Julie Déserti
email: julie.deserti@unibas.ch

DEGREE GROWTH OF BIRATIONAL MAPS OF THE PLANE

Keywords: 2010 Mathematics Subject Classification. -14E07, 37F10, 32H50 CONTENTS 1. Introduction 1 2. Elliptic

This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth.

The classification elements of infinite order with a bounded sequence of degrees is achieved, the case of elements of finite order being already known. The coefficients of the linear and quadratic growth are then described, and related to geometrical properties of the map. The dynamical number of base-points is also studied.

Applications of our results are the description of embeddings of the Baumslag-Solitar groups and GL(2, Q) into the Cremona group.

INTRODUCTION

A rational map of the complex projective plane P 2 = P 2 C into itself is a map of the following type φ : P 2 P 2 , (x : y : z) φ 0 (x, y, z) : φ 1 (x, y, z) : φ 2 (x, y, z) ,

where the φ i 's are homogeneous polynomials of the same degree without common factor. The degree deg φ of φ is by definition the degree of these polynomials. We will only consider birational maps, which are rational maps having an inverse, and denote by Bir(P 2 ) the group of such maps, classically called Cremona group.

We are interested in the behaviour of the sequence deg φ k k∈N . According to [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], the sequence is either bounded or has a linear, quadratic or exponential growth. We will say that φ is (1) elliptic if the growth is bounded;

(2) a Jonquières twist if the growth is linear;

(3) an Halphen twist if the growth is quadratic; Both authors were supported by the Swiss National Science Foundation grant no PP00P2_128422 /1.

(4) hyperbolic if the growth is exponential. This terminology is classical, and consistent with the natural action of Bir(P 2 ) on an hyperbolic space of infinite dimension, where Jonquières and Halphen twists are parabolic [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]Theorem 3.6]).

Recall that the first dynamical degree of φ ∈ Bir(P 2 ) is λ(φ) = lim k→+∞ (deg φ k ) 1/k ∈ R. This is an invariant of conjugation which allows to distinguish the first three cases (where λ(φ) = 1) from the last case (where λ(φ) > 1). There are plenty of articles on hyperbolic elements and the possible values for the algebraic integer λ(φ), we are here more interested in the growth of the first three cases.

The nature of the growth is invariant under conjugation, and induces geometric properties on φ, that we describe now.

An element φ ∈ Bir(P 2 ) is elliptic if and only if it is conjugate to an automorphism g ∈ Aut(S) of a smooth projective rational surface S such that g n belongs to the connected component Aut 0 (S) of Aut(S) for some n > 0 (see [13, Theorem 0.2, Lemma 4.1]). Reading this description, one would expect to find examples where g has infinite order and does not belong to the connected component. We will remove this possibility and refine the result of [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] in Section 2, by showing that, up to conjugation, either g has finite order or S = P 2 . The complete classification of elements of finite order of Bir(P 2 ) can be found in [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]; for elements of infinite order, one has (Proposition 2.3):

Theorem A. If φ ∈ Bir(P 2
) is elliptic of infinite order, then φ is conjugate to an automorphism of P 2 , which restricts to one of the following automorphisms on some open subset isomorphic to C 2 :

(1) (x, y) → (αx, βy), where α, β ∈ C * , and where the kernel of the group homomorphism Z 2 → C * given by (i, j) → α i β j is generated by (k, 0) for some k ∈ Z; (2) (x, y) → (αx, y + 1), where α ∈ C * .

The end of Section 2 is devoted to the description of the conjugacy classes of such maps (Proposition 2.4) and their centralizers in the Cremona group (Lemmas 2.7 and 2.8).

A birational map φ ∈ Bir(P 2 ) has a finite number b(φ) of base-points (that may belong to P 2 or correspond to infinitely near points). We will call the number

µ(φ) = lim k→+∞ b(φ k ) k ,
the dynamical number of base-points of φ. In Section 3, we study the sequence {b(φ k )} k∈N and deduce some properties on the number µ(φ); let us state some of them. It is a non-negative integer invariant under conjugation; it also allows us to give a characterisation of birational maps conjugate to an automorphism of a smooth projective rational surface (Proposition 3.5):

Theorem B. 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.

In the case where φ is a Jonquières twist, the number µ(φ) determines the degree growth of φ. A Jonquières twist preserves an unique pencil of rational curves [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]Theorem 0.2]. The sequence deg φ k k∈N grows as αk for some constant α ∈ R. The number α is not invariant under conjugation, but one can show that the minimal value is attained when the rational curves of the pencil are lines, and is an integer divided by 2. More precisely, one has (Proposition 4.4):

Theorem C. Let φ ∈ Bir(P 2 ) be a Jonquières twist. admits a minimum, which is equal to µ(φ) 2 ∈ 1 2 N. (2) There exists an integer a ∈ N such that

lim k→+∞ deg(φ k ) k = µ(φ) 2 • a 2 .
Moreover, a = 1 if and only if φ preserves a pencil of lines.

The case of Halphen twists is similar. An Halphen twist preserves an unique pencil of elliptic curves (see [START_REF] Gizatullin | Rational G-surfaces[END_REF], [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]Theorem 0.2]). Any such pencil can be sent by a birational map onto a pencil of curves of degree 3n with 9 points of multiplicity n, called Halphen pencil. We obtain the following (Proposition 5.1): Theorem D. Let φ ∈ Bir(P 2 ) be an Halphen twist.

(1) The set

lim k→+∞ deg(ψφ k ψ -1 ) k 2 ψ ∈ Bir(P 2 ) admits a minimum κ(φ) ∈ Q >0 .
(2) There exists an integer a ≥ 3 such that

lim k→+∞ deg(φ k ) k 2 = κ(φ) • a 2 9 .
Moreover, a = 3 if and only if φ preserves an Halphen pencil.

An application of our results is the description of birational maps whose two distinct iterates are conjugate, the non-existence of embeddings of Baumslag-Solitar groups into the Cremona group and the description of the embeddings of GL(2, Q) into the Cremona group ( § 6): Theorem E. Let φ denote a birational map of P 2 of infinite order. Assume that φ n and φ m are conjugate and that |m| = |n|. Then, φ is conjugate to an automorphism of C 2 of the form (x, y) → (αx, y + 1), where α ∈ C * such that α m+n = 1 or α m-n = 1.

In particular, if φ is conjugate to φ n for any positive integer n, then φ is conjugate to (x, y) → (x, y + 1).

Theorem F. If |m|, |n|, 1 are distinct, there is no embedding of BS(m, n) = r, s | rs m r -1 = s n , m, n ∈ Z, mn = 0 into the Cremona group.
Theorem G. Let ρ : GL(2, Q) → Bir(P 2 ) be an embedding. Up to conjugation by an element of Bir(P 2 ), there exists an odd integer k and an homomorphism χ :

Q * → C * (with respect to multiplication) such that ρ a b c d = x • χ(ad -bc) (cy + d) k , ay + b cy + d , ∀ a b c d ∈ GL(2, Q).
Remark 1.1. All these statements hold replacing C by an algebraically closed field of characteristic zero.

Acknowledgements. The authors would like to thank Yves de Cornulier and Ivan Marin for interesting discussions on group theory. Thanks also to Charles Favre for questions on elliptic maps of infinite order and to Serge Cantat for his remarks and references.

ELLIPTIC MAPS

2.1. Classification of elliptic maps of infinite order. If φ is a birational map of P 2 such that deg φ k k∈N is bounded, it is conjugate to an automorphism g of a smooth rational surface S such that the action of g on Pic(S) is finite [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]Lemma 4.1]. If g has finite order, the possible conjugacy classes are completely classified in [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]. Here we deal with the case of elements of infinite order, classifying the possibilities and describing its centralizers in Bir(P 2 ).

Proposition 2.1. Let g be an automorphism of a smooth rational surface S which has infinite order but has a finite action on Pic(S). Then, there exists a birational morphism S → X where X is equal either to P 2 or to an Hirzebruch surface F n for n = 1, which conjugates g to an automorphism of X.

Remark 2.2. The proof of this result follows from a study of possible minimal pairs, which is similar to the one made in [START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF] for finite abelian subgroups of Bir(P 2 ) (see [START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF]Lemmas 3.2,6.1,9.7]).

Proof. Contracting the possible sets of disjoint (-1)-curves on S which are invariant by g, we can assume that the action of g on S is minimal. The action of g on Pic(S) being of finite order, the process corresponds to applying a G-Mori program, where G is a finite group acting on Pic(S) (we only look at parts of the Picard group which are invariant). Then one of the following occurs ( [START_REF] Ju | Rational surfaces over perfect fields[END_REF][START_REF] Iskovskih | Minimal models of rational surfaces over arbitrary fields[END_REF]):

(1) Pic(S) g has rank 1 and S is a del Pezzo surface;

(2) Pic(S) g has rank 2, and there exist a conic bundle π : S → P 1 on S, together with an automorphism h of P 1 such that h • π = π • g.

We want to show that S is P 2 or an Hirzebruch surface F n for n = 1, and exclude the other cases.

In the case where Pic(S) g has rank 1, the fact that g has infinite order but finite action on Pic(S) implies that the kernel of the group homomorphism Aut(S) → Aut(Pic(S)) is infinite. So S is a del Pezzo surface of degree (K S ) 2 ≥ 6. The surface cannot be F 1 otherwise the exceptional section would be invariant. Similarly, it cannot be the unique del Pezzo surface of degree 7, which has exactly three (-1)-curves, forming a chain (one touches the two others, which are disjoint), because the curve of the middle (and also the union of the two others) would be invariant. The only possibilities are thus P 2 , P 1 × P 1 = F 0 , and the del Pezzo surface of degree 6.

If S is the del Pezzo surface of degree 6, any element h ∈ Aut(S) acting minimally on S has finite order [7, Lemma 9.7]. Let us recall the simple argument. The del Pezzo surface of degree 6 is isomorphic to

S = (x : y : z), (u : v : w) ∈ P 2 × P 2 | xu = yv = zw .
The projections π 1 , π 2 on each factor are birational morphisms contracting three (-1)-curves on p 1 = (1 : 0 : 0), p 2 = (0 : 1 : 0) and p 3 = (0 : 0 : 1). The group Pic(S) is generated by the six (-1)-curves of S, which are E i = (π 1 ) -1 (p i ) and F i = (π 2 ) -1 (p i ) for 1 ≤ i ≤ 3 and form an hexagon. In fact, the action on the hexagon gives rise to an isomorphism Aut(S) ≃ (C * ) 2 ⋊ (Sym 3 ⋊ Z/2Z) = (C * ) 2 ⋊ D 6 . The action of g on S being minimal, g permutes cyclically the curves, and either g or g -1 acts as

(E 1 → F 2 → E 3 → F 1 → E 2 → F 3 ).
This implies that g or g -1 is equal to (x : y : z), (u : v : w) → (αv : βw : u), (βy : αz : αβx) , for some α, β ∈ C * , and has order 6.

We can now assume the existence of an invariant conic bundle π : S → P 1 . If π has no singular fibre, then S is a Hirzebruch surface F n and n = 1 because of the minimality of the action. It remains to exclude the case where π has at least one singular fibre. The minimality of the action on S implies that the two components of any singular fibre F (which are two (-1)-curves) are exchanged by a power g k of g, and in particular that the whole singular fibre is invariant by g k . Note that k a priori depends on F.

We now prove that g k does not act trivially on the basis of the conic bundle. If g k acts trivially on the basis of the fibration, the automorphism g 2k acts trivially on Pic(S); taking a birational morphism S → F n which contracts a component in each singular fibre one conjugates g 2k to an automorphism of F n , which fixes pointwise at least one section. The pull-back on S of this section intersects only one component in each singular fibre and its image by g k gives thus another section, also fixed by g 2k . The action of g k on a general fibre of π exchanges the two points of the two sections and hence has order 2: contradiction.

The action of g k on the basis is non-trivial and fixes the point of P 1 corresponding to the singular fibre; so the same holds for g (recall that the fixed points of an element of Aut(P 1 ) and any of its non-trivial powers are the same). This implies that F is invariant by g, so its two components are exchanged by it (and thus k is odd).

In particular, g exchanges the two components of any singular fibre. This implies that the number of singular fibres of π is at most 2, so S is the blow-up of one or two points of an Hirzebruch surface.

The fact that the two components of at least one singular fibre are exchanged gives a symmetry on the sections, that will help us to determine S. Denote by -m the minimal self-intersection of a section of π and let s be one section which realises this minimum. Contracting the components in the singular fibres which do not intersect s, one has a birational morphism S → F n . The image of s is a section with minimal selfintersection, so m = n. If n = 0, then taking some section of F 0 = P 1 × P 1 of self-intersection 0 passing through at least one blown-up point, its strict transform on S would be a section of negative self-intersection, which contradicts the minimality of s 2 , so m = n > 0. Let us denote by s ′ the section g(s), which also has self-intersection -m on S but self-intersection -m + r on F m , where r is the number of singular fibres of π. Because any section of F m distinct from the exceptional section has self-intersection ≥ m, we get -m + r ≥ m, so 2 ≥ r ≥ 2m, which implies that m = 1 and r = 2.

The surface S is thus the blow-up of two points on F 1 , not lying on the exceptional section and not on the same fibre, so is a del Pezzo surface of degree 6. The fact that g acts minimally on S is impossible, as we already observed. Proposition 2.3. Let φ be a birational map of P 2 of infinite order, such that deg φ k k∈N is bounded. Then φ is conjugate to an automorphism of P 2 , which restricts to one of the following automorphisms on some open subset isomorphic to C 2 :

(1) (x, y) → (αx, βy), where α, β ∈ C * , and where the kernel of the group homomorphism Z 2 → C * given by (i, j) → α i β j is generated by (k, 0) for some k ∈ Z; (2) (x, y) → (αx, y + 1), where α ∈ C * .

Proof. According to Proposition 2.1 the map φ is conjugate to an automorphism of a minimal surface S, equal to either P 2 or an Hirzebruch surface.

Suppose first that S = P 2 . Looking at the Jordan normal form, any automorphism of P 2 is conjugate to -either (x : y : z) → (αx : βy : z), -or (x : y : z) → (αx : y + z : z), -or (x : y : z) → (x + y : y + z : z). This latter automorphism is conjugate to (x : y : z) → (x : y + z : z) in Bir(P 2 ) (for instance by (x : y : z) (xz -1 2 y(yz) : yz : z 2 ), as already observed in [6, Example 1]). It remains to study the case of diagonal automorphisms to show the assertion on the kernel stated in the proposition. As in the proof of [START_REF] Blanc | Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the Cremona groups[END_REF]Proposition 6],

we associate to each diagonal automorphism ψ : (x : y : z) → (αx : βy : z) the kernel ∆ ψ of the following homomorphism of groups:

δ ψ : Z 2 → C * , (i, j) → α i β j .
For any M = a b c d ∈ GL(2, Z), we denote by M(ψ) the diagonal automorphism

(x : y : z) → α a β b x : α c β d y : z ,
which is the conjugate of ψ by the birational map (x, y) x a y b , x c y d (viewed in the chart z = 1). We can check that

δ M(ψ) = δ ψ • t M, which implies that ∆ M(ψ) = t M -1 (∆ ψ ).
We can always choose M (by a result on Smith's normal form) such that ∆ M(φ) is generated by k 1 e 1 and k 1 k 2 e 2 , where e 1 , e 2 are the canonical basis vectors of Z 2 , and k 1 , k 2 are non-negative integers, and replace φ with M(φ), which is conjugate to it. Since φ and M(φ) have infinite order, we see that k 2 = 0, and get the assertion on the kernel stated in the proposition.

If S = F 0 = P 1 × P 1 , we can reduce to the case of P 2 by blowing-up a fixed point and contracting the strict transform of the members of the two rulings passing through the point.

Suppose now that S = F n for n ≥ 2. If g fixes a point of F n which is not on the exceptional section, we can blow-up the point and contract the strict transform of the fibre to go to F n-1 . We can thus assume that all points of F n fixed by g are on the exceptional section. The action of g on the basis of the fibration is, up to conjugation, x → αx or x → x + 1 for some α ∈ C * . Removing the fibre at infinity and the exceptional section, we get C 2 , where the action of g is -either (x, y) → (αx, βy + Q(x)), -or (x, y) → (x + 1, βy + Q(x)), where α, β ∈ C * and Q is a polynomial of degree ≤ n. The action on the fibre at infinity is obtained by conjugating by (x, y)

1 x , y
x n . In the first case, there is no fixed point on the fibre at infinity (except the point on the exceptional section) if and only if β = α n and deg Q = n. There is no fixed point on x = 0 if and only if Q(0) = 0 and β = 1. This implies that α is a primitive k-th rooth of unity, where k is a divisor of n. Conjugating by (x, y

+ γx d ) we replace Q(x) with Q(x) + γ(α d -1)x d , so we can assume that the coefficient of x d is trivial if d is not a multiple of k, which means that Q(x) = P(x k ) for some polynomial P ∈ C[x].
In particular, g is equal to (x, y) → (ξx, y + P(x k )) and is conjugate to (x, y) → ξx, y + 1 by (x, y)

x, y P(x k ) . In the second case, there is no fixed point on C 2 , and no point on the fibre at infinity if and only if β = 1 and deg Q = n. Conjugating g by (x, y) → (x, y + γx n+1 ) (which corresponds to performing an elementary link F n F n+1 at the unique fixed-point and then coming back with an elementary link at a general point of the fibre at infinity), we get

(x, y) → x + 1, y -γx n+1 + Q(x) + γ(x + 1) n+1 .
Choosing the right element γ ∈ C, we can decrease the degree of Q(x), and get (x, y) → (x+ 1, y) by induction.

Conjugacy classes of elliptic maps of infinite order.

Following [START_REF] Blanc | Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the Cremona groups[END_REF], we will call elements of the form (x, y) → (αx, βy), resp. (x, y) → (αx, y + 1) diagonal automorphisms, resp. almost-diagonal automorphisms of C 2 (or P 2 ). The conjugacy classes in each family are given by the following:

Proposition 2.4 ([6], Theorem 1).

(1) A diagonal automorphism and an almost-diagonal automorphism of C 2 are never conjugate in Bir(C 2 ).

(2) Two diagonal automorphisms (x, y) → (αx, βy) and (x, y) → (γx, δy) are conjugate in Bir(C 2 ) if and

only if there exists a b c d ∈ GL(2, Z) such that (α a β b , α c β d ) = (γ, δ).
(

) 3
for some m, n ∈ Z, |m| = |n|, then φ is conjugate to an automorphism of C 2 of the form (x, y) → (αx, y + 1), where α ∈ C * such that α m+n = 1 or α m-n = 1.
Proof. Note that mn = 0 since φ has infinite order. Then φ is conjugate to one of the two cases of Proposition 2.3. First of all, assume that up to conjugation φ is (x, y) → (αx, βy) and that the kernel ∆ φ of the group homomorphism Z 2 → C * given by (i, j) → α i β j is generated by (k, 0) for some k ∈ Z. Since φ m and φ n are conjugate there exists a matrix

N = a b c d ∈ GL(2, Z) such that (α m ) a (β m ) b , (α m ) c (β m ) d = (α n , β n ) (Proposition 2.4
). This means that α ma-n β mb , α mc β md-n = (1, 1), so (man, mb), (mc, mdn) belong to ∆ φ . In particular mb = mdn = 0, which implies that b = 0, so ad = ±1, which is impossible since m = ±n.

Assume now that φ is conjugate to (x, y) → (αx, y + 1) for some α in C * . The fact that φ m and φ n are conjugate implies that α m+n = 1 or α m-n = 1 (Proposition 2.4).

Centralisers of elliptic maps of infinite order.

If φ is a birational map of P 2 , we will denote by C(φ) the centraliser of φ in Bir(P 2 ):

C(φ) = ψ ∈ Bir(P 2 ) φψ = ψφ .
In the sequel, we describe the centralisers of elliptic maps of infinite order of Bir(P 2 ). The results are groups which contain the centralisers of some elements of PGL(2, C). We recall the following result, whose proof is an easy exercise. Recall that PGL(2, C) is the group of automorphisms of P 1 , or equivalently the group of Möbius transformations x ax+b cx+d . Lemma 2.6. For any α ∈ C * , we have

η ∈ PGL(2, C) η(αx) = αη(x) =    PGL(2, C) if α = 1 {x γx ±1 | γ ∈ C * } if α = -1 {x → γx | γ ∈ C * } if α = ±1
Lemma 2.7. Let us consider φ : (x, y) → (αx, βy) where α, β are in C * , and where the kernel of the group homomorphism Z 2 → C * given by (i, j) → α i β j is generated by (k, 0) for some k ∈ Z. Then the centraliser of φ in Bir(P 2 ) is

C(φ) = (x, y) (η(x), yR(x k )) R ∈ C(x), η ∈ PGL(2, C), η(αx) = αη(x) .
Proof. Let ψ : (x, y) (ψ 1 (x, y), ψ 2 (x, y)) be an element of C(φ). The fact that ψ commutes with φ is equivalent to

(⋆) ψ 1 (αx, βy) = αψ 1 (x, y)
and (⋄) ψ 2 (αx, βy) = βψ 2 (x, y).

Writing ψ i = P i Q i for i = 1, 2
, where P i , Q i are polynomials without common factors, we see that P 1 , P 2 , Q 1 , Q 2 are eigenvectors of the linear automorphism φ * of the C-vector space C[x, y] given by φ * : f (x, y) → f (αx, βy). This means that each of the P i , Q i is a product of a monomial in x, y with an element of

C[x k ]. Using (⋆) and (⋄), we get the existence of R 1 , R 2 ∈ C(x) such that ψ 1 (x, y) = xR 1 (x k ), ψ 2 (x, y) = yR 2 (x k ).
The fact that ψ is birational implies that

ψ 1 (x, y) is an element η(x) ∈ PGL(2, C); it satisfies η(αx) = αη(x)
because of (⋆).

C(φ) = (x, y) (η(x), y + R(x)) η ∈ PGL(2, C), η(αx) = αη(x), R ∈ C(x), R(αx) = R(x) .
Proof. Conjugating by (x, y) → (x, βy), we can assume that β = 1. Let ψ : (x, y) (ψ 1 (x, y), ψ 2 (x, y)) be a birational map of P 2 which commutes with φ. One has

(⋆) ψ 1 (αx, y + 1) = αψ 1 (x, y) and (⋄) ψ 2 (αx, y + 1) = ψ 2 (x, y) + 1.
Equality (⋆) implies that ψ 1 only depends on x (see [

B(αx) = B(x) + 1 -a, which implies that ∂B ∂x (αx) = α -1 ∂B ∂x (x)
, and thus that x ∂B ∂x (x) is invariant under x → αx. If α is not a root of unity, this means that ∂B ∂x = c/x for some c ∈ C; since B is a rational function, one gets c = 0 and B is a constant (or equivalently an element such that B(αx) = B(x)). It implies moreover a = 1 and we are done.

If α is a primitive k-th root of unity, the fact that ψ : (x, y) (η(x), ay + B(x)) commutes with

φ k : (x, y) (x, y + k) yields a(y + k) + B(x) = ay + B(x) + k, so a = 1. We again get B(αx) = B(x).

ON THE GROWTH OF THE NUMBER OF BASE-POINTS

If S is a projective smooth surface, any element φ ∈ Bir(S) has a finite number of base-points, which can belong to S or be infinitely near. We denote by b(φ) the number of such points. We will call the number In this section, we precise the properties of this number, and will in particular see that it is an integer.

µ(φ) = lim k→+∞ b(φ k ) k ,
If φ ∈ Bir(S) is a birational map, we will say that a (possibly infinitely near) base-point p of φ is a persistent base-point if there exists an integer N such that p is a base-point of φ k for any k ≥ N but is not a base-point of φ -k for any k ≥ N.

If p is a point of S or a point infinitely near, which is not a base-point of φ ∈ Bir(S), we define a point φ • (p), which will also be a point of S or a point infinitely near. For this, take a minimal resolution

Z π 2 1 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ π 1 Ð Ð ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ S φ G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ S,
where π 1 , π 2 are sequences of blow-ups. Because p is not a base-point of φ, it corresponds, via π 1 , to a point of Z or infinitely near. Using π 2 , we view this point on S, again maybe infinitely near, and call it φ • (p). Remark 3.1. If p is not a base-point of φ ∈ Bir(S) and φ(p) is not a base-point of ψ ∈ Bir(S), we have (ψφ)

• (p) = ψ • (φ • (p)). If p is a general point of S, then φ • (p) = φ(p) ∈ S.
Example 3.2. If S = P 2 , p = (1 : 0 : 0) and φ is the birational map (x : y : z)

(yz + x 2 : xz : z 2 ), the point φ • (p) is not equal to p = φ(p), but is infinitely near to it.
Using this definition, we put an equivalence class on the set of points that belong to S or are infinitely near, by saying that p is equivalent to q if there exists an integer k such that (φ k ) • (p) = q (this implies that p is not a base-point of φ k and that q is not a base-point of φ -k ). The set of equivalence classes is the generalisation of the notion of set of orbits for birational maps.

(φ k ) -νk k ≥ 0 ⊂ Z is bounded.
In particular, µ(φ) is an integer, equal to ν.

Proof. Let us say that a base-point q is periodic if (φ k ) • (q) = q for some k = 0, or if q is a base-point of φ k for any k ∈ Z \ {0} (which implies that (φ k ) • (q) is never defined for k = 0). Let us denote by P the set of periodic base-points of φ and by P the finite set of points equivalent to a point of P .

The number of base-points of φ and φ -1 being finite, there exists an integer N such that for any nonperiodic base-point p and for any j, j ′ ≥ N, p is a base-point of φ j (respectively of φ -j ) if and only if p is a base-point of φ j ′ (respectively of φ -j ′ ).

We decompose the set of non-periodic base-points of φ into four sets:

B ++ = p | p is a base-point of φ j
, and is a base-point of φ -j for j ≥ N , B +-= p | p is a base-point of φ j , but is not a base-point of φ -j for j ≥ N , B -+ = p | p is not a base-point of φ j , but is a base-point of φ -j for j ≥ N , B --= p | p is not a base-point of φ j , and is not a base-point of φ -j for j ≥ N .

Note that B +-is the set of persistent base-points of φ and that B -+ is the set of persistent base-point of φ -1 .

This decomposes the set of base-points of φ into five disjoint sets. Two base-points p, p ′ of φ which are equivalent belong to the same set.

We fix an integer k ≥ 2N and compute the number of base-points of φ k . Any such base-point being equivalent to a base-point of φ, we take a base-point p of φ, and count the number m p,k of base-points of φ k which are equivalent to p.

If p belongs to P , the number of points equivalent to p is less than #( P ), so m p,k ≤ #( P ). If p is not in P , any point equivalent to p is equal to (φ i ) • (p) for some i, and all are distinct, so we have

m p,k = #I p,k , where I p,k = i ∈ Z p is not a base-point of φ i , but p is a base-point of φ i+k . If p ∈ B ++ , since p is not a base-point of φ i one has -N < i < N, thus m p,k < 2N. A point p ∈ B --is a base-point of φ i+k hence -N < i + k < N and m p,k < 2N.
If p ∈ B -+ , the fact that p is not a base-point of φ i implies that -N < i and the fact that p is a base-point of φ i+k implies that i + k ≤ N. With these two inequalities one has -N < i ≤ Nk. Since k > 2N, we get m p,k = 0.

If p ∈ B +-, the fact that p is not a base-point of φ i implies that i < N and the fact that p is a base-

point of φ i+k implies that -N < i + k. This yields -N -k < i < N, so m p,k ≤ 2N + k. Conversely, if i ≤ -N and i + k ≥ N, p is not a base-point of φ i , but p is a base-point of φ i+k (or equivalently i ∈ I p,k ), so m p,k ≥ #[N -k, -N] = k -2N + 1. The two conditions together imply that m p,k -k ∈ [-2N, 2N].
Recall that B +-is the set of persistent base-points of φ. The above counting explains that the number of base-points of φ k , for k big, behaves like νk, where ν is the set of equivalence classes of of persistent basepoints of φ. More precisely, there exist two integers c, d which do not depend on k, such that the total number of base-points of φ k is between νk + c and νk + d, for any k ≥ 2N. Recalling that µ(φ) = lim k→+∞ b(φ k ) k , where b(φ k ) is the number of base-points of φ k , we obtain µ(φ) = ν.

Corollary 3.4. The dynamical number of base-points is an invariant of conjugation: if θ : S

Z is a birational map between smooth projective surfaces and φ ∈ Bir(S), then µ(φ) = µ(θφθ -1 ).

In particular, if φ is conjugate to an automorphism of a smooth projective surface, then µ(φ) = 0.

Proof. The map θ factorises into the blow-up of a finite number of base-points followed by the contraction of a finite number of curves. The number of equivalence classes of persistent base-points of φ and θφθ -1 is thus the same, and we get the result from Proposition 3.3.

It is thus clear that µ(φ) = 0 if φ is conjugate to an automorphism of a smooth projective surface.

Proposition 3.5. Let S be a smooth projective surface, and let φ ∈ Bir(S). The following conditions are equivalent:

(1) µ(φ) = 0;

(2) φ is conjugate to an automorphism of a smooth projective surface.

Proof. Corollary 3.4 yields (2) ⇒ (1). It suffices then to show (1) ⇒ (2). Denote by K the set of points, that belong to S as proper or infinitely near points, which are base-points of φ i and φ -j , for some i, j > 0. Let us prove that K is a finite set. Choosing the smallest possible i, j associated to p ∈ K, the point p is equivalent to (φ i-1 ) • (p), which is a base-point of φ, and the same holds replacing p with (φ k ) • (p), forj < k < i. By this way, we associate a finite number of points of K to each base-point of φ. This shows that K is finite. Observing that any point of K is either a proper point of S or is infinitely near, we can blow-up the set K, and obtain a birational morphism π : S → S.

By construction, the birational map φ ∈ Bir( S) given by π -1 φπ has less base-points than φ, and satisfies that no point of S is a base-point of φ i and φ -j , for some i, j > 0 (in the notation of [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], the map φ is now algebraically stable). If a (-1)-curve of S (a smooth curve isomorphic to P 1 and having self-intersection (-1)) is contracted by φ, we contract it, via a birational morphism η 1 : S → S 1 . The map φ 1 = η 1 • φ • (η 1 ) -1 is again algebraically stable, and we continue the process if a (-1)-curve of S 1 is contracted by φ 1 . At the end, we obtain a birational morphism η = η k • • • • • η 1 : S → S k = S which conjugates φ to φ k = φ, and such that no (-1)-curve of S is contracted by φ.

It remains to show that φ is an automorphism of S. Assuming the converse, we will deduce a contradiction. We write τ 1 : X 1 → S the blow-up of the base-points of φ, and write χ 1 : X 1 → S the morphism χ 1 = φτ 1 , which is the blow-up of the base-points of φ -1 . Denote by C 1 ⊂ X 1 a (-1)-curve contracted by χ 1 onto a base-point p of φ -1 (such a curve exists for each base-point). Because p is a base-point of φ -1 , it is not a basepoint of φ r for any r > 0. The map φ has no persistent base-point, because µ(φ) = µ( φ) = 0 (Proposition 3.3 and Corollary 3.4). In consequence, there exists k > 1 such that p is not a base-point of φ -k .

We write χ 2 : X 2 → X 1 the blow-up of the base-points of φ 1-k τ 1 , and denote by τ 2 : X 2 → S the morphism τ 2 = φ 1-k τ 1 χ 2 , which is the blow-up of the base-points of (τ 1 ) -1 φ k-1 ; this yields the following commutative diagram: X 2

τ 2 Ñ Ñ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ χ 2 2 2 ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ X 1 χ 1 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ τ 1 6⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ S φ k-1 G G ❴ ❴ ❴ ❴ ❴ ❴ S φ G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ S.
Because p is not a base-point of φ -k , the curve C 1 ⊂ X 1 has to be contracted by τ 2 • (χ 2 ) -1 , and the curve C 2 = (χ 2 ) -1 (C 1 ) has self-intersection -1 (otherwise φ -k would have a base-point infinitely near to p, and p itself would then be a base-point).

The curve τ 1 (C 1 ) is contracted by φ, and is then not a (-1)-curve. Because it is contracted by φ 1-k , there is a base-point q of φ 1-k that is a proper point of τ 1 (C 1 ). Since τ 2 is a morphism, τ 1 χ 2 blows-up all base-points of φ k-1 , and thus blows-up q. The fact that q is a base-point of φ 1-k , implies that it is not a base-point of φ, and thus that it is not blown-up by τ 1 . In consequence, (τ 1 ) -1 (q) is a point blown-up by ξ 2 , and which lies on C 1 . This is incompatible with the fact that C 2 = (χ 2 ) -1 (C 1 ) has self-intersection -1.

Remark 3.6. In [13, Theorem 0.4] one can find a characterisation of hyperbolic birational maps φ which are conjugate to an automorphism of a projective surface. If φ ∈ Bir(P 2 ) is hyperbolic, we conjugate it to a birational map of a smooth projective surface S where the action is algebraically stable (this means that the (φ| H 1,1 (S) ) n = (φ n )| H 1,1 (S) for each n); its action on H 1,1 (S) admits the eigenvalue λ(φ) > 1 with eigenvector θ + . The map φ is birationally conjugate to an automorphism if and only if (θ + ) 2 = 0. Proposition 3.5 gives another characterisation, for all maps φ ∈ Bir(P 2 ) (not only hyperbolic maps), depending only on µ(φ).

Example 3.7. In [START_REF] Bedford | Periodicities in linear fractional recurrences: degree growth of birational surface maps[END_REF][START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF][START_REF] Bedford | Continuous families of rational surface automorphisms with positive entropy[END_REF][START_REF] Bedford | Dynamics of rational surface automorphisms: rotations domains[END_REF][START_REF] Déserti | Special automorphisms of rational surfaces with positive topological entropy[END_REF], automorphisms with positive entropy are constructed starting from a birational map of P 2 . In [START_REF] Déserti | Special automorphisms of rational surfaces with positive topological entropy[END_REF] the authors take a birational map ψ ∈ Bir(P 2 ), and choose A ∈ Aut(P 2 ) such that Aψ is conjugate to an automorphism of a surface with dynamical degree > 1. The way to find A is exactly to ensure that Aψ has no persistent base-point (i.e. µ(Aψ) = 0). Let us give an example ( [START_REF] Déserti | Special automorphisms of rational surfaces with positive topological entropy[END_REF]):

Let φ = Aψ be the birational map given by

A : (x : y : z) αx + 2(1 -α)y + (2 + α -α 2 )z : -x + (α + 1)z : x -2y + (1 -α)z
with α ∈ C \ {0, 1} and ψ : (x : y : z) (xz 2 + y 3 : yz 2 : z 3 ).

The map ψ (resp. ψ -1 ) has five base-points, p = (1 : 0 : 0) and four points infinitely near; we will denote P 1 (resp. P 2 ) the collection of these points. The automorphism A is chosen such that:

• P 1 , A P 2 , AψA P 2 have distinct supports;

• P 1 = (Aψ) 2 A P 2 . In particular the base-points of φ are non-persistent, so φ is conjugate to an automorphism of a rational surface. More precisely φ is conjugate to an automorphism with positive entropy on P 2 blown up in P 1 , A P 2 and AψA P 2 (see [START_REF] Déserti | Special automorphisms of rational surfaces with positive topological entropy[END_REF]Theorem 3.1]).

GROWTH OF JONQUIÈRES TWISTS

Lemma 4.1. Let φ be a birational map of P 2 which preserves the pencil of lines passing through some point p 0 . The set

deg φ k -k • µ(φ) 2 k ≥ 0 ⊂ Z is bounded.
In particular, the sequence deg φ k k∈N grows linearly if and only if µ(φ) > 0 and its growth is given by µ(φ)

2 ∈ 1 2 N.
Remark 4.2. Conjugating φ by a map which preserves the pencil does not change the growth deg φ k k∈N , but conjugating it by a map which does not preserve the pencil can increase it (see Proposition 4.4).

Proof. For any k, φ k preserves the pencil of lines passing through p 0 . It implies that the linear system of φ k (which is the pull-back of the system of lines of P 2 by φ k ) has multiplicity deg φ k -1 at p 0 and has exactly 2(deg φ k -1) other base-points, all of multiplicity 1. In particular, deg

φ k = ⌊ b(φ k )
2 ⌋. The result follows then directly from Proposition 3.3.

Example 4.3.

Let us consider the family of birational maps studied in [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF] and defined as

f α,β : (x : y : z) ((αx + y)z : βy(x + z) : z(x + z)), α, β ∈ C * .
Any of the f α,β has three base-points: (1 : 0 : 0), (0 : 1 : 0) and (-1 : α : 1), and preserves the pencil of lines passing through (1 : 0 : 0). Checking that (-1 : α : 1) is the only one persistent base-point ([11, Theorem 1.6]), the growth of deg f k α,β k∈N is given by k 2 (see [START_REF] Déserti | Expériences sur certaines transformations birationnelles quadratiques[END_REF]Lemma 1.4]).

Proposition 4.4. Let φ ∈ Bir(P 2 ) be a Jonquières twist. There exists an integer a ∈ N such that

lim k→+∞ deg(φ k ) k = a 2 µ(φ) 2 .
Moreover, a = 1 if and only if φ preserves a pencil of lines.

Proof. Since φ is a Jonquières twist, there exists ψ ∈ Bir(P 2 ) such that φ = ψφψ -1 preserves the pencil of lines passing through some point p ∈ P 2 . Let π : F 1 → P 2 be the blow-up of p ∈ P 2 , and let φ ∈ Bir(F 1 ) be φ = π -1 φπ. Denote by L P 2 the linear system of lines of P 2 and by Λ the linear system on F 1 corresponding to the image by π -1 ψ of the system of lines of P 2 . The degree of φ k is equal to the free intersection of L P 2 with φ k (L P 2 ), which is the free intersection of Λ with φ(Λ).

On F 1 , Λ is linearly equivalent to aL + b f , where L = π -1 (L P 2 ), f is the divisor of a fibre and where a, b ∈ N. Note that φ k (L) is the transform on F 1 of the linear system of φ -k , and is thus equal to L + (d k -1) f , where d k is the degree of φ k (and of φ -k ). The system φ k (Λ) is then linearly equivalent to aL

+ (a(d k -1) + b) f , so the total intersection of φ k (Λ) with Λ is a 2 d k + 2ab. Because φ k (Λ) • f = a, each base-point of φ k (Λ)
has at most multiplicity a. By Lemma 4.1, lim

k→+∞ d k k = µ(φ) 2 .
The number of base-points of Λ being bounded, the free intersection of φ k (Λ) with Λ grows like a 2 µ(φ) 2 • k. It remains to see that if a = 1 then φ preserves a pencil (the other direction follows from Lemma 4.1). If a = 1, one gets f • Λ = 1. This implies that the free intersection of φ -1 π( f ) with φ -1 π(Λ) = L P 2 is 1; so φ -1 π( f ) is a pencil of lines, invariant by φ. We can also derive the following new results. Proof. The fact that deg φ k k∈N grows linearly implies that φ preserves a pencil of rational curves [13, Theorem 0.2]. In particular φ is conjugate to a birational map of P 2 which preserves the pencil of lines passing through some fixed point p 0 . According to Lemma 4.1, one finds µ(φ) > 0.

As φ m and φ n are conjugate in Bir(P 2 ) one has µ(φ m ) = µ(φ n ) (Corollary 3.4). Since µ(φ k ) = |k • µ(φ)| for any k, we get |m| = |n|.

GROWTH OF HALPHEN TWISTS

In Section 4 (especially Lemma 4.1), we described the degree growth of a Jonquières twist φ, and showed that it is given by µ(φ), a birational invariant given by the growth of base-points. For an Halphen twist, the dynamical number of base-points is trivial, but the growth can also be quantified by an invariant. Recall that an Halphen twist φ preserves an unique pencil of elliptic curves. By [14, Theorem 2 and Proposition 7, page 127], a power of φ preserves any member of the pencil, and acts on this via a translation. Proposition 5.1. Let φ ∈ Bir(P 2 ) be an Halphen twist.

(1) The set

lim k→+∞ deg(ψφ k ψ -1 ) k 2 ψ ∈ Bir(P 2 )
admits a minimum, which is a positive rational number κ(φ) ∈ Q. If φ acts via a translation on each member of its invariant pencil of elliptic curves, then κ(φ) ∈ 9N.

(2) There exists an integer a ≥ 3 such that lim

k→+∞ deg(φ k ) k 2 = κ(φ) • a 2 9 .
(3) The following conditions are equivalent:

(a) a = 3;

(b) φ preserves an Halphen pencil, i.e. a pencil of (elliptic) curves of degree 3n passing through 9 points with multiplicity n.

Proof. An Halphen twist preserves an unique elliptic fibration, so there exists an element ψ in Bir(P 2 ) such that φ ′ = ψφψ -1 preserves an Halphen pencil. Denoting by π : S → P 2 the blow-up of the 9 base-points of the pencil, φ = π -1 φ ′ π is an automorphism of S, which preserves the elliptic fibration S → P 1 given by | -mK S | for some positive integer m. Replacing φ by some power if needed, we can assume that φ is a translation on a general fibre. As explained in the proof of [14, Proposition 9, page 132], this yields the existence of an element ∆ ∈ Pic(S) (depending on φ) with ∆ • K S = 0 such that the action of φ on Pic(S) is given by

D → D -m(D • K S ) • ∆ + γK S ,
where γ is an integer depending on D which can be computed using the self-intersection:

γ = - m 2 2 (D • K S ) • ∆ 2 + m(D • ∆).
We denote by L the linear system of lines of P 2 and by Λ = π -1 ψ(L) its transform on S. The degree of φ n is equal to the free intersection of L with φ n (L), which is equal to the free intersection of Λ with φ n (Λ).

The map φ n acts on Pic(S) as

D → D -m(D • K S ) • (n∆) + - m 2 2 (D • K S ) • (n∆) 2 + m(D • (n∆)) K S .
This yields

Λ • φ n (Λ) = Λ 2 -m(Λ • K S ) • (n∆ • Λ) + -m 2 2 (Λ • K S )(n∆) 2 + m(nΛ • ∆) (K S • Λ) = Λ 2 + -m 2 2 (Λ • K S ) 2 (∆) 2 • n 2 .
The free intersection between Λ and φ n (Λ) is thus equal to

Λ 2 + - m 2 2 (Λ • K S ) 2 (∆) 2 • n 2 - k ∑ i=1 µ i (Λ) • µ i ( φ n (Λ)),
where µ i (Λ) and µ i ( φ n (Λ)) denote the multiplicities of respectively Λ and φ n (Λ) at the r base-points of Λ. Since φ is an automorphism of S, the contribution given by the base-points is bounded, so we find that

lim n→+∞ deg(φ n ) n 2 = m 2 (Λ • K S ) 2 • -∆ 2 2 .
Note that Λ is the lift by π -1 of the homaloidal linear system ψ(L), so Λ • (-K S ) ≥ 3, and equality holds if and only if Λ has no base-point. This shows that the minimum among all homaloidal systems is attained when Λ has no base-point; we get κ

(φ) = 9m 2 • -∆ 2 2 and a = Λ • K S .
Let us prove that -∆ 2 is a positive even number. To do this, we take the orthogonal basis L, E 1 , . . . , E 9 of Pic(S), where L is the pull-back of a line of P 2 and E 1 , . . . , E 9 are the exceptional divisors associated to the points blown-up. Writing

∆ = dL - 9 ∑ i=1 a i E i , the equality ∆•K S = 0 implies that ∑ a i = 3d. Modulo 2, we have ∑(ai) 2 ≡ ∑ a i ≡ 3d ≡ d 2 , so ∆ 2 = d 2 -∑(a i ) 2
is even. Applying Cauchy-Schwarz to (1, . . . , 1) and (a 1 , . . . , a 9 ), we get 9 ∑(ai) 2 ≥ (∑ a i ) 2 , and equality holds only when all a i are equal. This latter would imply that ∆ is a multiple of K S , and thus that φ acts trivially on Pic(S). Hence we have 9 ∑(ai) 2 > (∑ a i ) 2 = (3d) 2 , which implies that -∆ 2 = ∑(ai) 2d 2 > 0.

The fact that -∆ 2 is a positive even number implies that κ(φ) = 9m is equivalent to the fact that Λ has no base-point, which corresponds to say that ψ -1 π is a birational morphism, or equivalently that the unique pencil of elliptic curves invariant by φ is an Halphen pencil. The last two points are the following: the point p 8 is infinitely near to p 6 , corresponding to the tangent direction of the line y = iz, and p 9 is infinitely near to p 7 , corresponding to the tangent direction of the line y = -iz.

The surface S inherits an elliptic fibration S → P 1 , and the lift of α yields an automorphism α = παπ -1 of S. Denote by E i ∈ Pic(S) the divisor of self-intersection -1 corresponding to the point p i . If i = 6, 7, then E i corresponds to a (-1)-curve of S; and E 6 , E 7 correspond to two reducible curves of S.

For any ∆ ∈ Pic(S) satisfying ∆ • K S = 0, we denote by ι ∆ ∈ Aut(S) the automorphism which restricts on a general fibre C to the translation given by the divisor ∆| C . If ∆ 2 = -2, the action of ι ∆ on Pic(S) is given by (see the proof of Proposition 5.1)

D → D -(D • K S ) • ∆ + (D • (K S + ∆)) • K S .
For any automorphism σ ∈ Aut(S), one can check that ι σ(∆) = σ ι ∆ σ -1 . In particular, we have

( α ι ∆ ) 4 = ( α ι ∆ α -1 )( α 2 ι ∆ α -2 )( α 3 ι ∆ α -3 )ι ∆ = ι α(∆) ι α 2 (∆) ι α 3 (∆) ι ∆ = ι α(∆)+ α 2 (∆)+ α 3 (∆)+∆ .
Because of the action of α on the points p i , we have

α(E 6 ) = E 7 , α(E 7 ) = E 6 , α(E 2 ) = E 3 , α(E 3 ) = E 4 , α(E 4 ) = E 5 , α(E 5 ) = E 2 .
We now fix ∆ ∈ Pic(S) to be the divisor E 2 -E 6 (that satisfies ∆ • K S = 0 and ∆ 2 = -2), and obtain

α(∆) + α 2 (∆) + α 3 (∆) + ∆ = E 2 + E 3 + E 4 + E 5 -2E 6 -2E 7 ,
which has square -8. In particular, the number κ associated to ( αι ∆ ) 4

= ι α(∆)+ α 2 (∆)+ α 3 (∆)+∆ is -9 (ι α(∆)+ α 2 (∆)+ α 3 (∆)+∆ ) 2 2 = 36.
This shows, by Corollary 5.2, that κ(φ) = 9 4 , where φ is the birational map of P 2 conjugate to α ι ∆ by π -1 , namely απ ι ∆ π -1 .

6. APPLICATIONS 6.1. Birational maps having two conjugate iterates. Lemma 6.1. Let φ denote a birational map of P 2 . Assume that φ n and φ m are conjugate and assume that |m| = |n|. Then, φ is an elliptic and satisfies λ(φ) = 1 and µ(φ) = 0.

Proof. The map φ m is conjugate to φ n in Bir(P 2 ) so one gets λ(φ) |m| = λ(φ) |n| and |m| • µ(φ) = |n| • µ(φ). This yields λ(φ) = 1 and µ(φ) = 0.

The fact that λ(φ) = 1 implies that φ is elliptic, or a Jonquières or Halphen twist. The Jonquières and Halphen cases are impossible (Corollaries 4.6 and 5.2). Proposition 6.2. Let φ denote a birational map of P 2 of infinite order. Assume that φ n and φ m are conjugate and assume that |m| = |n|. Then, φ is conjugate to an automorphism of C 2 of the form (x, y) → (αx, y + 1), where α ∈ C * such that α m+n = 1 or α m-n = 1.

In particular, if φ is conjugate to φ n for any positive integer n, then φ is conjugate to (x, y) → (x, y + 1).

Proof. Follows from Corollary 2.5, Corollary 5.2 and Lemma 6.1. Recall that if G is a group, the derived groups of G are

Morphisms of

G (0) = G, G (i) = G (i-1) , G (i-1) = ghg -1 h -1 | g, h ∈ G (i-1) for all i ≥ 1,
and that G is solvable if there exists an integer N such that G (N) = {id}.

The groups BS(m, n) (resp. the subgroups of finite index of BS(m, n)) are solvable if and only if |m| = 1 or |n| = 1 (see [START_REF] Souche | Quasi-isométrie et quasi-plans dans l'étude des groupes discrets[END_REF]Proposition A.6

]).

A group G is said to be residually finite if for any g in G \ {id} there exist a finite group H and a group homomorphism Θ : G → H such that Θ(g) belongs to H \ {id}. The group BS(m, n) is residually finite if and only if |m| = 1 or |n| = 1 or |m| = |n| (see [START_REF] Meskin | Nonresidually finite one-relator groups[END_REF]). Let V be an affine algebraic variety; according to [START_REF] Bass | Automorphisms of groups and of schemes of finite type[END_REF] any subgroup of finite index of the automorphisms group of V is residually finite. Therefore if |m| = |n| and |m|, |n| = 1 there is no embedding of BS(m, n) into the group of polynomial automorphisms of the plane. There is an other proof using the amalgated structure of the group of polynomial automorphisms of the plane and the fact that BS(m, n) is not solvable ([10, Proposition 2.2]). Lemma 6.3. Let ρ be a homomorphism from BS(m, n) = r, s | rs m r -1 = s n to Bir(P 2 ). Assume that |m|,|n| and 1 are distinct. If ρ(s) has infinite order, the image of the subgroup of finite index r, s m | rs m 2 r -1 = s mn of BS(m, n) is solvable.

Proof. By Proposition 6.2, one can conjugate ρ so that ρ(s) : (x, y) (αx, y+1) where α ∈ C * and α m-n = 1 or α m+n = 1. Denoting respectively by ψ the map x, n m y or x -1 , n m y one has ψρ(s) m ψ -1 = ρ(s) n . So ρ(r) = ψτ where τ commutes with ρ(s) m : (x, y) → (α m x, y + m).

According to Lemma 2.8, one then has τ = (η(x), y + R(x)) for some η ∈ PGL(2, C), η(α m x) = α m η(x), and some R ∈ C(x) satisfying R(α m x) = R(x). And one gets ρ(r) = η(x) ±1 , n m (y + R(x)) . The group generated by ρ(s m ) and ρ(r) is thus solvable. Proof. Let us set

t q = 1 q 0 1 & d m,n = m 0 0 n , q, m, n ∈ Q.
Remark that t 1 is conjugate to t n in GL(2, Q), for any n ∈ Z {0}; we can then assume, after conjugation, that ρ(t 1 ) = (x, y + 1) (Proposition 6.2). As ρ(t 1/n ) commutes with ρ(t 1 ) there exist

A n in PGL(2, C) and R n in C(x) such that ρ(t 1/n ) = (A n (x), y + R n (x))
(see Lemma 2.8). Let us prove now that A n (x) = x. Since t n 1/n = t 1 the element A n is of finite order so A n is conjugate to some ξx where ξ is some root of unity. Hence ρ

(t 1/n ) is conjugate to (ξx, y + Q(x)) where Q ∈ C(x) satisfies Q(x) + Q(ξx) + • • • + Q(ξ n-1 x) = 1. The map (ξx, y + Q(x)) is then conjugate to ξx, y + 1 n by x, y -∑ n-1 i=1 iQ(ξ i x) n
. Since t 1/n is conjugate to t 1 , Proposition 2.4 implies that ξ = 1, which achieves to show that A n (x) = x. This implies, with equality ρ(t 1/n ) n = ρ(t 1 ), that R n (x) = 1/n. We thus have for any q in Q ρ(t q ) = (x, y + q). given by a → χ(a 2 ) a k is injective. It remains to observe that k can be chosen to be positive. Indeed, otherwise one conjugates by 1 x , y and replaces χ with 1 χ to replace k with -k.

One can see that GL(n, Q) does not embedd into Bir(P 2 ) as soon as n ≥ 3. Indeed, Theorem 6.7 implies that the diagonal matrices are sent onto diagonal elements of PGL(3, C) = Aut(P 2 ), which is impossible, by considering the involutions. One can also find another less obvious corollary: The map ρ is injective if and only if χ(a 2 ) = τ(a k ) for any a ∈ C * {1}.

( 1 )

 1 The setlim k→+∞ deg(ψφ k ψ -1 ) k ψ ∈ Bir(P 2 )

Lemma 2 . 8 .

 28 Let us consider φ : (x, y) → (αx, y + β) where α, β ∈ C * . The centraliser of φ in Bir(P 2 ) is equal to

  the dynamical number of base-points of φ. Since b(φψ) ≤ b(φ) + b(ψ) for any φ, ψ ∈ Bir(S), we see that µ(φ) is a non-negative real number. Moreover, b(φ -1 ) and b(φ) being always equal, we get µ(φ k ) = |k • µ(φ)| for any k ∈ Z.

Proposition 3 . 3 .

 33 Let φ be a birational map of a smooth projective surface S. Denote by ν the number of equivalence classes of persistent base-points of φ. Then, the set b

Lemma 4 .Corollary 4 . 5 .

 445 1 and the second assertion of Corollary 3.4 imply the following statement of [13, Theorem 0.2.] Let φ be a Jonquières twist; then φ is not conjugate to an automorphism.

Corollary 4 . 6 .

 46 Let φ be a Jonquières twist. If φ m and φ n are conjugate in Bir(P 2 ) for some m, n ∈ Z, then |m| = |n|.

Corollary 5 . 2 .Example 5 . 3 . 2 : 2 :

 525322 Let φ ∈ Bir(P 2 ) be an Halphen twist. The integer κ(φ) is a birational invariant which satisfies κ(φ m ) = m 2 κ(φ) for any m ∈ Z. In particular, the maps φ n and φ m are not conjugate if |m| = |n|.Proof. Is a direct consequence of Proposition 5.1. Let us give an example where κ(φ) is not an integer. Let Λ be the pencil of cubic curves of P 2 given by λ(x 2 y + z 3 + y 2 z) + µ(x 2 z + y 3 + yz 2 ) = 0, (λ : µ) ∈ P 1 . The pencil is invariant by the automorphism α ∈ Aut(P 2 ) defined by (x : y : z) → (ix : -y : z).Denote by π : S → P 2 the blow-up of the base-points of Λ, which are 7 proper points of P 2 and 2 infinitely near points. More precisely, the 7 proper points arep 1 = (1 : 0 : 0), p 2 = (i √ 2 : 1 : 1), p 3 = α(p 2 ) = (-√ -1 : 1), p 4 = α(p 3 ) = (-i √ 2 :1 : 1), p 5 = α(p 4 ) = ( √ -1 : 1), p 6 = (0 : i : 1), p 7 = α(p 6 ) = (0 : -i : 1).

  Baumslag-Solitar groups in the Cremona group. For any integers m, n such that mn = 0, the Baumslag-Solitar group BS(m, n) is defined by the following presentation BS(m, n) = r, s | rs m r -1 = s n .

2 ,

 2 From d m,n t 1 d-1 m,n = t m/n one gets (using again Lemma 2.8) thatρ(d m,n ) = η m,n (x), m n y + R m,n (x) , η m,n ∈ PGL(2, C), R m,n ∈ C(x).The map(Q * ) 2 → PGL(2, C) given by (m, n) → η m,n is a homomorphism, which cannot be injective. There exists thus one element d m,n with (m, n) = (1, 1) such that ρ(d m,n ) = x, m n y + R m,n (x) . Note that m = n since the centralizers of ρ(d m,n ) and ρ(t 1 ) are different. Conjugating by x, y + R m,n (x)m/n-1 , we can assume that ρ(d m,n ) = x, m n y . From d m,n d a,b = d a,b d m,n one gets for any a, b in Q ρ(d a,b ) = η a,b (x), a b y , η a,b ∈ PGL(2, C).The homomorphism Q * → PGL(2, C) given by a → η a,a is injective, so up to conjugation by an element of PGL(2, C) we can assume that for any a ∈ Q {0, 1} there exists χ a,a ∈ C {0, 1} such that η a,a (x) = χ a,a x. This implies the existence ofχ a,b ∈ C * for any a, b in (Q * ) 2 , such that η a,b (x) = χ a,b x.We now compute the image ofM = 0 1 -1 0 . Since ρ(M) commutes with ρ(d 2,2 ) = (χ 2,2 x, y) where χ 2,2 ∈ C * is of infinite order, there exist R ∈ C(y) and ν ∈ PGL(2, C) such that ρ(M) = (xR(y), ν(y)) (Lemma 2.8). For any a ∈ Q * , equality Md a,1 = d 1,a M yields χ a,1 x • R(ay), ν(ay) = χ 1,a x • R(y), 1 a ν(y) .This implies that R(y) = αy -k and ν(y) = β y , for some α, β ∈ C * , k ∈ Z, i.e. ρ(M) = α x y k , β y . We use now equality (Mt 1 ) 3 = id: the second component of (ρ(M)ρ(t 1 ))3 beingβ(y + β + 1) (β + 1)y + 2β + 1 ,we find β = -1 and compute (ρ(M)ρ(t 1 )) 3 = (xα 3 (-1) k , y) so α 3 = (-1) k . Since ρ(M) 2 = (xα 2 (-1) k , y) has order 2, we have -α 2 = (-1) k , thus α = -1 and k is odd. Writing χ(a) = χ a,1 for any a ∈ Q, the map Q * → C * given by a → χ(a) is a homomorphism, and one gets ρ(d a,1 ) = (χ(a)x, ay). The group GL(2, Q) is generated by the maps d a,1 , t a and M, so Q). This yields an embedding if and only if the homomorphism from Q * to C *

Corollary 6 . 8 .Remark 6 . 9 .

 6869 Let ρ : GL(2, C) → Bir(P 2 ) be an embedding of GL(2, C) into the Cremona group. There exist a positive odd integer k, a field homomorphism τ : C → C and a group homomorphism χ :C * → C * such that ρ a b c d = x • χ(adbc) (τ(c)y + τ(d)) k , τ(a)y + τ(b) τ(c)y + τ(d) , One sees that in the description above, ρ is an embedding if and only if the group homomorphism C * → C * given by a → χ(a 2 ) τ(a) k is injective. This happens for instance by taking χ(a) = τ(a) k±1 2 , any positive odd integer k and any field homomorphism τ : C → C. Proof. The map ρ induces an embedding of GL(2, Q) into the Cremona group. According to Theorem 6.7 one has a description of ρ |GL(2,Q) . Up to conjugacy, there exists an odd positive integer k and an homomorphism χ : Q * → C * such that ρ a b c d = x • χ(adbc) (cy + d) k , , b ∈ C * . For any a ∈ C * , the matrix d a commutes with all diagonal matrices with entries in Q; this implies, with the description above, that ρ(d a ) = (χ(a)x, τ(a)y) for some χ(a), τ(a) in C * (Lemma 2.7). This yields two group homomorphisms χ, τ : C * → C * . Observe that χ is an extension of χ, i.e. χ(a) = χ(a) for any a ∈ Q. The equality d a t 1 d a -1 = t a implies that ρ(t a ) = (x, y + τ(a)) , ∀a ∈ C * . In particular, τ extends to an (injective) field homomorphism C → C. The group GL(2, C) being generated by GL(2, Q) and d a a ∈ C * , one has ρ a b c d = x • χ(adbc) (τ(c)y + τ(d)) k , τ(a)y + τ(b) τ(c)y + τ(d) ,

  Let φ ∈ Bir(P 2 ) be an elliptic map which has infinite order. If φ m is conjugate to φ n in Bir(P 2 )

Two almost diagonal automorphisms (x, y) → (αx, y + 1) and (x, y) → (γx, y + 1) are conjugate in Bir(C 2 ) if and only if α = γ ±1 . Corollary 2.5.

  2

  Corollary 6.4. If |m|, |n| and 1 are distinct, there is no embedding of BS(m, n) into the Cremona group.Proof.Lemma 6.3 shows that the image of any embedding would be virtually solvable, impossible when |m|, |n| and 1 are distinct. 6.3. Embeddings of GL(2, Q) into the Cremona group. To simplify the notation, we will denote in this last section by (φ 1 (x, y), φ 2 (x, y)) the rational map (x, y) (φ 1 (x, y), φ 2 (x, y)) from C 2 to C 2 . Let us first give examples of embeddings of GL(2, Q) into the Cremona group. Let k be an odd integer and let χ :Q * → C * be a homomorphism such that a → χ(a 2 )a k is injective. The morphism ρ from GL(2, Q) to the Cremona group given by

	Example 6.5. ρ (cy + d) given by a b c d = x • χ(ad -bc)
	ρ	a b c d	=	x cy + d	,	ay + b cy + d	,
	which is obviously conjugate (by extending the actions to P 2 ) to the classical embedding
	ρ	a b c d	= (ax + by, cx + dy) .

k , ay + b cy + d is an embedding. Note that ρ(GL(2, Q)) is conjugate to a subgroup of automorphisms of the k-th Hirzebruch surface F k . Changing k gives then infinitely many non conjugate embeddings in the Cremona group. Remark 6.6. Taking k = 1 and χ the trivial map, Example 6.5 yields the embedding ρ : GL(2, Q) → Bir(P 2 ) Theorem 6.7. Let ρ : GL(2, Q) → Bir(P 2 ) be an embedding of GL(2, Q) into the Cremona group, then up to conjugation ρ is one of the embeddings described in Example 6.5.