Aut(C n ).
The subgroup Tame n ⊂ Aut(C n ) generated by J n and Aff n is called the group of tame automorphisms. For n = 2 one has Tame 2 = Aut(C 2 ), this follows from the fact that Aut(C 2 ) = J 2 * J 2 ∩Aff 2 Aff 2 (see [START_REF] Jung | Über ganze birationale Transformationen der Ebene[END_REF]). The group Tame 3 does not coincide with Aut(C 3 ): the Nagata automorphism is not tame ( [START_REF] Shestakov | The tame and the wild automorphisms of polynomial rings in three variables[END_REF]). Derksen gives a set of generators of Tame n (see [START_REF] Van Den Essen | Polynomial automorphisms and the Jacobian conjecture[END_REF] for a proof):

+ z 2 1 , z 1 , z 2 , . . . , z n-1 belongs to G n (C). But z 0 z n + z 2 1 : z 1 z n : z 2 z n : . . . : z n-1 z n : z 2 n = g 1 σ n g 2 σ n g 3 σ n g 2 σ n g 4
where

g 1 = z 2 -z 1 + z 0 : 2z 1 -z 0 : z 3 : z 4 : . . . : z n : z 1 -z 0 ,
g 2 = z 0 + z 2 : z 0 : z 1 : z 3 : z 4 : . . . : z n , g 3 =z 1 : z 0 + z 2 -3z 1 : z 0 : z 3 : z 4 : . . . : z n ,

g 4 = z 1 -z n : -2z n -z 0 : 2z n -z 1 :
-z 2 : -z 3 : . . . : -z n-1 .

Free groups and G n (C).

-Following the idea of [14, Proposition 5.7] we prove that:

Proposition 4.3.
-Let g 0 , g 1 , . . ., g k be some generic elements of Aut(P n C ). The group generated by g 0 , g 1 , . . ., g k , and σ n is the free product k+1 Z * . . . * Z * (Z/2Z), the g i 's and σ n being the generators for the factors of this free product.

In particular the subgroup g 0 σ n , g 1 σ n , . . . , g k σ n of G n (C) is a free group. If g, σ n is not isomorphic to Z * Z/2Z, then there exists a word M g in Z * Z/2Z such that M g (g, σ n ) = id. Note that the set of words M g is countable, and that for a given word M the set

R M = g M(g, σ n ) = id is algebraic in Aut(P n C ).
Consider an automorphism g written in the following form αz 0 + βz 1 : γz 0 + δz 1 : z 2 : z 3 : . . . : z n where α β γ δ ∈ PGL(2; C). Since the pencil z 0 = tz 1 is invariant by both σ n and g, one inherits a linear representation g, σ n → PGL(2; C) defined by

g : t → αt + β γt + δ , σ n : t → 1 t .
But the group generated by α β γ δ and 0 1 1 0 is generically isomorphic to Z * Z/2Z

(see [START_REF] De La Harpe | Topics in geometric group theory[END_REF]). Hence the complements R C M are dense open subsets, and their intersection is dense by Baire property.

Some algebraic properties of G n (C)

5.1. The group G n (C) is perfect. -If G is a group, and if g is an element of G, we denote by

N(g; G) = f g f -1 | f ∈ G .
the normal subgroup generated by g in G.

Proposition 5.1. -The following assertions hold:

1. N(g; PGL(n + 1; C)) = PGL(n + 1; C) for any g ∈ PGL(n + 1; C) {id}; 2. N(σ n ; G n (C)) = G n (C); 3. N(g; G n (C)) = G n (C) for any g ∈ PGL(n + 1; C) {id}.
Proof. -Let us work in the affine chart z n = 1.

1. Since PGL(n + 1; C) is simple one has the first assertion.

2. Let φ be in G n (C); there exist g 0 , g 1 , . . .,

g k in Aut(P n C ) such that φ = (g 0 ) σ n g 1 σ n . . . σ n g k (σ n ). As PGL(n + 1; C) is simple N(-id; PGL(n + 1; C)) = PGL(n + 1; C),
and for any 0 ≤ i ≤ k there exist f i,0 , f i,1 , . . ., f i,ℓ i in PGL(n + 1; C) such that

g i = f i,0 -id f -1 i,0 f i,1 -id f -1 i,1 . . . f i,ℓ i -id f -1
i,ℓ i . We conclude by using the fact that -id and σ n are conjugate via an element of G n (C) (see the proof of Proposition 3.1).

3. Fix g in PGL(n + 1; C) {id}. Since N(g; PGL(n + 1; C)) = PGL(n + 1; C), the involution -id can be written as a composition of some conjugates of g. The maps -id and σ n being conjugate one has

σ n = ( f 0 g f -1 0 ) ( f 1 g f -1 1 ) . . . ( f ℓ g f -1 ℓ ) for some f i in G n (C). So N(σ n ; G n (C)) ⊂ N(g; G n (C))
, and one concludes with the second assertion.

Corollary 5.2. -The group G n (C) satisfies the following properties:

1. G n (C) is perfect, i.e. [G n (C), G n (C)] = G n (C); 2. for any φ in G n (C) there exist g 0 , g 1 , . . ., g k automorphisms of P n C such that φ = (g 0 σ n g -1 0 )(g 1 σ n g -1 1 ) . . . (g k σ n g -1 k )
Proof. -1. The third assertion of Proposition 5.1 implies that any element of G n (C) can be written as a composition of some conjugates of t = z 0 : z 1 + z n : z 2 + z n : . . . : z n-1 + z n : z n . As t = z 0 : 3z 1 : 3z 2 : . . . : 3z n-1 : z n , (2z 0 : z 1 + z n : z 2 + z n : . . . :

z n-1 + z n : 2z n ) , the group G n (C) is perfect.
2. For any α 0 , α 1 , . . ., α n in C * set d(α 0 , α 1 , . . . , α n ) = (α 0 z 0 : α 1 z 1 : . . . : α n z n ), and let us define H as follows:

H = g 0 σ n g -1 0 g 1 σ n g -1 1 . . . g ℓ σ n g -1 ℓ | g i ∈ PGL(n + 1; C), ℓ ∈ N .
The second assertion of the Corollary is then equivalent to H = G n (C). Let us remark that H is a group that contains σ n , and that PGL(n + 1; C) acts by conjugacy on it. One can check that

d α σ n d -1 α = d 2 α σ n = σ n d -2 α .
(5.1) Hence for each g in PGL(n + 1; C) we have gd α σ n d -1 α g -1 = (gd 2 α g -1 )(gσ n g -1 ), so gd 2 α g -1 belongs to H. Since any automorphism of P n C can be written as a product of diagonalizable matrices, PGL(n + 1; C) ⊂ H. Indeed, write C as the algebraic closure of a purely transcendental extension Q(x i , i ∈ I) of Q; if f : I → I is an injective map, then there exists a field morphism κ :

On the restriction of automorphisms of the group birational maps to

C → C x i → x f (i) .
Note that such a morphism is surjective if and only if f is onto.

In 2006, using the structure of amalgamated product of Aut(C 2 ), the automorphisms of this group have been described:

Theorem 5.3 ([20]

). -Let ϕ be an automorphism of Aut(C 2 ). There exist a polynomial automorphism ψ of C 2 , and a field automorphism κ such that

ϕ( f ) = κ (ψ f ψ -1 ) ∀ f ∈ Aut(C 2 ).
Then, in 2011, Kraft and Stampfli show that every automorphism of Aut(C n ) is inner up to field automorphisms when restricted to the group Tame n :

Theorem 5.4 ([31]

). -Let ϕ be an automorphism of Aut(C n ). There exist a polynomial automorphism ψ of C n , and a field automorphism κ such that

ϕ( f ) = κ (ψ f ψ -1 ) ∀ f ∈ Tame n .
Even if Bir(P 2 C ) hasn't the same structure as Aut(C 2 ) (see Appendix of [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF]) the automorphisms group of Bir(P 2 C ) can be described, and a similar result as Theorem 5.3 is obtained ( [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF]). There is no such result in higher dimension; nevertheless in [START_REF] Cantat | Morphisms between Cremona groups and a characterization of rational varieties[END_REF] Cantat classifies all (abstract) homomorphisms from PGL(k + 1; C) to the group Bir(M) of birational maps of a complex projective variety M, provided k ≥ dim C M. Before recalling his statement let us introduce some notation. Given g in Aut(P n C ) = PGL(n + 1; C) we denote by t g the linear transpose of g. The involution g → g ∨ = ( t g) -1

determines an exterior and algebraic automorphism of the group Aut(P n C ) (see [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF]). Theorem 5.5 ([11]). -Let M be a smooth, connected, complex projective variety, and let n be its dimension. Let k be a positive integer, and let ρ : Aut(P k C ) → Bir(M) be an injective morphism of groups. Then n ≥ k, and if n = k there exists a field morphism κ : C → C, and a birational map

ψ : M P n C such that either ψ ρ(g) ψ -1 = κ g ∀ g ∈ Aut(P n C ) or ψ ρ(g) ψ -1 = ( κ g) ∨ ∀ g ∈ Aut(P n C ); in particular M is rational. Moreover, κ is an automorphism of C if ρ is an isomorphism.
Let us give the proof of Theorem D: Theorem 5.6. -Let ϕ be an automorphism of Bir(P n C ). There exists a birational map ψ of P n C , and a field automorphism κ such that ϕ

(g) = κ (ψgψ -1 ) ∀ g ∈ G n (C).
Proof. -Let us consider ϕ ∈ Aut(Bir(P n C )). Theorem 5.5 implies that up to birational conjugacy and up the action of a field automorphism either ϕ

(g) = g ∀ g ∈ Aut(P n C ) or ϕ(g) = g ∨ ∀ g ∈ Aut(P n C ). (5.3)
In other words up to birational conjugacy and up to the action of a field automorphism one cas assume that either ϕ |Aut(P n C ) : g → g, or ϕ |Aut(P n C ) : g → g ∨ . Now determine ϕ(σ n ). Let us work in the affine chart z n = 1. For 0 ≤ i ≤ n -2 denote by τ i the automorphism of P n C that permutes z i and z n-1

τ i = z 0 , z 1 , . . . , z i-1 , z n-1 , z i+1 , z i+2 , . . . , z n-2 , z i .
Let η be given by

η = z 0 , z 1 , . . . , z n-2 , 1 z n-1 .
One has

σ n = τ 0 ητ 0 τ 1 ητ 1 . . . τ n-2 ητ n-2 η so ϕ(σ n ) = ϕ(τ 0 )ϕ(η)ϕ(τ 0 ) ϕ(τ 1 )ϕ(η)ϕ(τ 1 ) . . . ϕ(τ n-2 )ϕ(η)ϕ(τ n-2 ) ϕ(η).
Since any τ i belongs to Aut(P n C ) one can, thanks to (5.3), compute ϕ(τ i ), and one gets: ϕ(τ i ) = τ i . Let us now focus on ϕ(η). We will distinguish the two cases of (5.3). Assume that ϕ |PGL(n+1;C) = id. For any α = (α 0 , α 1 , . . . , α n-1 ) in (C * ) n set d α = (α 0 z 0 , α 1 z 1 , . . . , α n-1 z n-1 ); the involution η satisfies for any α = (α 0 , α 1 , . . . , α n-1 ) ∈ (C * ) n d β η = ηd α where β = (α 0 , α 1 , . . . , α -1 n-1 ). Hence ϕ(η) = ±z 0 , ±z 1 , . . . , ±z n-2 , α z n-1 for α ∈ C * . As η commutes with t = z 0 + 1, z 1 + 1, . . . , z n-2 + 1, z n-1 , the image ϕ(η) of η commutes to ϕ(t) = t. Therefore ϕ(η) = z 0 , z 1 , . . . , z n-2 , α z n-1 .

If

h n = z 0 z 0 -1 , z 0 -z 1 z 0 -1 , z 0 -z 2 z 0 -1 , . . . , z 0 -z n-1 z 0 -1 then ϕ(h n ) =

  and conclude with Lemma 2.1.

14 ,

 14 Proposition 5.1]). The proof of this statement is based on the following Lemma due to Birkhoff ([2, Lemma 1]): if a, b and c are three elements of GL(n; C) such that [a, b] = c, [a, c] = [b, c] = id, c p = id for some p prime then p ≤ n. Assume that there exists an injective homomorphism ρ from Bir(P 2C ) to GL(n; C). For any p > n prime consider in the affine chart z 2 = 1 the maps (exp(2iπ/p)z 0 , z 1 ), (z 0 , z 0 z 1 ), (z 0 , exp(-2iπ/p)z 1 ).

Remark 4 . 4 .

 44 -The meaning of "generic" is explained in the proof below. Proof. -Let us show the statement for k = 0 (in the general case it is sufficient to replace the free product Z * Z/2Z by Z * Z * . . . * Z * Z/2Z).

  G n (C). -If M is a projective variety defined over a field k ⊂ C the group Aut k (C) of automorphisms of the field extension C/k acts on M(C), and on both Aut(M) and Bir(M) as follows κ ψ(p) = (κψκ -1 )(p) (5.2) for any κ in Aut k (C), any ψ in Bir(M), and any point p in M(C) for with both sides of (5.2) are well defined. Hence Aut k (C) acts by automorphisms on Bir(M). If κ : C → C is a morphism field, this contruction gives an injective morphism Aut(P n C ) → Aut(P n C ) g → g ∨ .

5 . 3 .

 53 h n , and (h n σ n )3 = id implies that ϕ(σ n ) = σ n . If ϕ |PGL(n+1;C) coincides with g → g ∨ , a similar argument yields ϕ(h n )ϕ(σ n ) 3 = id. Simplicity of G n (C). -An algebraic family of Bir(P n C ) is the data of a rational map φ : M × P n C P n C , where M is a C-variety, defined on a dense open subset U such that -for any m ∈ M the intersection U m = U ∩ ({m} × P n C ) is a dense open subset of {m} × P n C , -and the restriction of id × φ to U is an isomorphism of U on a dense open subset of M × P n C . For any m ∈ M the birational map z φ(m, z) represents an element φ m in Bir(P n C ); the map M → Bir(P n C ), m → φ m is called morphism from M to Bir(P n C ). These notions yield the natural Zariski topology on Bir(P n C ), introduced by Demazure ([18]) and Serre ([45]): the subset Ω of Bir(P n C ) is closed if for any C-variety M, and any morphism M → Bir(P n C ) the preimage of Ω in M is closed. Note that in restriction to Aut(P n C ) one obtains the usual Zariski topology of the algebraic group Aut(P n C ) = PGL(n + 1; C). Let us recall the following statement: Proposition 5.7 ([5]). -Let n ≥ 2. Let H be a non-trivial, normal, and closed subgroup of Bir(P n C ). Then H contains Aut(P n C ) and PSL 2; C(z 0 , z 1 , . . . , z n-2 ) . In our context we have a similar statement: Proposition 5.8. -Let n ≥ 2. Let H be a non-trivial, normal, and closed subgroup of G n (C). Then H contains Aut(P n C ) and σ n .

  Theorem 4.1. -Let n ≥ 3 be a natural integer. The group Tame n is generated by Aff n , and the Jonquières map z 0 + z 2 1 , z 1 , z 2 , . . . , z n-1 . Proposition 4.2. -The group G n (C) contains the group of tame polynomial automorphisms of C n .

	Proof. -The inclusion Aff n ⊂ Aut(P n C ) is obvious; according to Theorem 4.1 we thus just have
	to prove that z 0

Proof. -A similar argument as in [START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF] allows us to prove that Aut(P n C ) is contained in H. The fact -id and σ n are conjugate in G n (C) (see Proof of Proposition 3.1) yields the conclusion.

The proof of Proposition E follows from Proposition 5.8 and Corollary 5.2.