Serge Cantat
email: cantat@univ-rennes1.fr

DYNAMICAL DEGREES OF BIRATIONAL TRANSFORMATIONS OF PROJECTIVE SURFACES

The dynamical degree λ( f ) of a birational transformation f measures the exponential growth rate of the degree of the formulae that define the n-th iterate of f . We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of λ( f ) and the structure of the conjugacy class of f . For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.

INTRODUCTION

Given a birational transformation f : X X of a projective surface, defined over a field k, its dynamical degree λ( f ) is a positive real number that measures the complexity of the dynamics of f . For instance, if k is the field of complex numbers, log(λ( f )) provides an upper bound for the topological entropy of f : X(C) X(C) and is equal to it under natural assumptions (see [START_REF] Bedford | Energy and invariant measures for birational surface maps[END_REF][START_REF] Dinh | Une borne supérieure pour l'entropie topologique d'une application rationnelle[END_REF]). Our goal is to study the structure of the set of all dynamical degrees λ( f ), when f runs over the group of all birational transformations Bir(X) and X over the collection of all projective surfaces.

The dynamical degree λ( f ) is invariant under conjugacy. An important feature of our results may be summarized by the following slogan: Precise knowledge on λ( f ) provides useful information on the conjugacy class of f . In particular, we shall obtain effective, quantitative bounds for the solutions of certain equations in Bir(X), like the conjugacy problem asking for a solution h of the equation h f h -1 = g.

Another motivation of the present paper is to develop the "dictionary" between groups of birational transformations of projective surfaces and mapping class groups of higher genus, closed, orientable surfaces. The dynamical degree λ( f ) plays a role which is similar to the dilatation factor λ(ϕ) of pseudo-Anosov mapping classes (see § 8 below). As we shall see, our main results should be compared to two theorems proved by W. Thurston. The first one describes explicitly the set of topological entropies of post-critically finite, continuous, multimodal transformations of the unit interval as the set of logarithms of "weak Perron numbers". The second describes the structure of the set of volumes of hyperbolic manifolds of dimension 3; this set is a countable, nondiscrete, and well ordered subset of the real line.

1.1. Dynamical degrees, Pisot numbers, and Salem numbers.

1.1.1. Dynamical degrees. Let X be a projective surface defined over an algebraically closed field k. In what follows, NS(X) denotes the Néron-Severi group of X. Given a ring A, NS A (X) stands for NS(X) ⊗ Z A; hence, NS Z (X) coincides with NS(X).

Let f be a birational transformation of X defined over k. Then f determines an endomorphism f * : NS(X) → NS(X) of NS(X). The dynamical degree λ( f ) of f is defined as the spectral radius of the sequence of endomorphisms ( f n ) * , as n goes to +∞. More precisely, once a norm • has been chosen on the real vector space End(NS R (X)), one defines

λ( f ) = lim n→∞ ( f n ) * 1/n ;
this limit exists, and does not depend on the choice of the norm. Moreover, for every ample divisor D ⊂ X

λ( f ) = lim n→∞ (D • ( f n ) * D) 1/n ,
where C •D denotes the intersection number between divisors or divisor classes. By definition, f is loxodromic if λ( f ) > 1.

The dynamical spectrum of X is defined as the set

Λ(X) = {λ( f ) | f ∈ Bir(X)}.
If one wants to specify the field k, one may denote the dynamical spectrum by Λ(X, k).

Example 1.1. The Néron-Severi group of P 2 k coincides with the Picard group Pic(P 2 k ), has rank 1, and is generated by the class e 0 of a line:

NS(P 2 k ) = Pic(P 2 k ) = Ze 0 .
Fix a choice of homogeneous coordinates [x : y : z] on the projective plane P 2 k . Let f be an element of Cr 2 (k). One can then find three homogeneous polynomials P, Q, and R in the variables (x, y, z), of the same degree d, and without common factor of positive degree, such that f ([x : y : z]) = [P(x, y, z) : Q(x, y, z) : R(x, y, z)].

This degree d does not depend on the choice of homogeneous coordinates; it is denoted by deg( f ) and called the degree of f . On Pic(P 2 k ), f acts by multiplication by its degree deg( f ); thus, we have λ( f ) = lim deg( f n ) 1/n . For instance, the standard quadratic involution satisfies deg(σ n ) = 1 or 2, according to the parity of n; hence λ(σ) = 1.

1.1.2. Pisot and Salem numbers (see [START_REF]Pisot and Salem numbers[END_REF]). By definition, a Pisot number is an algebraic integer λ ∈ ]1, ∞[ whose other Galois conjugates lie in the open unit disk; Pisot numbers include integers d ≥ 2 as well as reciprocal quadratic integers λ > 1. A Salem number is an algebraic integer λ ∈ ]1, ∞[ whose other Galois conjugates are in the closed unit disk, with at least one on the boundary; hence, the minimal polynomial of λ has at least two complex conjugate roots on the unit circle, and the degree of λ is at least 4. We denote by Pis the set of Pisot numbers and by Sal the set of Salem numbers. It is known that Pis is a closed subset of the real line. It is contained in the closure of Sal, and its infimum is equal to λ P 1.324717, the unique root λ P > 1 of the cubic equation x 3 = x + 1; this Pisot number is known as the plastic number, or padovan number. The smallest accumulation point of Pis is the golden mean λ G = (1 + √ 5)/2; all Pisot numbers between λ P and λ G have been listed.

Our present knowledge of Salem numbers is much weaker. Conjecturally, the infimum of Sal is larger than 1, and should be equal to the Lehmer number, i.e. to the Salem number λ L 1.176280 obtained as the unique root > 1 of the irreducible polynomial x 10 + x 9x 7x 6x 5x 4x 3 + x + 1.

1.1.3. Dynamical degrees and algebraic stability. One says that f ∈ Bir(X) is algebraically stable when the endomorphism f * of the Néron-Severi group NS(X) satisfies

( f n ) * = ( f * ) n (1.1)
for all positive integers n. If f is algebraically stable, then f -1 is also algebraically stable and λ( f ) is the spectral radius of the endomorphism f * of NS(X); in particular, λ( f ) is an algebraic integer. Diller and Favre proved in [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] that every birational transformation of a projective surface X is conjugate by a birational morphism π : Y → X to an algebraically stable transformation π -1 • f • π. From this fact and the Hodge index theorem, they obtained the following result.

Theorem 1.2 (Diller and Favre). Let k be a field and let f be a birational transformation of a projective surface defined over k. If λ( f ) is different from 1, then λ( f ) is a Salem or a Pisot number.

In this article we initiate the study of the dynamical spectrum Λ(X). By Diller-Favre Theorem, Λ(X) splits in two parts, its Pisot part Λ P (X) and its Salem part Λ S (X). The problem is to describe which numbers can appear in each of these sets, as well as the relationship between these two sets.

Example 1.3. When f is an algebraically stable transformation of P 2 k , one gets λ( f ) = deg( f ). For instance, the automorphism h of the affine plane defined by h(X,Y ) = (Y, X + Y d ) extends to a birational map of the projective plane such that deg(h n ) = d n for all n ≥ 0. In particular, Λ(P 2 k ) contains all integers d ≥ 1, for all fields k. is an element of GL 2 (Z) and (X,Y ) denotes the coordinates on k * × k * , the automorphism associated to A is defined by f A (X,Y ) = (X a Y b , X c Y d ). This provides an embedding of GL 2 (Z) in the automorphism group Aut(k * × k * ), and thus in Bir(P 2 k (k)). For every A in GL 2 (Z), the dynamical degree of f A is equal to the spectral radius of the matrix A, i.e. to the modulus of its unique eigenvalue λ with |λ| ≥ 1; this implies that f A is not an algebraically stable transformation of P 2 k as soon as λ( f A ) > 1, because λ( f A ) is not an integer in that case.

As a byproduct of this example, the dynamical spectrum of the plane contains all reciprocal quadratic integers, i.e. all roots λ > 1 of equations x 2 + 1 = tx with t in Z.

1.2. Salem numbers and automorphisms. The dynamical degree of an automorphism, if different from 1, is either a quadratic number or a Salem number (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]). Here we prove a converse statement.

Theorem A. Let k be an algebraically closed field. Let f be a birational transformation of a projective surface X, defined over k. If λ( f ) is a Salem number, there exists a projective surface Y and a birational mapping ϕ : Y X such that ϕ -1 • f • ϕ is an automorphism of Y .

Thus, one can decide whether a birational transformation is conjugate to an automorphism by looking at its dynamical degree, except when this degree is 1 or a quadratic integer. For the quadratic case, Examples 2.2 and 2.3 show that there are quadratic integers which are simultaneously realized as dynamical degrees of automorphisms, and of birational transformations that cannot be conjugate to an automorphism. See Remark 2.4 for birational transformations with dynamical degree equal to 1.

Once Theorem A is proved, three corollaries can be deduced from results of McMullen and the second author (see [START_REF] Curtis | Dynamics on blowups of the projective plane[END_REF] and [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). The first corollary (see § 2.6) is a spectral gap property for dynamical degrees: There is no dynamical degree in the interval ]1, λ L [. The second corollary does not seem to be related to values of dynamical degrees, but the simple proof given here makes use of the spectral gap. It asserts that the centralizer, in the group Bir(X), of a loxodromic element f is finite by cyclic (see § 4.3). The third consequence is an effective and explicit bound for the optimal degree of a conjugacy (see § 4.4):

Corollary 1.5. Two loxodromic elements f , g ∈ Bir(P 2 k ) of degree ≤ d are conjugate if and only if they are conjugate by an element h of degree ≤ (2d) 57 .

1.3. From projective surfaces to the projective plane. Non rational surfaces are easily handled with.

Theorem B. Let k be an algebraically closed field. Let X be a projective surface defined over k. If X is not rational, then

(1) Λ(X) = {1} if X is not birationally equivalent to an abelian surface, a K3 surface, or an Enriques surface; (2) Λ(X) \ {1} is made of quadratic integers and of Salem numbers of degree at most 6 (resp. 22, resp. 10) if X is an abelian surface (resp. a K3 surface, resp. an Enriques surface). The union of all dynamical spectra Λ(X) where X runs over the set of nonrational projective surfaces defined over k, and k runs over the set of all fields, is a closed discrete subset of the real line.

Remark 1.6. When the characteristic of the field k vanishes, the degree bounds of Assertion (2) become 4, 20, and 10 (in place of 6, 22, and 10). This result, proved in Section 3, shows that the most interesting case is provided by rational surfaces. Thus, in the following statements, one can assume that X is birationally equivalent to the projective plane P 2 k ; the dynamical spectrum is then equal to the set Λ(P 2 k ) of dynamical degrees of elements of the Cremona group Cr 2 (k) = Bir(P 2 k ). 1.4. Degrees and conjugacy classes.

1.4.1. Minimal degree in the conjugacy class. Given an element f of Bir(P 2 k ), define the minimal degree of f in its conjugacy class as the positive integer

mcdeg( f ) = min deg(g • f • g -1 )
where g describes Bir(P 2 k ). The function mcdeg is constant on conjugacy classes, and λ( f ) ≤ mcdeg( f ) ≤ deg( f ) for all birational transformations of the plane. One of our main goals is to provide the following reverse inequality 1 .

Theorem C. Let k be an algebraically closed field. Let f be a birational transformation of the plane

P 2 k . (1) If λ( f ) ≥ 10 6 then mcdeg( f ) ≤ 4700 λ( f ) 5 . (2) If λ( f ) > 1, then mcdeg( f ) ≤ cosh(18 + 345 log(λ( f ))).
On the other hand, there are sequences of elements f n ∈ Bir(P 2 k ) such that mcdeg( f n ) goes to +∞ with n while λ 1 ( f n ) = 1 for all n. k and the dynamical degrees λ( f n ) decrease with n, then λ( f n ) becomes eventually constant).

Theorem D. Let k be an algebraically closed field. The dynamical spectrum Λ(P 2 k ) ⊂ R is well ordered, and it is closed if k is uncountable. In Theorem 7.4, we also show that Λ P (P 2 k ) is contained in the closure of Λ S (P 2 k ) if k is algebraically closed and of characteristic 0. From Theorem B and Theorem D, one obtains the existence of gaps in the dynamical spectrum of projective surfaces that is, small intervals of real numbers that contain infinitely many Pisot and Salem numbers, but do not contain any dynamical degree.

Corollary 1.7. Let Λ be the set of all dynamical degrees of birational transformations of projective surfaces, defined over any field. Then,

(1) Λ is a well ordered subset of R + ;

(2) if λ is an element of Λ, there is a real number ε > 0 such that ]λ, λ + ε] does not intersect Λ; (3) there is a non-empty interval ε > 0 such that ]λ G , λ G + ε] on the right of the golden mean that contains infinitely many Pisot and Salem numbers, but does not contain any dynamical degree.

In fact, gaps as in the third assertion of this corollary occur infinitely often, because there are infinitely many Pisot numbers that are limits of Pisot numbers from the right. 1.5. Organization of the paper. Section 2 provides a proof of Theorem A and its first corollary, the absence of dynamical degree between 1 and λ L 1.17628. Theorem B is proved in Section 3; this may be skipped on a first reading. Section 4 introduces the bubble space and an infinite dimensional hyperbolic space on which Bir(X) acts by isometries; as a first application, we obtain two new corollaries of Theorem A. Section 5 contains preliminary results on the infinite Weyl group W ∞ : This group is a Coxeter group on an infinite set of generators, and plays a crucial technical role in the study of the Cremona group Cr 2 (k). The proof of Theorem C is quite difficult even if, in spirit, it is a variation on Noether-Castelnuovo proof of the fact that PGL 3 (k) and the standard quadratic involution σ generate Bir(P We are also grateful to the referee for his careful reading and his suggestions.

SALEM NUMBERS AND AUTOMORPHISMS

This section is devoted to the proof of Theorem A. On our way, we introduce basic definitions that are used all along this article.

2.1. Indeterminacy points, homaloidal nets and base points. Let X be a projective surface defined over an algebraically closed field k. Let f be a birational transformation of X. We denote by Ind( f ) the set of indeterminacy points of f ; by convention, it is a proper subset of X and does not include infinitely near points.

The base points of f are defined as follows. Let D be a very ample divisor on X and |D| be the complete linear system containing D. The image of |D| by f is a linear system on X (which, in general, is not complete) ; when f is an element of the Cremona group and D is a line in P 2 k , this linear system f * |D| is the homaloidal net of f -1 (see [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]). The set of base points of f -1 (resp. the base ideal of f -1 ) is defined as the set (resp. the ideal) of base points of this linear system: Base points may be infinitely near, and come with a multiplicity.

The notion of base point does not depend on the choice of a very ample divisor, but the multiplicities of the base points depend on this choice.

This distinction between base points and indeterminacy points is just used to emphasis the arguments for which it is important to know whether the point is a proper point of X or not.

2.2. Algebraic stability and the intersection form. One says that f is algebraically stable if the sequence ( f n ) * of endomorphisms of NS(X) satisfies ( f n ) * = ( f * ) n for all integers n (cf. §1.1.3). As explained in [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], f is not algebraically stable if, and only if there is an indeterminacy point q of f -1 and a non-negative integer k such that f is well defined at q, f (q), ..., f k-1 (q), and f k (q) is an indeterminacy point of f . Blowing-up q, ..., f k (q), the number of such points decreases and, in a finite number of steps, one constructs a birational morphism π : X → X such that π -1 • f • π is algebraically stable (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] for this proof).

Lets us now assume that f is algebraically stable. The dynamical degree λ( f ) is then equal to the spectral radius of f * ∈ End(NS(X)) and also to the spectral radius of f * = f -1 * because these endomorphisms are adjoint for the intersection form:

f * C • D = C • f * D for all pairs (C, D) of divisor classes. Factorize f as f = ε • π -1
where π : Z → X and ε : Z → X are birational morphisms. Write π as a composition π 1 • . . . • π m of (inverse of) point blowups, and denote by F j ⊂ Z the total transform of the indeterminacy point of π -1 j under the map π j • . . . • π m . Then, denote by E j the direct image of F j by ε, for 1 ≤ j ≤ m. Each E j , if not zero, is an effective divisor. According to [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], Theorem 3.3, one has

f * f * C = C + m ∑ j=1 (C • E j )E j (2.1)
for all curves (resp. divisor class) C in X; this formula corresponds to the following fact: The preimage of C goes through the base points p j of f with multiplicity (C • E j ); thus, the total transform of f -1 C by f contains both C and ∑ j (C • E j )E j . Taking intersection, and using that f * and f * are adjoint endomorphisms of NS(X) for the intersection form, one gets

f * C • f * C = C •C + m ∑ j=1 (E j •C) 2 . (2.2)
In particular, f * increases self-intersections. This property and Hodge index theorem, according to which the intersection form has signature (1, ρ(X) -1), are responsible for λ( f ) being a Pisot or Salem number (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], Theorem 5.1).

2.3.

Eigenvectors and automorphisms. Since X has dimension 2, one easily shows that f * and f * preserve the pseudo-effective and nef cones of NS R (X).

Assume that the dynamical degree λ( f ) is larger than 1. Perron-Frobenius theorem assures the existence of an eigenvector

Θ + X ( f ) for f * in the nef cone of NS R (X) such that f * Θ + X ( f ) = λ( f )Θ + X (
f ); moreover, this vector is unique up to scalar factor (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]).

Theorem 2.1 (Diller-Favre). Let X be a projective surface, and f be a birational transformation of X, both defined over an algebraically closed field k. Assume that the dynamical degree λ( f ) is larger than 1.

Then (1) Θ + X ( f ) • Θ + X ( f ) = 0 if and only if Θ + X ( f ) • E j = 0 for all E j ; (2) If Θ + X ( f ) • Θ + X ( f ) = 0, there exists a birational morphism η : X → Y , such that η • f • η -1 is an automorphism of Y .
Sketch of the proof. Equation (2.2) and the eigenvector property

f * Θ + X ( f ) = λ( f )Θ + X ( f ) imply that (λ( f ) 2 -1)Θ + X ( f ) • Θ + X ( f ) = m ∑ j=1 (E j • Θ + X ( f )) 2 .
Hence, all divisors E j are orthogonal to Θ + X ( f ) if, and only if, Θ + X ( f ) is an isotropic vector.

Let us now prove the second assertion. By the first assertion, every E j is orthogonal to Θ + X ( f ); since the E j are effective and Θ + X ( f ) is nef, all irreducible components of the E j are orthogonal to Θ + X ( f ); in other words, the Q-vector subset of NS Q (X) generated by the irreducible components of the divisors E j is contained in the orthogonal complement Θ + X ( f ) ⊥ of the isotropic vector Θ + X ( f ) ⊥ . On Θ + X ( f ) ⊥ , the intersection form is negative and its kernel is the line generated by Θ + X ( f ). From Equation (2.1), one gets

f k * Θ + X ( f ) = λ( f ) -k Θ + X ( f ). Since λ( f ) > 1 and f * preserves the lattice NS Z (X) one deduces that Θ + X ( f ) is irrational: no scalar multiple of Θ + X ( f ) is contained in NS Z (X)
. Thus, the intersection form is negative definite on the Q-vector space generated by all classes of irreducible components of the divisors E j .

From Grauert-Mumford contraction theorem (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF], thm. 2.1 p. 91), there is a birational morphism η 0 : X → Y 0 which contracts simultaneously all these components. Let f 0 be the birational transformation η 0 • f • η -1 0 . Since Θ + X ( f ) does not intersect the curves which are contracted by η 0 , the class (η 0 ) * Θ + X ( f ) ∈ NS R (Y ) is both isotropic and an eigenvector for ( f 0 ) * with eigenvalue λ( f ). One can thus iterate this process until f -1 0 does not contract any curve, i.e. f 0 is an automorphism of Y 0 . If Y 0 is singular, and Y is a minimal desingularization of Y 0 , f 0 lifts to an automorphism f Y of Y ; one can then show that there

is an intermediate birational morphism η : X → Y such that η • f • η -1 = f Y .
This concludes the proof.

Example 2.2. Let E be the elliptic curve associated to the lattice of Gaussian (resp. Eisenstein) integers:

E = C/Z[i] (resp.C/Z[j])
where i 2 = -1 (resp. j 3 = 1, j = 1). Let A be the abelian surface

E × E. The group GL 2 (Z[i]) (resp. GL 2 (Z[j]
)) acts by automorphisms on A, and commutes to ν(x, y) = (ix, iy) (resp. ν(x, y) = (jx, jy)). As a consequence PGL 2 (Z[i]) (resp. PGL 2 (Z[j])) acts by automorphisms on the (singular) rational surface X 0 = A/ν, and on its minimal desingularisation X. The surface X being rational, this construction provides an embedding of

PGL 2 (Z[i]) (resp. PGL 2 (Z[j])) into the Cremona group. If M is an element of the linear group GL 2 (Z[i]) (resp. GL 2 (Z[j]
)), the associated birational transformation g M has dynamical degree

λ(g M ) = λ(M) 2
where λ(M) is the spectral radius of the matrix M. Its spectral radius is the Golden mean λ G . The square of λ G can be realized as the dynamical degree of the monomial map f C 2 associated to the second power C 2 of C, as described in Example 1.4. It is also realized as the dynamical degree of the transformation g C from Example 2.2. The birational transformation f C 2 is not conjugate to an automorphism of a rational surface Y , while g C is.

Remark 2.4. The previous two examples show that Theorem A does not extend to quadratic integers.

If f is a birational transformation of a projective surface X with λ( f ) = 1, then ( f n ) * is bounded, or it grows linearly with n, or it grows quadratically. In the first and third cases, f is conjugate to an automorphism of a projective surface Y by some birational transformation ϕ : Y X. In the second case, f is not conjugate to an automorphism (see Section 4.2.2). Thus, again, the "degree growth" determines whether f is, or not, conjugate to an automorphism.

Proof of Theorem A.

Let us now prove Theorem A. Assume λ( f ) is a Salem number. Let χ(t) ∈ Z[t] be the minimal polynomial of λ( f ). By assumption, there exists a root α of χ with modulus 1; one can thus fix an automorphism σ of the field of complex numbers such that σ(λ( f )) = α.

By Diller-Favre Theorem, we may assume that f is algebraically stable. The eigenvector Θ + X ( f ) corresponds to the eigenvalue λ( f ); as such, it may be taken in NS L (X), where L is the splitting field of χ. Our goal is to show that Θ + X ( f ) is orthogonal to all E j , 1 ≤ j ≤ m; the conclusion will follow from Theorem 2.1.

The automorphism σ of the field C acts on NS C (X), preserving NS(X) pointwise. Apply σ to both sides of

f * Θ + X ( f ) = λ( f )Θ + X ( f ); since f * is defined over Z, one obtains f * Ψ = αΨ, with Ψ = σ(Θ + X ( f )).
Since the divisor classes of the E j are in NS(X), all of them are σ-invariant. Thus, applying σ to Equation (2.1) we get

f * f * Ψ = Ψ + m ∑ j=1 (Ψ • E j )E j .
Taking intersection with the complex conjugate Ψ of Ψ, and using f * Ψ = αΨ, we get

(α ᾱ)Ψ • Ψ = f * Ψ • f * Ψ = Ψ • Ψ + m ∑ j=1 |E j • Ψ| 2 .
Since α has modulus 1, all intersections E j • Ψ vanish and, applying σ again, we deduce that Θ + X ( f ) • E j = 0 for all 1 ≤ j ≤ m. This concludes the proof.

2.5. Salem numbers in Λ(P 2 k ). Let f be an element of Cr 2 (k) such that λ( f ) is a Salem number. According to Theorem A, f is conjugate to an automorphism g of a smooth rational surface X; according to Kantor and Nagata [START_REF] Nagata | On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1[END_REF][START_REF] Nagata | On rational surfaces[END_REF], X is a blow-up of P 2 k with Picard number ρ(X) ≥ 11. Thus, the study of Λ S (P 2 k ) reduces to the following question: Which Salem numbers can be realized as spectral radii of linear transformations

g * ∈ End(NS R (X))
where X describes the set of blow-ups of P 2 k and g runs over the group Aut(X)? Recent results answer this question.

Write X as a blow-up of the plane at n points p 1 , p 2 , ..., p n ; some of them can be infinitely near points; we choose indices in such a way that j ≥ i if p j is infinitely near p i . Denote by π : X → P 2 k the birational morphism corresponding to this sequence of blow-ups. Let e i ∈ NS(X) denotes the Néron-Severi class of the total transform of p i under π (for 1 ≤ i ≤ n), and let e 0 ∈ NS(X) be the class of the total transform of a line in

P 2 k . Then NS(X) = Pic(X) = Ze 0 ⊕ Ze 1 ⊕ . . . ⊕ Ze n ,
and the basis (e 0 , e 1 , . . . , e n ) is orthogonal with respect to the intersection form on Pic(X). More precisely, we have e 0 • e 0 = 1, e i • e i = -1 if i ≥ 1, and e i • e j = 0 if i = j.

The canonical class of X is k X = -3e 0 + e 1 + e 2 + . . . + e n .

The automorphism group Aut(X) acts linearly on Pic(X), preserves k X , and preserves the intersection form. As a consequence, Aut(X) preserves the orthogonal complement k ⊥ X of k X in Pic(X). The elements

v 0 = e 0 -e 1 -e 2 v i = e i -e i+1 , 1 ≤ i ≤ n -1
form a basis of k ⊥ X , with respect to which the intersection form is given by the Dynkin diagram T 2,3,n-3 :

• • • • • • • . . . 1 2 3 4 n -2 n -1 0 FIGURE 1. Coxeter-Dynkin diagram of type T 2,3,n-3
In other words,

v k • v k = -2, for all indices k, v i • v j = 0,
if the vertices i and j are not linked by an edge, v i • v j = 1, if the vertices i and j are the endpoints of an edge.

The Weyl (or Coxeter) group W X of X is the group of orthogonal transformations of Pic(X) generated by the involutions

s i : u → u + (u • v i )v i , 0 ≤ i ≤ n -1.
This group preserves the orthogonal decomposition Pic(X) = Zk X ⊕ k ⊥ X and is isomorphic to the Coxeter group W n of the Dynkin diagram T 2,3,n-3 . It turns out that the definition of W X does not depend on the choice of the realization of X as a blow-up of the plane; as an abstract group, W X depends only on the Picard number of X.

Theorem 2.5 (Nagata, McMullen, Uehara). Let k be an algebraically closed field.

(1) Let X be a rational surface obtained from the projective plane P 2 k by a sequence of blow-ups. The image of Aut(X) in GL(NS(X)) is contained in the Weyl group W X .

(2) If char(k) = 0 and if Φ is an element of W n , there exists a rational surface Y with Picard number n + 1 and an element g of Aut(Y ) such that the dynamical degree λ(g) of g is equal to the spectral radius λ(Φ) of Φ.

(3) There are Salem numbers which are not contained in Λ(P 2 k ) (resp. in Λ(X) for any projective surface X).

When char(k) = 0, this theorem shows that the Salem part of Λ(P 2 k ) is described in purely algebraic terms: It coincides with the set of spectral radii λ(Φ) > 1, with Φ in some W n , n ≥ 10, and this set does not exhaust all Salem numbers.

Remark 2.6. Assertion (1) is due to Nagata (see [START_REF] Nagata | On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1[END_REF][START_REF] Nagata | On rational surfaces[END_REF]). Assertion (2) is due to Uehara, based on previous works by McMullen and Bedford and Kim (see [START_REF] Uehara | Rational surface automorphisms with positive entropy[END_REF]). When the characteristic p of the field is positive, Harbourne proves a similar result, but for Φ in a normal subgroup W n (p) of W n of finite index (the index goes to +∞ with n, see Example 3.4 in [START_REF] Harbourne | Automorphisms of K3-like rational surfaces[END_REF]). Assertion (3) makes use of Theorem A to extend a former result of McMullen. More precisely, McMullen proves that there are Salem numbers between λ L and λ P which are not realized by eigenvalues of elements in the Coxeter groups W n (see [START_REF] Curtis | Coxeter groups, Salem numbers and the Hilbert metric[END_REF]), and deduce from this that there are Salem numbers which are not realized by dynamical degrees of automorphisms of surfaces (see [START_REF] Curtis | Dynamics on blowups of the projective plane[END_REF]); Theorem A implies that Mc-Mullen's result holds for dynamical degrees of birational transformations.

2.6. Gaps in the dynamical spectrum. As announced in the introduction, we can now prove the following corollary to Theorem A.

Corollary 2.7. Let k be an algebraically closed field.

(1) If f is a birational transformation of a projective surface X defined over k and λ( f ) is in the interval ]1, λ P [, then f is conjugate to an automorphism of a projective surface Y by a birational mapping φ : X Y .

(2) There is no dynamical degree in the interval ]1, λ L [.

(3) If char(k) = 0, the minimum of the dynamical degree λ( f ) > 1 among all birational transformations of projective surfaces defined over k (resp. of P 2 k ) is equal to the Lehmer number λ L 1.176280.

Proof. Let f be a birational transformation of a projective surface X defined over an algebraically closed field k. Assume that the dynamical degree λ( f ) is a Salem number. From Theorem A, f is conjugate to an automorphism of a smooth projective surface. Thus, Assertion (1) follows from the fact that λ( f ) is a Salem number if 1 < λ( f ) < λ P 1.324717. From Theorem 1.2 in [START_REF] Curtis | Dynamics on blowups of the projective plane[END_REF], we deduce that λ( f ) ≥ λ L , where λ L is the Lehmer number. Since all Pisot numbers are larger than λ L , this proves assertion [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF].

If char(k) = 0, there is an automorphism g of a rational surface X such that λ(g) = λ L (see [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] and [START_REF] Curtis | Dynamics on blowups of the projective plane[END_REF]); McMullen recently announced that such an example also exists on a projective K3 surface (see [START_REF] Curtis | Dynamics with small entropy on projective k3 surfaces[END_REF]). In particular, the infimum of all dynamical degrees is a minimum, and is equal to the Lehmer number. This proves (3).

SURFACES WHICH ARE NOT RATIONAL

In this section we prove Theorem B and provide an example of a K3 surface with automorphisms f whose dynamical degrees λ( f ) are degree 22 Salem numbers.

Let X be a projective surface defined over an algebraically closed field k. In order to prove Theorem B, we consider the Kodaira dimension of X and refer to the classification of surfaces in Kodaira dimension 0 and -∞ (see [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]).

3.1. Ruled surfaces. If the Kodaira dimension of X is -∞ but X is not rational, then X is ruled in a unique, Bir(X)-invariant way. This implies that all elements of Bir(X) have dynamical degree 1 (see § 4.2.2 and Theorem 4.4 below).

3.2. Minimal models and automorphisms. If the Kodaira dimension of X is non negative, X admits a unique minimal model X . From now on, we replace X by X , so that we now have Bir(X) = Aut(X). In particular, all elements of Λ(X) \ {1} are Salem numbers, obtained from eigenvalues of linear transformations of NS(X) (preserving the intersection form).

Positive Kodaira dimension.

If the Kodaira dimension is equal to 2, the automorphism group is finite, and Λ(X) reduces to {1}. If the Kodaira dimension of X is equal to 1, the Kodaira-Iitaka fibration provides an Aut(X)equivariant fibration X → B from X to a curve B. The divisor class of the generic fiber of this fibration is an isotropic vector in NS(X). This vector is Aut(X)-invariant and, consequently, all elements f in Bir(X) are elliptic or parabolic. This implies that λ( f ) = 1 for all f in Bir(X).

Vanishing Kodaira dimension.

Let us now assume that (X is minimal and) the Kodaira dimension of X is equal to 0. According to the classification of surfaces, X is either (i) an abelian surface;

(ii) a hyperelliptic surface, obtained as a quotient of an abelian surface by a fixed point free group of automorphisms; (iii) a K3 surface; (iv) or an Enriques surface. Hyperelliptic surfaces don't have automorphisms with λ( f ) > 1, as shown in [START_REF] Cantat | Dynamique des automorphismes des surfaces projectives complexes[END_REF]. In cases (i), (iii), and (iv), X has Picard number bounded from above by 4, 22 and 10 respectively. This shows that λ( f ) is a Salem number of degree at most 22. Moreover, the Picard number is at most 20 if the characteristic of k vanish, so that λ( f ) is an algebraic integer of degree at most 20 in this case. Proposition 3.1. In characteristic 2, there are examples of pairs (X, f ) where X is a K3 surface, f : X → X is an automorphism, and λ( f ) is a Salem number of degree 22. To construct such an example, we make use of one of the main results of [START_REF] Dolgachev | A supersingular K3 surface in characteristic 2 and the Leech lattice[END_REF]. Let k be an algebraically closed field of characteristic 2. There exists a K3 surface X, defined over k, such that (i) the Picard number of X is equal to 22; (ii) the automorphism group of X is infinite, and does not preserve any proper subspace of NS R (X).

Let O R (NS(X)) be the Lie group of orthogonal endomorphisms of the Néron-Severi space with respect to the intersection form. This group is an algebraic group, and we denote by O 0 R (NS(X)) its irreducible component that contains the identity. From the second property, we deduce that the image Aut(X) of

Aut(X) in GL R (NS(X)) intersects O 0 R (NS R (X)) on a Zariski dense subgroup; indeed, if G ⊂ O 0 R (NS(X)
) is not Zariski dense, then G preserves a non-trivial, strict subspace of NS R (X) (see [START_REF] Benoist | Adhérence de Zariski des groupes de Coxeter[END_REF] for instance).

As Aut(X) is Zariski dense, we can now prove that the characteristic polynomial of a "general" element of Aut(X) is irreducible (over Z), its degree is equal to 22, and its larger root is a Salem number. The proof relies on the following remark: If g * is an element of Aut(X) , then g * preserves the integral structure of NS(X), and preserves the intersection form, the signature of which is equal to (1, 21); hence,

• if g * has no eigenvalue of modulus > 1, the roots of χ g * are algebraic integers of modulus at most 1 and, by Kronecker Lemma, are roots of 1; thus χ g * splits as a product of cyclotomic polynomials; • if g * has an eigenvalue of modulus > 1, it is unique and is either quadratic or a Salem number; hence, if χ g * (t) splits as a product of two non-constant polynomials q(t) and r(t) in Z[t], then χ g * is divisible by a cyclotomic polynomial.

Thus, either there are elements g with the required properties, or χ g * is divisible by a cyclotomic polynomial of degree at most 22 for every g in Aut(X).

Since their degree is bounded by 22, there are only finitely many cyclotomic polynomials to consider. Let V 22 ⊂ R[t] be the set of all monic polynomials of degree 22. Given a cyclotomic polynomial r(t), the subset

V 22 (r) = {χ(t) | r(t) divides χ(t)}
is a proper algebraic subset of positive codimension; moreover, the image of O 0 R (NS(X)) in V 22 by the characteristic polynomial mapping is not contained in this set, because there are elements of O 0 R (NS(X)) without any eigenvalue being a root of unity (here we use that 22 is even). Since Aut(X) is Zariski dense in O 0 R (NS(X)), we conclude that there are elements f * of Aut(X) such that χ f * (t) is not contained in any V 22 (r); the characteristic polynomial of such an element is irreducible (over Z), its degree is equal to 22, and its larger root is a Salem number.

Remark 3.2. This argument has now been extended to other examples of K3

surfaces in positive characteristic by Esnault, Oguiso, and Yu (see [START_REF] Esnault | Automorphisms of elliptic K3 surfaces and salem numbers of maximal degree[END_REF]).

3.3.

Discrete spectrum. To conclude the proof of Theorem B, we need to show that the union of all dynamical spectra Λ(X) where X runs over the set of non-rational projective surfaces defined over k, and k runs over the set of all fields, is a discrete subset of the real line. This follows from the upper bounds 22 for the degrees of Salem numbers in Λ(X) and the following lemma. Lemma 3.3. Let B be a positive number and Sal B be the set of Salem numbers of degree at most B. Then Sal B is a closed discrete subset of the real line.

Proof. Let λ be such a Salem number, contained in the interval [a -1 , a], with a > 1. Its minimum polynomial χ(t) ∈ Z[t] has integer coefficients, and all of them are symmetric polynomials in λ, 1 λ and its conjugates of modulus 1. Since all these numbers have modulus at most a, all coefficients of χ are bounded by C B a B , where C B depends only on B. Since the coefficients of χ are integers, there is a finite list of possible coefficients, a finite list of possible minimum polynomials χ, and therefore a finite list of Salem numbers λ ∈

[a -1 , a] of degree ≤ B.
Thus, the intersection of Sal B with any compact interval [a -1 , a] ⊂ R * + is finite, and Sal B is discrete.

BLOW-UPS, BUBBLES, ISOMETRIES

When X is a projective surface, the group Bir(X) acts faithfully by isometries on a hyperbolic space H X , the dimension of which is infinite when X is ruled or rational. This construction is described in [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF] and [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF] ; in this section, we summarize the main facts and apply them to control centralizers and conjugacy classes in Bir(X). The reader may consult [START_REF] Burger | Equivariant embeddings of trees into hyperbolic spaces[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], and [START_REF] Favre | Le groupe de Cremona et ses sous-groupes de type fini[END_REF] for the results which are summarized in the paragraphs 4.1 and 4.2.

4.1.

Bubbles and Picard-Manin space. Let X be a projective surface, defined over an algebraically closed field k. If π : Y → X is a birational morphism, one obtains an embedding of Néron-Severi groups π

* : NS(X) → NS(Y ). Given two birational morphisms π 1 : Y 1 → X and π 2 : Y 2 → X, one says that π 2 is above π 1 (or covers π 1 ) if π -1 1 • π 2 is regular.
Starting with two birational morphisms π 1 : Y 1 → X and π 2 : Y 2 → X, one can always find a third birational morphism π 3 : Y 3 → X which covers π 1 and π 2 . It follows easily that the inductive limit of all groups NS(Y i ), for all surfaces Y i above X, is well defined. This limit is the Picard-Manin space Z X of X; the intersection form determines a scalar product on Z X , which we denote by (v, w) → v • w.

The bubble space B(X) of X is defined as follows. Consider all surfaces Y above X, i.e. all birational morphisms π : Y → X. Given p 1 on Y 1 and p 2 on Embed NS(X) as a subgroup of the Picard-Manin space. This finite dimensional lattice is orthogonal to e(p) for all p in B(X), and the Picard-Manin space coincides with the direct sum

Y 2 , identify p 1 with p 2 if π -1 1 • π 2 is a local isomorphism in a neighborhood
Z X = NS(X) ⊕ p Ze(p)
where p runs over the bubble space. Thus, each element v of the Picard-Manin space can be written as a finite sum

v = v X + ∑ p a(p)e(p).
The canonical form on Z X is a linear form Ω : Z X → Z, which is defined by

Ω(v) = k X • v X -∑ p a(p),
where k X is the canonical divisor of X.

There is a completion process, for which the completion Z X of Z X ⊗ Z R is represented by square integrable sums:

Z X = {w + ∑ p a(p)e(p) | w ∈ NS R (X), and ∑ p a(p) 2 < ∞}
The intersection form extends as a scalar product with signature (1, ∞) on this space, but the canonical form Ω doesn't.

The hyperbolic space H X of X is then defined by

H X = {w ∈ Z X | w • w = 1, and w • a > 0 for all ample classes a ∈ NS(X)}.
This space H X is an infinite dimensional analogue of the classical hyperbolic spaces

H n : The distance dist on H X is defined by cosh(dist(v, w)) = v • w for all pairs of elements of H X ; it is complete. If H X is cut with a subspace
of Z X of dimension n, and the intersection is not empty, the result is a totally geodesic hyperbolic space of dimension n -1. In particular, geodesics are intersections of H X with planes. The projection of H X in the projective space P(Z X ) is one to one, and the boundary of its image is the projection of the cone of isotropic vectors of Z X . Thus, we denote by ∂H X the set 1.-The hyperbolic space H X is log(3)-hyperbolic, in the sense of Gromov (see [START_REF] Coornaert | Géométrie et théorie des groupes[END_REF]).

∂H X = {R + v ∈ Z X | v • v = 0,
2.

-Since H X is Gromov hyperbolic, one can approximate configurations of points in H X by configuration of points in metric trees (see [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF] for instance).

3.-The set ∂H X coincides with the Gromov boundary of the hyperbolic space H X (note that H X is not locally compact).

4.2.

Isometries and dynamical degrees. The important fact is that Bir(X)

acts faithfully on Z X by continuous linear endomorphisms, preserves the intersection form, the effective cone, the nef cone; it also preserves the subset Z X and canonical form k : Z X → Z (this linear form does not extend to Z X ). In particular, it preserves the hyperbolic space H X .

Remark 4.2. Intuitively, elements of Bir(X) behave like automorphisms on Z X , because all points have been blown-up to define Z X , so that all indeterminacy points have been resolved. When the Kodaira dimension of X is nonnegative and X is minimal, then Bir(X) coincides with Aut(X). The space H X can be replaced by the subset of NS(X, R) of elements v with v • v = 1 and v • a > 0 for all ample classes a; the action of Bir(X) = Aut(X) is not always faithful but the kernel coincides with the connected component Aut 0 (X) up to finite index (see [START_REF] David | Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]).

Let f be an element of Bir(X). Denote by f • its action on Z X :

f • : Z X → Z X
is a linear isometry with respect to the intersection form. We also denote by f • the isometry of H X that is induced by this endomorphism of Z X .

4.2.1.

Translation length and types of isometries. The translation length of an isometry g of H X is defined, as for all hyperbolic spaces, by

L(g) = inf{dist(v, g(v)) | v ∈ H X }.
If this infimum is a minimum, either it is equal to 0 and g has a fixed point in H X , in which case g is elliptic, or it is positive and g is loxodromic (also called hyperbolic). If g is loxodromic, the set of points x ∈ H X such that dist(x, g(x)) is equal to the translation length of g is a geodesic line Ax(g) ⊂ H X ; its boundary points are represented by isotropic vectors a(g) and b(g) in Z X such that g(a(g)) = e L(g) a(g) and g(b(g)) = e -L(g) b(g).

The axis of g is the intersection of H X with the plane containing a(g) and b(g).

Normalize the choice of a(g) and b(g) in such a way that a(g) • b(g) = 1. Let x be a point of H X , or a point of the isotropic cone of Z X that intersects all ample classes positively; then, the orbit g n (x) converges to the boundary point R + a(g)

when n goes to +∞, and to R + b(g) when n goes to -∞. More precisely, in Z X

we have 1 e nL(g) g n (x) → (x • b(g)) a(g), and

1 e nL(g) g -n (x) → (x • a(g)) b(g) as n goes to +∞.
When the infimum is not realized, L(g) is equal to 0, and g is parabolic: g fixes a unique line in the isotropic cone of Z X ; this line is fixed pointwise, and all orbits g n (x) in H X accumulate to the corresponding boundary point when n goes to ±∞ (see [START_REF] Burger | Equivariant embeddings of trees into hyperbolic spaces[END_REF] for examples of accumulation without convergence).

4.2.2.

Types of birational transformations. This classification of isometries into three types hold for all isometries of H X . For isometries f • induced by birational transformations of X, there is a dictionary between this classification and the geometric properties of f . To state it, let us introduce the following definitions: A birational transformation f of a projective surface X is (i) virtually isotopic to the identity if there is a positive iterate f n of f and a birational mapping

φ : Z X such that φ -1 • f n •φ is an element of Aut(Z) 0 ;
(ii) a Halphen twist if f preserves a one parameter family of genus one curves on X but f is not virtually isotopic to the identity; (iii) a Jonquières twist if f preserves a one parameter family of rational curves on X but f is not virtually isotopic to the identity. When f is a Halphen or a Jonquières twist then, after conjugacy by a birational mapping φ : Z X, f permutes the fibers of a genus one or rational fibration π : Z → B. Let z be the divisor class of the generic fiber of this fibration. Then z is an isotropic vector in Z X that is fixed by f • ; in particular, f • can not be loxodromic. Remark 4.3. Let f : X X be a Halphen twist, and let φ : Z → X be a modification of X on which the f -invariant family of genus one curves form a fibration π : Z → B. Let Z be a relative minimal model of this genus one fibration (Z is obtained from Z by blowing down exceptional divisors of the first kind that are contained in fibers of π, and iteration of this process). Then, f becomes an automorphism of Z . On the other hand, Jonquières twists are not conjugate to automorphisms of projective surfaces (see [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]). Theorem 4.4 (Gizatullin, Cantat, Diller-Favre, see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). Let k be an algebraically closed field. Let X be a projective surface defined over k. Let f be birational transformation of X, let f • be the isometry of H X determined by f , and let x be a point of H X .

(1) f • is elliptic if and only if f is virtually isotopic to the identity.

(

) If f • is parabolic, either x • f n • (x) 2
grows linearly with n, and f is a Jonquières twist, or x • f n • (x) grows quadratically with n, and f is a Halphen twist.

(3) f • is loxodromic if and only if the dynamical degree λ( f ) is > 1. In all cases, the translation length L( f • ) is equal to the logarithm of λ( f ).

f • (e 0 ) = deg( f )e 0 -∑ a(p)e(p)
where deg( f ) is the degree of f and a(p) is the multiplicity of the homaloidal net f * |O(1)| at the point p (p may be "infinitely near"). Since e 0 does not intersect any of the e(p), one gets cosh(dist(e 0 , f

• (e 0 ))) = e 0 • f • (e 0 ) = deg( f ).
This establishes the link between deg( f n ) and dist(e 0 , f n p (e 0 )) which leads to the equality L( f ) = log(λ( f )) (see § 6.1 for details and complements).

Example 4.6. An element f of the Cremona group is virtually isotopic to the identity if and only if f has finite order or f is conjugate to an element of Aut(P 2 k ) = PGL 3 (k) (see [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]).

Centralizers.

Corollary 4.7. Let f be a birational transformation of a projective surface X. If f is loxodromic, the infinite cyclic group generated by f is a finite index subgroup of the centralizer of f in Bir(X).

Proof. Let f be a loxodromic birational transformation of the surface X. Then f acts on the hyperbolic space H X as a hyperbolic isometry, with an invariant axis Ax( f ). The endpoints of Ax( f ) correspond to two eigenvectors b( f ) and

a( f ) in the isotropic cone of Z X , with f • (b( f )) = 1 λ( f ) b( f ), and f • (a( f )) = λ( f )a( f ).
These vectors are unique up to scalar multiplication. Let Cent( f ) denote the centralizer of f in the group Bir(X). It preserves the eigenlines Rb( f ) and Ra( f ), acting on each of them by scalar multiplication. This provides a morphism θ :

Cent( f ) → R * + such that g • (a( f )) = θ(g)a( f )
for all g in Cent( f ). Moreover, θ(g) or its inverse coincides with the dynamical degree of g because, if g is loxodromic, then g • has exactly two fixed points on the boundary of H X .

The image of θ is a subgroup of R * + which is contained in Λ(X) ∪ {1} ∪ Λ(X) -1 . From the spectral gap property, this image does not intersect the interval ]1, λ L [, and is consequently a discrete subgroup of R * + . Since all infinite discrete subgroups or R * + are cyclic, the image θ(Cent( f )) is cyclic.

Let Cent( f ) 0 be the kernel of θ. All we need to prove is that Cent( f ) 0 is finite because, then, the exact sequence

1 → Cent( f ) 0 → Cent( f ) θ → Z → 0 proves that Cent( f ) is finite by cyclic.
The group Cent( f ) 0 preserves Ax( f ) and fixes a( f ). It must therefore fix Ax( f ) pointwise. Let q be a point of Ax( f ) and let ∆ be its distance to e 0 in H X (where, as above, e 0 is the class of a multiple of an ample divisor D on X with e 2 0 = 1). Let h be an element of Cent( f ) 0 . Then dist(e 0 , h • (e 0 )) ≤ 2dist(e 0 , q) = 2∆; hence

h • (e 0 ) • e 0 ≤ cosh(2∆).
This shows that the degree of h ∈ Cent( f ) 0 ⊂ Bir(X) with respect to the polarization D is uniformly bounded by some explicit constant M = cosh(2∆).

Assume that the Kodaira dimension Kod(X) is non-negative. Changing X into its unique minimal model, we assume that X is minimal; this implies Bir(X) = Aut(X) because Kod(X) ≥ 0. Thus, Aut(X) contains a loxodromic element (determined by f ) and X is either an abelian surface, a K3 surface, or an Enriques surface. The group Cent( f ) 0 is, now, a group of automorphisms of X with bounded degree with respect to a fixed polarization D on X. This implies that the intersection of Aut(X) 0 with Cent( f ) 0 is a finite index subgroup of Cent( f ) 0 . Thus, either Cent( f ) 0 is finite, or it contains a connected algebraic subgroup G ⊂ Aut(X) 0 of positive dimension. In the latter case, X is an abelian variety and G acts by translations on X, because Aut(X) is discrete for K3 and Enriques surfaces. Let G 1 be a closed, one dimensional subgroup of G: Its orbits form a fibration of X by elliptic curves. Since f commutes to G 1 , it preserves this fibration of X. This contradicts the fact that f is loxodromic, and proves that Cent( f ) 0 is finite.

Assume that Kod(X) is negative. Since Bir(X) contains a loxodromic element f , the surface X is rational, and we can suppose that X is the projective plane and e 0 is the class of a line in P 2 k . The group Cent( f ) 0 is a subgroup of Bir(P 2 k ) of bounded degree. From Corollaries 2.8 and 2.18 of [START_REF] Blanc | Topologies and structures on the Cremona group[END_REF], we deduce that its Zariski closure in Bir(P 2 k ) is an algebraic subgroup of Bir(P 2 k ). Denote by G the connected component of the identity in this group. If Cent( f ) 0 is infinite, the dimension of G is positive, and a result of Enriques shows that G is contained, after conjugation, in the group of automorphisms of a minimal, rational surface (see [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF][START_REF] Enriques | Sui gruppi continui di trasformazioni cremoniane nel piano[END_REF]). As a consequence, G contains a Zariski closed abelian subgroup A whose orbits have dimension 1 in X. Those orbits are organized in a pencil of curves that is invariant under the action of f . This contradicts λ( f ) > 1 and shows that Cent( f ) 0 is finite.

4.4.

Conjugacy between loxodromic transformations. Assertion (1) of Corollary 2.7 can be rephrased as follows: For all loxodromic elements f in Bir(P 2 k ) and all points x in

H P 2 k , dist(x, f • (x)) ≥ log(λ L )
where λ L is the Lehmer number.

Lemma 4.8. For all loxodromic elements f in Bir(P 2 k ) and all points x in Proof. Let y be the projection of the point x on the axis of f • . Let n be the least positive integer which satisfies dist(y, f n • (y)) ≥ 8 log(3). Consider the geodesic quadrilateral with vertices x, y, f n • (y), and

H P 2 k , dist(x, Ax( f • )) ≤ 28 • dist(x, f • (x)).
f n • (x). By hyperbolicity, the geodesic segment [x, f n • (x)] is contained in the (2 log(3)
)neighborhood of the other three, and its length is at least 8 log(3); hence, its middle point m is at most

(2 log(3))-away from [y, f n • (y)]. Let m be the projec- tion of m on the segment [y, f n • (y)].
Then the distance from x to m is equal to the sum of the distances from x to y and from y to m , up to an error of 2 log(3). The same estimate in the triangle (m , f n

• (y), f n • (x)) provides the inequality : dist(x, f n • (x)) ≥ dist(x, y) + dist(y, f n • (y)) + dist( f n • (y), f n • (x)) -8 log(3)
. Since f • is an isometry and y is the projection of x on its axis, the choice for n

implies n • dist(x, f • (x)) ≥ 2dist(x, Ax( f • )
). On the other hand, Corollary 2.7 shows that n can be chosen to be the smallest integer above 8 log(3)/ log(λ L ) 54.13, that is n = 55. Theorem 4.10. Let f and g be two loxodromic elements of Bir(P 2 k ). If f is conjugate to g, one can find an element h of Bir(P 2 k ) such that f = hgh -1 and deg(h) ≤ 2 57 (deg( f )deg(g)) 29 Proof. Let x be the projection of e 0 on the axis of f , and y be its projection on the axis of g. Let h 0 be an element of Bir(P 2 k ) that conjugates f to g; it maps y onto a point z

0 := (h 0 ) • (y) of Ax( f • ). Let k be an integer such that dist( f k p (z 0 ), x) ≤ log(λ( f ))
. Such a k exists because f • acts by translation of length log(λ( f )) on its axis. Changing h 0 into h = f k • h 0 , we obtain a new conjugacy from g to f that maps y onto a point z = h(y) at distance at most log(λ( f )) from x. Now,

dist(e 0 , h • (e 0 )) ≤ dist(e 0 , x) + dist(x, h • (y)) + dist(h • (y), h • (e 0 )). Since h • is an isometry we get dist(e 0 , h • (e 0 )) ≤ dist(e 0 , Ax( f • )) + log(λ( f )) + dist(e 0 , Ax(g • )).
The previous lemma can then be applied to e 0 , and gives

dist(e 0 , h • (e 0 )) ≤ log(λ( f )) + 28 • (dist(e 0 , f • (e 0 )) + dist(e 0 , g • (e 0 ))).
The result follows from log(λ( f • )) ≤ dist(e 0 , f • (e 0 )), cosh(dist(e 0 , f • (e 0 ))) = deg( f ) and easy estimates for the reciprocal function of cosh(•).

Let us add the following complement to Theorem 4.10, which shows that the hypothesis on f and g (being loxodromic) in the theorem can be checked in a finite number of steps. Proof. Let us write g = f 200 , which is loxodromic if and only if f is.

One direction has been proved by Junyi Xie in [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF]:

If deg(g 2 ) ≥ 3 19 deg(g), then dist(e 0 , g 2
• (e 0 )) > dist(e 0 , g • (e 0 )) + 18 log(3), and this implies that g • is a loxodromic isometry because H X is log(3)-hyperbolic.

In the other direction, if f • is loxodromic, the translation length of f • is at least log(λ L ). Hence, the translation length of g is larger than 200 log(λ L ). Then, as in the proof of Lemma 4.8, dist(e 0 , g 2

• (e 0 )) ≥ dist(e 0 , g • (e 0 )) + L -8 log(3), and this implies that deg(g

2 ) ≥ 3 19 deg(g) because λ 200 L > 3 8+19 .
Example 4.12. Let m be a positive integer and a be a non-zero element of the field k. Let (x, y) be affine coordinates of the plane. Consider the transformation f a : (x, y) → (ax, y). This automorphism is conjugate to g a,m : (x, y) → (ax, a m y) by the monomial transformation h(x, y) = (x, x m • y). The degree of f a and of g a,m is equal to 1, but the degree of h is m + 1. If a Z is a Zariski dense subgroup of k * , one easily shows that there is no conjugacy h of degree < m + 1. Thus, the degree of the conjugacy is not bounded by the degree of f a and g a,m if one does not assume λ( f ) > 1.

THE WEYL GROUP W ∞

We now define, and study, a group of linear transformations of Z P 2 k , with integer coefficients, preserving the intersection form. This group of isometries W ∞ is a subgroup of Isom(Z P 2 k ) that contains the image of Bir(P 2 k ). Let p and q be two elements of B(P 2 k ). The element e(p)e(q) of Z P 2 k has self-intersection -2; as a consequence, the linear transformation τ p,q : x → x + (x • (e(p)e(q)))(e(p)e(q)) is the orthogonal reflection that maps e(p)e(q) to its opposite. The group generated by all these reflections is the subgroup of elements of Sym(B(P 2 k )) with finite support. Similarly, σ 0 corresponds to the orthogonal reflection associated to e 0e(p 1 )e(p 2 )e(p 3 ). This explains why W ∞ is called infinite Weyl group (or Coxeter group).

By construction, W ∞ preserves the intersection form, the canonical form Ω, and extends as a group of isometries of H P 2 each e(q), q ∈ B(P 2 k ) \ {e(p 1 ), e(p 2 ), e(p 3 )}, to some e(q ) with q ∈ B(P 2 k ) \ {e(p 1 ), e(p 2 ), e(p 3 )}. This implies that f • is the composition of σ 0 with an element of Sym B(P 2 k ) , so that f • is in W ∞ . The result follows from Noether-Castelnuovo theorem, which asserts that Bir(P 2 k ) is generated by Aut(P 2 k ) and the standard quadratic transformation, when k is algebraically closed (see [START_REF] Kollár | Rational and nearly rational varieties[END_REF] or [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]).

Remark 5.2.

1.-The group W ∞ is strictly larger than Cr 2 (k) because the elements of Sym(B(P 2 k )) fix e 0 but most of them are not induced by projective linear transformations of the plane.

2.-Lemma 5.1 is implicitly contained in Noether's original "proof" of Noether-Castelnuovo theorem. , it is either elliptic, parabolic, or loxodromic; moreover, the dynamical degree

λ(h) = lim k→+∞ (deg(h k ) 1/k )
is well defined, and its logarithm is the translation length L(h).

It is not a priori clear that the multiplicities a(p) are non-negative. For example, Λ = 3e 0 + e(p 1 ) -∑ 7 i=2 e(p i ) satisfies Λ 2 = 1 and intersects the canonical form as e 0 does. To show that such an element cannot be sent onto e 0 by an element of W ∞ , we need the following lemma.

Lemma 5.3. Let v ∈ Z P 2 k be one of the following vectors: e 0 , e(q 1 ), e 0e(q 1 ), 3e 0 -l ∑ i=1 e(q i ) for some distinct points q 1 , . . . , q l ∈ B(P 2 k ). For any h ∈ W ∞ , the following holds:

(1) There exists n ≥ 1 and s 1 , . . . ,

s n ∈ Sym(B(P 2 k )) satisfying h(v) = s n σ 0 s n-1 σ 0 . . . s 2 σ 0 s 1 (v) (s i σ 0 . . . σ 0 s 1 )(v) • e 0 > (s i-1 σ 0 . . . σ 0 s 1 )(v) • e 0
for all i = 2, . . . , n.

(2) Either h(v) = e(q) for some q ∈ B(P 2 k ), or there exists k ≥ 0, nonnegative integers d, a 1 , . . . , a k , and a finite set of points r 1 , . . . , r k ∈

B(P 2 k ), such that h(v) = de 0 - k ∑ i=1 a i e(r i ).
Remark 5.4. One can list elements of degree 1 or 2 in W ∞ . This will be useful for the proof of Lemma 5.3.

• Let h be an element of degree 1 in W ∞ . This means that h(e 0 ) = e 0 -∑ a(p)e(p) where a(p) ∈ Z vanishes for all but a finite number of points p ∈ B(P 2 k ). Since the self-intersection is preserved under the action of h, the sum ∑ p a(p) 2 is equal to 0; this implies that h(e 0 ) = e 0 . Let p be an element of

B(P 2 k ). From h(e(p)) • e 0 = h(e(p)) • h(e 0 ) = 0
we deduce that h(e(p)) = ∑ b(q)e(q) with b(q) ∈ Z, and then that h(e(p)) = ±e(q) for some point q in B(P 2 k ), because the self-intersection of h(e p ) is -1. Since the canonical linear form is preserved, one concludes that h(e p ) = e q . In other words, h is an element of Sym(B(P 2 k )).

• Say that h ∈ W ∞ is quadratic if its degree is equal to 2. Write h(e 0 ) = 2e 0 - ∑ a(p)e(p) with a(p) ∈ Z. The invariance of the self-intersection provides 4 -∑ p a(p) 2 = 1 ;
hence, there are exactly three base points, each with multiplicity 1. Composing h with an element of Sym(B(P 2 k )), one may assume that these three base-points coincide with the base-points p 1 , p 2 , and p 3 of σ 0 . Then σ 0 h has degree 1. We conclude that quadratic elements are composition sσ 0 s with s and s in Sym(B(P 2 k )). Proof. The proof of this lemma parallels classical facts from Coxeter group theory.

We first prove that (1) implies (2). The proof proceeds by induction on the minimum number n ≥ 1 for which h satisfies assertion [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]. If n = 1 then h(v) = s 1 (v) for some element s 1 ∈ Sym(B(P 2 k )) , and assertion (2) follows. Let us now assume that n ≥ 2 and apply the induction hypothesis to s n-1 σ 0 . . . s 2 σ 0 s 1 . Let w = s n-1 σ 0 . . . s 2 σ 0 s 1 (v). If w is equal to e q for some q ∈ B(P 2 k ), we apply s n σ 0 : h(v) = s n σ 0 (w) is either e(q ) or e 0e(q )e(q ) for some q , q ∈ B(P 2 k ), and the first case is in fact impossible by [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]. Otherwise, we write w = de 0 -∑ k i=1 c i e(r i ) for some points r 1 , . . . , r k ∈ B(P 2 k ) and some non-negative integers d, c i . Ordering the points and adding, if necessary, points with trivial coefficients c i = 0, we assume that r 1 , r 2 , and r 3 are the three base-points p 1 , p 2 and p 3 of σ 0 . By definition of σ 0 , we find

σ 0 (w) = (d + d )e 0 - 3 ∑ i=1 (c i + d )e(r i ) - m ∑ i=4 c i e(r i ), where d = d -c 1 -c 2 -c 3 . Since d + d = s n σ 0 . . . s 2 σ 0 s 1 (v) • e 0 > d,
we obtain d > 0 and see that c i + d is non-negative for i = 1, 2, 3. This proves that (1) ⇒ (2) by induction on n.

We now prove assertion [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]. By definition of W ∞ , we can always write h as a composition s m σ 0 s m-1 σ 0 . . . s 2 σ 0 s 1 for some s 1 , . . . , s m ∈ Sym(B(P 2 k )). For i = 1, . . . , m, write

w i = s i σ 0 . . . σ 0 s 1 (v) and d i = w i • e 0 .
Our aim is to replace the sequence (s i ) m i=1 , keeping s m σ 0 s m-1 σ 0 . . . s 2 σ 0 s 1 (v) = h(v), in order to assure that the sequence (d i ) m i=1 increases strictly. Let s and s be elements of Sym(B(P 2 k )). One easily checks, for all possibilities of v, that

s σ 0 s(v) • e 0 > v • e 0 or s σ 0 s(v) = s (v)
for some s ∈ Sym(B(P 2 k )). We can then change the sequence (s i ) m i=1 to assure that m = 1, in which case the result is obvious, or d 2 > d 1 .

We set

S = {i ∈ {2, . . . , m -1} | d i-1 < d i ≥ d i+1 } and write D = max{d i |i ∈ S} with the convention D = 0 if S = / 0.
We then denote by l the number of elements i ∈ {2, . . . , m -1} such that d i = D and prove assertion (1) by induction on the pairs (D, l), ordered lexicographically.

If D = 0 or l = 0, then S = / 0 and we have

d 1 < d 2 < • • • < d m ,
which achieves the proof. We can thus assume that there is some k ∈ S such that

d k = D > 0. Let us recall that d k-1 < d k ≥ d k+1 .
Since k < m, the induction hypothesis and the proof of (1) ⇒ (2) yield the existence of points p 1 , . . . , p r such that

w k = s k σ 0 . . . s 2 σ 0 s 1 (v) = d k e 0 - r ∑ i=1 a i e(p i )
for some non-negative integers a 1 , . . . , a r . We again assume that p 1 , p 2 , p 3 are the three base-points of σ 0 . Since w k+1 = s k+1 σ 0 (w k ) and w k-1 = σ 0 (s k ) -1 (w k ), we find

d k+1 = e 0 • σ 0 (w k ) = σ 0 (e 0 ) • w k = 2d k - 3 ∑ i=1 e(p i ) • w k = 2d k - 3 ∑ i=1 a i , and
d k-1 = e 0 • σ 0 (s k ) -1 (w k ) = s k σ 0 (e 0 ) • w k = 2d k - 3 ∑ i=1 e(q i ) • w k , where q i = s k (p i ) for i = 1, 2, 3. This implies 3 ∑ i=1 e(q i ) • w k > d k and 3 ∑ i=1 e(p i ) • w k ≥ d k .
Let t + 2 be the number of points of the set {p 1 , p 2 , p 3 , q 1 , q 2 , q 3 }. We can define t sets of 3 points P 1 , . . . , P t ⊂ B(P 2 k ), such that P 1 = {q 1 , q 2 , q 3 }, P t = {p 1 , p 2 , p 3 }, and

• P i ∩ P i-1 contains 2 points and • ∑ x∈P i w k • e(x) > d k for i = 1, . . . ,t -1.
For each i = 1, . . . ,t, we fix an element s i ∈ Sym(B(P 2 k )) that sends {p 1 , p 2 , p 3 } onto P i ; for i = 1, we choose s 1 to be s k and for i = t we choose s t = Id. We then write w i = σ 0 (s i ) -1 (w k ) and g i = σ 0 (s i+1 ) -1 s i σ 0 . To illustrate this, we make a picture in the case where t = 4:

w k σ 0 (s 4 ) -1 B B σ 0 (s 1 ) -1 s s σ 0 (s 3 ) -1 ( ( σ 0 (s 2 ) -1 Ò Ò w k-1 = w 1 g 1 @ @ w 4 w 2 g 2 G G w 3 g 3 W W For i = 1, . . . ,t -1, the inequality ∑ x∈P i w k • e(x) > d k yields e 0 • w i < d k .
Moreover, the fact that P i ∩ P i-1 contains two points implies that g i is an element of W ∞ of degree 2; as such, it is equal to a i σ 0 b i for some a i , b i ∈ Sym B(P 2 k ) (see Remark 5.4).Because s t preserves P t = {p 1 , p 2 , p 3 }, s k+1 σ 0 s t σ 0 is equal to some a t ∈ Sym B(P 2 k ) . We can then write s k+1 σ 0 s k σ 0 ∈ W ∞ , that sends w k-1 onto w k+1 , as s k+1 σ 0 s k σ 0 = (s k+1 σ 0 s t σ 0 )σ 0 (s t ) -1 s 1 σ 0 = a t (σ 0 (s t ) -1 s t-1 σ 0 ) . . . (σ 0 (s 3 ) -1 s 2 σ 0 )(σ 0 (s 2 ) -1 s 1 σ 0 ) = a t g t g t-1 . . . g 1 . For i = 1, . . . ,t -1, g i . . . g 1 (w k-1 ) = w i , which has intersection with e 0 smaller than d k . The replacement above in the decomposition (writing each g i as a i σ 0 b i and rearranging the terms) either decreases D or decreases l, without changing D.

Noether inequality.

Let h be an element of W ∞ of degree d. By Lemma 5.3, there is a finite subset of points q i ∈ B(P 2

k ) i = 2, . . . , k such that h(e 0 ) = de 0 - k ∑ i=1 a i e(q i ),
where the a i are positive integers. Computing h(e 0 ) 2 = e 2 0 = 1 and applying the canonical form Ω to h(e 0 ), we get the classical Noether equalities

k ∑ i=1 (a i ) 2 = d 2 -1, k ∑ i=1 a i = 3d -3.
(5.1)

Lemma 5.5 (Noether inequality). Let h be an element of W ∞ of degree d ≥ 2, and let a 1 , . . . , a k be the multiplicities of the base-points of h.

(1) The following equality is satisfied.

(d -1)(a 1 + a 2 + a 3 -(d + 1)) = (a 1 -a 3 )(d -1 -a 1 )+ (a 2 -a 3 )(d -1 -a 2 )+ ∑ k i=4 a i (a 3 -a i ) (2)
For any i, j with 1 ≤ i < j ≤ k, we have a i + a j ≤ d.

(3) Ordering the a i such that a 1 ≥ a 2 ≥ a 3 ≥ a 4 . . . we have

a 1 + a 2 + a 3 ≥ d + 1.
Proof. To prove assertion (1), multiply the second Noether relation by a 3 and subtract it from the first; then rearrange the terms.

Assertion (2) is equivalent to (e 0e(q i )e(q j )) • (h -1 (e 0 )) ≥ 0. Take an element s ∈ Sym B(P 2 k ) that sends q i and q j onto p 1 = [1 : 0 : 0] and p 2 = [0 : 1 : 0]. This implies that σ 0 s maps e 0e(q i )e(q j ) onto e(p 3 ), where p 3 = [0 : 0 : 1]. The inequality is now equivalent to (σ 0 sh -1 )(e 0 ) • e(p 3 ) ≥ 0, and follows from Lemma 5.3.

Assertion (2) implies that d -1a i ≥ 0 for all i, since the number of basepoints is bigger than 2. Then, assertion (3) follows from the first one, because the right-hand side of the equality is non-negative. Lemma 5.6. Let h ∈ W ∞ , and let p 1 , q 1 be points of B(P 2 k ) such that h(e 0e(p 1 )) = e 0e(q 1 ). Let m be the degree of h. There exists two subsets of 2m -2 points {p 2 , . . . , p 2m-1 } and {q 2 , . . . , q 2m-1 } in B(P 2 k ), such that q 1 = q i and p 1 = p i for i ≥ 2, and such that the following hold: h(e 0 ) = me 0 -(m -1)e(q 1 ) -∑ 2m-1 i=2 e(q i );

h -1 (e 0 ) = me 0 -(m -1)e(p 1 ) -∑ 2m-1 i=2 e(p i ); h(e(p 1 )) = (m -1)e 0 -(m -2)e(q 1 ) -∑ 2m-1 i=2 e(q i ); h -1 (e(q 1 )) = (m -1)e 0 -(m -2)e(p 1 ) -∑ 2m-1
i=2 e(p i ); h(e(p i )) = e 0e(q 1 )e(q i ) for i = 2, . . . , 2m -1; h -1 (e(q i )) = e 0e(p 1 )e(p i ) for i = 2, . . . , 2m -1.

Proof. Write h(e 0 ) = me 0 -∑ k i=1 a i e(q i ), where the a i are positive integers. Since ma 1 = (e 0e(q 1 )) • h(e 0 ) = (e 0e(p 1 )) • e 0 = 1, we obtain a 1 = m -1. From Noether equalities, one obtains ∑ k i=2 a i = ∑ k i=2 (a i ) 2 = 2m -2. In particular, all a i are equal to 1 and k = 2m -2:

h(e 0 ) = me 0 -(m -1)e(q 1 ) -2m-2 ∑ i=2 e(q i ).

From h(e(p 1 )) = h(e 0 ) -(e 0e(q 1 )) we deduce h(e(p 1 )) = (m -1)e 0 -(m -2)e(q 1 ) -∑ 2m-2 i=2 e(q i ). Apply now Lemma 5.3 for the elements e(q i ), i = 1, . . . , 2m -2. One finds a subset {p 1 , . . . , p l } of B(P 2 k ) and non-negative integers b i , c i1 , . . . , c il such that h -1 (e(q i )) = b i e 0 -l ∑ j=1 c i j e(p j ).

Since e 0 • h -1 (e(q i )) = h(e 0 ) • e(q i ) = 1 and e(p 1 ) • h -1 (e(q i )) = h(e(p 1 )) • e(q i ) = 1, we get b i = c i1 = 1. From (h -1 (e(q i ))) 2 = -1 follows that h -1 (e(q i )) = e 0 -e(p 1 ) -e(p i ) for some point p i ∈ B(P 2 k ) distinct from p 1 . Doing this for each i, this defines 2m -1 points p 2 , . . . , p 2m-1 . Then h(e(p i )) = h(e 0e(p 1 ))h(e 0e(p 1 )e(p i )) = e 0e(q 1 )e(q i ).

It remains to observe that mh(e 0 ) -(m -1)h(e(p 1 )) -∑ 2m-1 i=2 h(e(p i )) = e 0 , so h -1 (e 0 ) = me 0 -(m -1)e(p 1 ) -∑ 2m-1 i=2 e(p i ); the value of h -1 (e(q 1 )) follows now directly from h(e 0e(p 1 )) = e 0e(q 1 ). Lemma 5.7. Let h 1 , h 2 ∈ W ∞ be Jonquières elements with respect to the same

point p ∈ B(P 2 k ). Then deg(h 1 h 2 ) < deg(h 1 ) + deg(h 2 ).
In particular, the sequence {deg(h 1 ) n } n∈N grows at most linearly and h 1 is not loxodromic.

Remark 5.8. Let m ≥ 2 be an integer and h be an isometry of a finite dimensional hyperbolic space H m ; here, H m is one of the two connected components of the affine quadric x 2 0 = x 2 1 + . . . + x 2 m in R m+1 , and h is the restriction of an element of O 1,m (R) preserving H m . Assume that h is parabolic ; this means that the linear transformation h is not contained in a compact subgroup of O 1,m (R) but does not have any eigenvalue of modulus > 1. Then, h n grows quadratically with n. In other words, given any base point e 0 in H m , the sequence of distances dist(e 0 , h n (e 0 )) grows like cosh(cn 2 ) for some positive constant c.

Lemma 5.7 shows that the behavior of parabolic transformations may be different if H n is replaced by its infinite dimensional sibling. For instance, consider the birational transformation f : (x, y) → (xy, y). Then f • determines a Jonquières element of W ∞ with deg( f n

• ) = n; equivalently, dist(e 0 , f n p (e 0 )) grows like cosh(cn).

Proof. Note that deg(h 1 h 2 ) = h 1 h 2 (e 0 ) • e 0 = h 2 (e 0 ) • (h 1 ) -1 (e 0 ). Let q 2 , . . . , q k be the base-points of h 1 or (h 2 ) -1 which are distinct from p. Because (e 0e(p)) • (h i ) ±1 (e 0 ) = (e 0e(p)) • e 0 = 1, we get

(h 1 ) -1 (e 0 ) = d 1 e 0 -(d 1 -1)e(p) -∑ k i=2 a i e(q i ), h 2 (e 0 ) = d 2 e 0 -(d 2 -1)e(p) -∑ k i=2 b i e(q i ),
for some non-negative integers d 1 , d 2 , a i , and b i . Moreover,

d 1 = deg(h 1 ), d 2 = deg(h 2 ). Hence, deg(h 1 h 2 ) = d 1 d 2 -(d 1 -1)(d 2 -1) -∑ a i b i ≤ deg(h 1 ) + deg(h 2 ) -1.
Proposition 5.9. Let h be an element of W ∞ of degree d ≥ 3. Let q 1 , . . . , q k be the points which are base-points of either h or h -1 . Let a i and b i be the multiplicities of the base points: a i = e(q i ) • h -1 (e 0 ) and b i = e(q i ) • h(e 0 ). Then one of the following properties holds:

(1) d < 3(d -a i )(d -b i );
(2) a i = d -1 = b i for at least one base-point q i ; in that case h is a Jonquières element with respect to e 0e(q i ).

In particular if h is not a Jonquières element of W ∞ , then d -(a i + b i )/2 > d/3
for all i = 1, . . . , k.

Proof. To simplify the notation, write e i = e(q i ) for i = 1, . . . , k. Suppose that one of the a i is equal to d -1, and order the points to assume a 1 = d -1. Since a 1 = h(e 1 ) • e 0 , we have h(e 0e 1 ) • e 0 = 1; this implies that h(e 0e 1 ) = e 0e j for some j and we deduce b j = d -1. If j = 1, then h is a Jonquières element with respect to e 0e 1 . Assume now that j = 1. Lemma 5.6 implies that the a i for i = 1 and the b i for i = j are equal all to 0 or 1. In particular, for each i, we get (da i )(db i ) ≥ d -1 > d/3. This proves that either (1) or (2) is satisfied when some a i is equal to d -1.

Assume now that a j < d -1 for all indices j. In particular h(e 0e j ) is distinct from e 0e j . One only needs to show that d < 3(da i )(db i ) for all i. Reordering the points one may assume i = 1.

We have h(e 0e 1 ) • e 0 = (e 0e 1 ) • h -1 (e 0 ) = da 1 ≥ 2 and we can write h(e 0e 1 ) = (da 1 )e 0 -∑ k i=1 r i e i for some coefficients r i ≥ 0. Moreover

1 ≤ h(e 0 -e 1 ) • (e 0 -e 1 ) = d -a 1 -r 1 .
because e 0e 1 and h(e 0e 1 ) are two isotropic elements of Z P 2 k in the boundary of H P 2 k that are not orthogonal (hence their intersection is a positive integer). As h preserves the canonical linear form,

k ∑ i=1 r i = 3(d -a 1 ) -2.
From h(e 0 ) • h(e 0e 1 ) = e 0 • (e 0e 1 ) = 1 we get

d(d -a 1 ) -b 1 r 1 = ∑ k i=2 b i r i + 1 where b i = e i • h -1 (e 0 ) ≥
0 is the multiplicity of p i as a base-point of h. Applying Lemma 5.5 to h we have b

1 + b i ≤ d for i = 2, . . . , k, so b i ≤ d -b 1 . Because d(d -a 1 ) -b 1 r 1 = d(d -a 1 -r 1 ) + (d -b 1 )r 1 ≥ d, we get d ≤ 1 + (d -b 1 ) k ∑ i=2 r i ≤ 1 + (d -b 1 ) • (3(d -a 1 ) -2) < 3(d -b 1 )(d -a 1 ).
This concludes the proof of the alternative. Then, assertion (1) implies

(d -(a i + b i )/2) = (d-a i )+(d-b i ) 2 ≥ (d -a i )(d -b i ) > d/3.
This concludes the proof of the proposition.

5.5. Halphen elements. Given nine distinct points q i in B(P 2 k ), the class

K = 3e 0 - 9 ∑ i=1 e(q i ) is an isotropic vector of Z P 2 k
. An element h of W ∞ is an Halphen element with respect to such a class K if h(K) = K. Lemma 5.10 (Growing of Halphen type maps). If h 1 and h 2 are Halphen elements of W ∞ with respect to the same isotropic class K then

deg(h 1 h 2 ) < deg(h 1 ) + deg(h 2 ).
In particular, the sequence {deg(h 1 ) n } n∈N grows at most quadratically and h 1 is not loxodromic.

Proof. Note that deg(h

1 h 2 ) = h 1 h 2 (e 0 ) • e 0 = h 2 (e 0 ) • (h 1 ) -1 (e 0 ). For i = 1, 2, write d i = deg(h i ) and define v i by (h i ) -1 (e 0 ) = d i 3 K + v i .
This decomposition satisfies e 0 • v i = 0,

3 = K • e 0 = K • (h i ) ±1 (e 0 ) = K • v i because K • K = 0, and 1 = h i (e 0 ) 2 = (v i ) 2 + 2 d i 3 (K • v i ) = (v i ) 2 + 2d i .
Writing v 1 = ∑ a(p)e(p) and v 2 = ∑ b(p)e(p), one gets

(v 1 • v 2 ) 2 = ( ∑ a(p)b(p)) 2 ≤ ∑ a(p) 2 • ∑ b(p) 2 = (v 1 ) 2 • (v 2 ) 2 .
Hence,

deg(h 1 h 2 ) = ( d 1 3 K + v 1 ) • ( d 1 3 K + v 2 ) = d 1 + d 2 + v 1 • v 2 ≤ d 1 + d 2 + (2d 1 -1)(2d 2 -1) < ( d 1 + d 2 ) 2 .
Lemma 5.11. Let h be an element of W ∞ , let p 1 , . . . , p m the points which are base-points of either h or h -1 . Let a k and b k be the multiplicities of the base points:

a k = h(e 0 ) • e(p k ), b k = h -1 (e 0 ) • e(p k ). If d/3 ≥ 3 + (3 + m ∑ j=10 b j ) 9 max i=1 |3a i -d| + m ∑ j=10 a j .
then h is an Halphen element with respect to K = 3e 0 -∑ 9 i=1 e(p i ). Proof. To simplify the notation, we write e k = e(p k ) for k = 1, . . . , m. Thus,

h(e 0 ) = de 0 - 9 ∑ i=1 a i e i - m ∑ j=10
a j e(q j ) h -1 (e 0 ) = de 0 -

c k ≤ n -1, ∀1 ≤ k ≤ m.
Since h preserves the intersection form, Hodge index theorem implies that h(K) = K if and only if h(K) • K = 0, if and only if h(K) is proportional to K. We now assume that h does not fix K; this implies that h(K) • K is positive, hence that

1 ≤ 3n - 9 ∑ i=1 c i and 9 ∑ i=1 c i ≤ 3n -1 Since h(K) • e 0 = K • h -1 (e 0 ), Noether equalities imply n = 3d - 9 ∑ i=1 b i = 3 + m ∑ j=10 b j ≥ 3.
We now compute h(e 0 ) • h(K), and get

3 = dn - 9 ∑ i=1 a i c i - m ∑ j=10 a j c j = d 3 (3n - 9 ∑ i=1 c i ) - 9 ∑ i=1 (a i - d 3 )c i - m ∑ j=10 a j c j .
Then

d/3 ≤ 3 + 9 ∑ i=1 (a i - d 3 )c i + m ∑ j=10 a j c j ≤ 3 + 9 ∑ i=1 c i 9 max i=1 a i - d 3 + max j≥10 c j m ∑ j=10 a j ≤ 3 + (3n -1) 9 max i=1 |a i - d 3 | + (n -1) m ∑ j=10 a j ≤ 3 + m ∑ j=10 b j 3( 9 max i=1 |a i - d 3 |) + m ∑ j=10 a j + R where R = 3 - 9 max i=1 |a i - d 3 | - m ∑ j=10 a j < 3.
This concludes the proof.

5.6. Base points of Jonquières transformations: From W ∞ to Bir(P 2 k ). Elements of Jonquières type in W ∞ are not all realized by birational transformations of the plane. The precise constraints that the base points must satisfied are listed in the following proposition. Both the statement and its proof are necessary to obtain Theorem D. Proposition 5.12. Let p 1 , . . . , p 2m-1 ∈ B(P 2 k ) be 2m -1 distinct points. There exists a Jonquières element f ∈ Bir(P 2 k ) whose base-points are p 1 , . . . , p 2m-1 , and such that f

• -1 (e 0 ) = me 0 -(m -1)e(p 1 ) - 2m-1 ∑ i=2 e(p i )
if and only if the points p i can be ordered so as to satisfy the following properties:

(1) p 1 is a proper point of P 2 k ; (2) for any i ≥ 2, p i is either a proper point of P 2 k or in the first neighbourhood of p j for some j < i;

(3) for all i > j ≥ 2, there is no line of P 2 k which passes through p 1 , p i , p j ; (4) for all triples k > j > i ≥ 2, at least one of the two points p j , p k does not belong, as proper or infinitely near point, to the exceptional divisor associated to p i ; (5) the number of points in {p 2 , . . . , p 2m-1 } that belong, as proper or infinitely near points, to the exceptional divisor associated to p 1 , is at most m -1 (these are points "proximate" to p 1 ); (6) for any k ≥ 1, each curve of P 2 k of degree k with multiplicity k -1 at p 1 passes through at most k + m -1 of the points {p 2 , . . . , p 2m-1 }.

Proof. This result is well known to specialists (see [START_REF] Kh | Defining relations for the Cremona group of the plane[END_REF][START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]), but it is hard to extract this precise statement from the literature.

We first verify that the six properties are necessary. The case m = 1 corresponds to linear projective transformations and is easily handled with. So, assume that f is a Jonquières transformation with base points p i , degree m ≥ 2, and

f • -1 (e 0 ) = me 0 -(m -1)e(p 1 ) - 2m-1 ∑ i=2 e(p i ).
Let C be a curve of degree k with multiplicity k -1 at p 1 . Let I be the set of indices i with p i ∈ C. The class ke 0 -(k -1)e(p 1 ) -∑ i∈I p i is effective, and the class f -1

• (e 0 ) is numerically effective; their intersection is equal to

mk -(m -1)(k -1) -|I| = m + k -1 -|I|.
Since this number is non-negative, Property ( 6) is satisfied. Properties (3) and ( 4) are proved along the same lines. To prove (2), assume that p i is not a proper point of the plane, and is not in the first neighbourhood of any other base point p i . Then we find a point q which is not a base point such that e(q)e(p i ) is effective. Intersecting with f • -1 (e 0 ) one gets -1 ≥ 0, a contradiction. Property (1) and ( 5) are proved in a similar way (for Property [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF], the case m = 2 is special: there are three base points of multiplicity 1, and at least one of them is a proper point of the plane; we chose such a point and call it p 1 ).

We now prove that Properties (1) to ( 6) are sufficient to construct such a Jonquières transformation. Denote by π : X 2 → P 2 k the blow-up of p 1 . The surface X 2 is the Hirzebruch surface F 1 : it admits a morphism η 2 : X 2 → P 1 , whose fibres correspond to the lines of P 2 k through the point p 1 . By property (2), the point p 2 is a proper point of X 2 . Consider the fiber

F 2 = (η 2 ) -1 (η 2 (p 2 ));
since this curve is the strict transform of the line through p 1 and p 2 , property (3) implies that F 2 does not contain any of the p i , i ≥ 3, as proper or infinitely near point. Denote by X 2 X 3 the birational map which consists of the blow-up of p 2 , followed by the contraction of the strict transform of F 2 . By construction X 3 is a Hirzebruch surface F 0 or F 2 , and p 3 is now a proper point of X 3 by property [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]. The pencil of lines through p 1 correspond to the ruling η 3 : X 3 → P 1 and the assumptions ( 3) and ( 4) imply that the fiber F 3 of η 3 through p 3 does not contain any p i with i ≥ 4.

Iterating this process, one constructs a sequence of maps

X 3 X 4 • • • X 2m-1 .
For j = 2, . . . , 2m -1, the surface X j is a Hirzebruch surface and comes with a morphism η j : X j → P 1 , the fibers of which correspond to the lines of P 2 k through p 1 . Moreover, the point p j is a proper point of X j , and no other point of the fibre F j = (η j ) -1 (η j (p j )) is one of the p i ; this latter condition is given by (3) and (4).

By construction, X 2m-1 is isomorphic to F r , for some odd integer r. We claim that r = 1; this amounts to show that no section of η 2m-1 has selfintersection ≤ -3. If this section corresponds to the curve of X 2 = F 1 contracted by π onto p 1 (its exceptional curve), it implies that at least m of the points p 2 , . . . , p 2m-1 belong, as proper or infinitely near, to the section, contradicting hypothesis [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF]. We can thus assume that the hypothetic section of self-intersection ≤ -3 corresponds to a curve of P 2 k of degree k passing through p 1 with multiplicity k -1. Such a curve has self-intersection 2k -1 on X 2 = F 1 . It must pass through l of the points p 2 , . . . , p 2m-1 and it must have self-intersection (2k -1)l + (2m -2l) = 2(k + ml) -3 on X 2m-1 = F r . Assumption [START_REF] Benedetti | Lectures on hyperbolic geometry[END_REF] implies that l < k + m: this shows that the self-intersection is ≥ -1.

Therefore, X 2m-1 is isomorphic to F 1 . Contracting the exceptional divisor, we obtain a birational morphism X 2m-1 on a surface which is isomorphic to P 2 k ; hence, we can identify this surface with the initial plane P 2 k and assume that the section is contracted to the point p 1 . The composition of the maps

P 2 k = X 1 X 2 • • • X 2m-1 → P 2
k is a birational map that preserves the pencil of lines through p 1 , and whose base-points are exactly p 1 , . . . , p 2m-1 . Classical Noether equalities imply that the degree of the map is m, the multiplicity of p 1 is m -1 and the other multiplicities are 1. This achieves the proof.

DYNAMICAL DEGREES AND CONJUGACY CLASSES

Our goal is to prove Theorem C from the introduction. Thus, given a birational transformation f of the projective plane with large dynamical degree λ( f ), we want to conjugate f by an element g of Bir(P 2 k ) to obtain deg(g f g -1 ) ≤ C st λ( f ) 5 (where the constant C st does not depend on f and will be here 4700).

Given f ∈ Bir(P 2 k ) with λ( f ) > 1, the main arguments may be summarized as follows. The degree of f is large, compared to λ( f ), if and only if the axis Ax( f • ) is far away from the base point e 0 of the hyperbolic space H P 2 k . Thus, we want to conjugate f by g so that the axis g • (Ax( f p )) of g f g -1 becomes closer to e 0 . A similar problem occurs in the proof of Noether-Castelnuovo theorem. When one wants to prove that quadratic transformations of the plane generate Bir(P 2 k ), one starts with an element f in Bir(P 2 k ) and then looks for a quadratic map h such that deg(h f ) < deg( f ); on H P 2 k , the problem is to find a quadratic map such that g • ( f • (e 0 )) is closer to e 0 than f • (e 0 ) is. We follow the same strategy as in Noether's proof. In other words, we first work with elements of W ∞ , and produce elements h ∈ W ∞ such that h(Ax( f • )) is close to e 0 ; then, as Castelnuovo did to correct Noether's error, we have to modify h slightly in order to realize it as g • for some g in Bir(P 2 k ). Proposition 5.12 is used for this purpose. Remark 6.1. The proof makes use of basic ideas from hyperbolic geometry (on the metric space (H P 2 k , dist)). The distance is given by cosh(dist(u, v)) = u • v, and it becomes rapidly annoying to transfer inequalities from distances to intersection numbers, and vice versa. This is the reason why, there is minimum reference to dist in what follows.

6.1. Axis, degree, distance to e 0 .

Isotropic eigenvectors of loxodromic elements.

Let h be a loxodromic element of W ∞ , and let λ(h) be its dynamical degree. We refer to Section 4.2.1 for the basic properties of loxdromic isometries of hyperbolic spaces.

Write

h(e 0 ) = de 0 -∑ i a i e(p i )
where the a i ≥ 0 are the multiplicities of the base points p i ∈ B(P 2 k ). The positive integer d is the degree of h: d = h(e 0 ) • e 0 . As explained in § 4.2.1, h preserves two isotropic lines Rv + and Rv -, where v + and v -are elements of

Z P 2 k and h(v + ) = λ(h)v + , h(v -) = v -/λ(h).
With the normalization v + • e 0 = v -• e 0 = 1, the vectors v + and v -are uniquely defined. Moreover, one has

v + = lim n→∞ h n (e 0 ) λ n , v -= lim n→∞ h -n (e 0 )
λ n . Thus, we can write

v + = e 0 -∑ i α i e(p i ), v -= e 0 -∑ i β i e(p i )
where the p i form a countable subset of B(P 2 k ) ; the set {p i } is contained in the union of base points of all iterates h n for n ∈ Z. The canonical form Ω is h-invariant, hence Ω(v + ) = Ω(v -) = 0; since v + and v -are isotropic, we obtain

∑ i α 2 i = ∑ i β 2 i = 1 and ∑ α i = ∑ β i = 3.
Since v + and v -are limits of sequences h n (e 0 )λ -|n| , Lemma 5.3 implies the following positivity statement. for some distinct points p 1 , . . . , p l ∈ B(P 2 k ), then u•v ≥ 0 for all v ∈ Ax(h),where Ax(h) is here given by the intersection of the plane generated by v ± and H P 2

k . If h = f • for some element f ∈ Bir(P 2 k ) and C ∈ Z P 2 k is an effective divisor, then C • v ≥ 0 for any v ∈ Ax(h).
6.1.2. Axis, and translation length. The proof of the following lemma is straightforward (see [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF][START_REF] Burger | Equivariant embeddings of trees into hyperbolic spaces[END_REF]).

Ax(h) = 1 √ 2v + • v - tv + + v - t t ∈ R >0 ⊂ H P 2 k .
(iv) The distance δ from e 0 to the axis Ax(h) satisfies

cosh(δ) = 2 v + • v - .
It is realized by the projection of e 0 on the axis, i.e. by the point

E = 2 v + •v -• v + +v - 2 .
6 , defined by

u = a 2 0 + ∑ p∈B(P 2 k ) a 2 p .
for u = a 0 e 0 + ∑ a p e(p). If the degree of h is large, the euclidean norm of h(e 0 )/dv + must be small: Lemma 6.4 (Approximation of the axis). Let h ∈ W ∞ be a loxodromic element of degree d and dynamical degree λ. Then

1 d h -1 (e 0 ) -v -< 2 λd ; 1 d h(e 0 ) -v -< 2λ d 1 d h -1 (e 0 ) -v + < 2λ d ; 1 d h(e 0 ) -v + < 2 λd . Moreover, h(e 0 ) + h -1 (e 0 ) 2d - v + + v - 2 < 2 λd , and (λ -1 λ ) 2 2d 2 < v + • v -< 1 d 1 λ + λ + 2 .
In particular, Lemma 6.3 implies . From a geometric point of view, it would be better to scale them (by the square root of their self-intersection), but the formulas would be difficult to read.

2d 1 λ + λ + 2 < cosh(dist(e 0 , Ax(h))) < 2d λ -1 λ . ( 6
Proof. Let us derive the top four inequalities. From b i = e i • h -1 (e 0 ) = h(e i ) • e 0 we get

λ -1 = h(v -) • e 0 = v -• h -1 (e 0 ) = d -∑ i b i β i . With Noether equality ∑(bi) 2 = d 2 -1 and the relation ∑(βi) 2 = 1 we deduce that ∑ i b i d -β i 2 = ∑(bi) 2 d 2 - 2 ∑ b i β i d + ∑ (β i ) 2 = 2 λd - 1 d 2 < 2 λd . This means that h -1 (e 0 ) d -v -< 2
λd . If one replaces v -by v + , then λ -1 is changed into λ; replacing h with h -1 yields the three other inequalities.

The fifth inequality follows by the triangular inequality. We now estimate the intersection product v + • v -. On one hand,

v + -v - 2 = 2v + • v -= 2 -2 ∑ α i β i
because v + and v -are isotropic. On the other hand, the above inequalities yield

v + -v - 2 < 2 λd + 2λ d 2 = 2 d 1 λ + λ + 2 .
Thus, altogether, one gets

v + • v -= 1 -∑ α i β i < 1 d 1 λ + λ + 2 .
In the other direction, note that 1 λ = d -∑ i a i α i and λ = d -∑ i a i β i , and deduce λ -

1 λ = ∑ a i (α i -β i ). Cauchy-Schwarz inequality yields (λ - 1 λ ) 2 ≤ ∑ (a i ) 2 • ∑ (α i -β i ) 2 = (d 2 -1) • (2v + • v -), so that v + • v -≥ (λ -1 λ ) 2 2(d 2 -1)
.

This concludes the proof.

6.2. Decreasing the distance from e 0 to the axis. We keep the same notation, h being a loxodromic element of degree d, dynamical degree λ, ... In particular, E is the projection of e 0 to the axis Ax(h), a i = e(p i ) • h(e 0 ) and b i = e(p i ) • h -1 (e 0 ). The middle point of h(e 0 ) and h -1 (e 0 ) is

de 0 -∑ i c i e(p i ), with c i = a i + b i 2 .
6.2.1. Strategy. The following lemma provides a strategy to decrease the distance between Ax(h) and e 0 by conjugacy with a quadratic element g of W ∞ . Similarly, if γ i = (α i + β i )/2 then 1 2

(v + + v -) = e 0 -∑ γ i e(p i ).
Lemma 6.6. Let p 1 , p 2 , p 3 be three distinct points of B(P

2 k ). If 3 ∑ i=1 c i ≥ d + 5 2 d λ , then (1) (e
(p 1 ) + e(p 2 ) + e(p 3 ) -e 0 ) • E > √ 2( 5 2 - √ 6) λ( 1 λ +λ+2) (2) cosh(dist(e 0 , g(E))) < cosh(dist(e 0 , E)) - √ 2( 5 2 - √ 6) λ( 1 λ +λ+2)
, for any quadratic element g ∈ W ∞ with base-points at p 1 , p 2 , p 3 .

Proof. If necessary, we enlarge the set of base points {p i } ⊂ B(P 2 k ) to include the three points p 1 , p 2 , and p 3 (allowing multiplicities equal to 0). From Lemma 6.4 we know that

h -1 (e 0 ) + h(e 0 ) 2d - v + + v - 2 < 2 λd ,
which may be written as

∑ c i d -γ i 2 < 2 λd .
Apply Cauchy-Schwarz inequality for the scalar product between the vector (1, 1, 1) ∈ R 3 and the vector

((c i /d) -γ i ) 3 i=1 , to get 3 ∑ i=1 | c i d -γ i | < 6 λd .
By assumption, we have (∑ 3

i=1 c i d ) -1 ≥ 5 2 √ λd
; hence

γ 1 + γ 2 + γ 3 -1 > 5 2 - √ 6 √ λd
.

Since E = 2 v + •v - v + +v - 2 and v + • v -< 1 d 1 λ + λ + 2 (Lemma 6.4), we obtain (e(p 1 )+e(p 2 )+e(p 3 )-e 0 )•E > 2 1 d 1 λ + λ + 2 5 2 - √ 6 √ λd = √ 2( 5 2 - √ 6) λ 1 λ + λ + 2 If g is a quadratic element of W ∞ with base-points p 1 , p 2 , p 3 , then g -1 (e 0 ) = 2e 0 -e(p 1 ) -e(p 2 ) -e(p 3 ). Consequently, cosh(dist(e 0 , E)) -cosh(dist(e 0 , g(E))) = e 0 • E -e 0 • g(E) = e 0 • E -g -1 (e 0 ) • E = (e(p 1 ) + e(p 2 ) + e(p 3 ) -e 0 ) • E
and the conclusion follows from the previous inequality.

Noether inequality for the axis of h.

Lemma 6.7. The coefficients c i = (a i + b i )/2 of (h(e 0 ) + h -1 (e 0 ))/2 satisfy

∑ k i=1 c i = 3d -3; ∑ k i=1 (c i ) 2 > (d 2 -1) -d 2 λ -1 + λ + 2 ;
and

(d -1)(c 1 + c 2 + c 3 -(d + 1)) > (c 1 -c 3 )((d -1) -c 1 )+ (c 2 -c 3 )((d -1) -c 2 )+ ∑ k i=4 c i (c 3 -c i ) -d 2 λ -1 + λ + 2 Proof.
The first equality directly follows from Lemma 5.5, which asserts that

∑ k i=1 a i = ∑ k i=1 b i = 3d -3. By Lemma 6.4, we have k ∑ i=1 a i d - b i d 2 < 2 λd + 2λ d 2 = 2 d 1 λ + λ + 2 . Since ∑(ai) 2 = ∑(bi) 2 = d 2 -1, we get d 2 -1 -∑ a i b i < d 1 λ + λ + 2 , hence ∑(ai + b i ) 2 = 2(d 2 -1) + 2 ∑ a i b i > 4(d 2 -1) -2d 1 λ + λ + 2 .
Dividing by 4, we obtain the first inequality. Then, subtract

c 3 ∑ c i = c 3 • 3(d - 1) to obtain successively k ∑ i=1 (c i ) 2 -c 3 k ∑ i=1 c i > (d -1)((d + 1) -3c 3 ) - d 2 λ -1 + λ + 2 and (d -1)(3c 3 -(d + 1)) + 3 ∑ i=1 c i (c i -c 3 ) > k ∑ i=4 c i (c 3 -c i ) - d 2 λ -1 + λ + 2 .
The inequality follows by rearranging the terms as in the proof of Noether's inequality.

6.2.3. Decreasing the distance to the axis by conjugacy in W ∞ .

Proposition 6.8. Let h ∈ W ∞ be of degree d and dynamical degree λ > 10 6 . Let p 1 , . . . , p k the base-points of h or h -1 , and c i = (a i + b i )/2 be the average of their multiplicities. Order the c i in such a way that c

1 ≥ c 2 ≥ c 3 ≥ • • • ≥ c k . If d > 24λ 3 , then c 1 + c 2 + c 3 ≥ d + 5 2 d λ .
Remark 6.9. Together with Lemma 6.6, Assertion (2), this proposition provides a way to conjugate a loxodromic element h of W ∞ by a quadratic involution g ∈ W ∞ so as to decrease the distance from e 0 to the axis.

Proof. We use the inequality of Lemma 6.7 and observe that (c 2c 3 )(d -1c 2 ) can be removed, as it is non-negative. Indeed, by hypothesis we have c 2 ≥ c 3 , and Noether equalities (5.1) imply that a 2 ≤ d -1 and b 2 ≤ d -1. This yields

∑ 3 i=1 c i -d > 1 + (c 1 -c 3 )(d-1-c 1 ) d-1 + ∑ k i=4 c i (c 3 -c i ) d-1 -d(λ -1 +λ+2) 2(d-1) . (6.2)
As a consequence, the result follows from

(c 1 -c 3 )(d -1 -c 1 ) + k ∑ i=4 c i (c 3 -c i ) > 1 2 d(λ -1 + λ + 2) + (d -1) 5 2 d λ .
The hypothesis on λ implies 5 2 d λ < √ d/400; thus, it suffices to prove that

(c 1 -c 3 )((d -1) -c 1 ) + k ∑ i=4 c i (c 3 -c i ) > d √ d 200 . (6.3) Note that 2(d -1 -c 1 ) = ((d -1) -a 1 ) + ((d -1) -b 1 ) is non-negative be- cause d -1 ≥ a 1 and d -1 ≥ b 1 (
this follows, for instance, from Lemma 5.5).

Since c 1 ≥ c 3 ≥ c i for i ≥ 4, every term of the left sum is non-negative. We do a case by case study, and show that Inequality (6.3) holds in each case.

Step 1.-Assume, first, that c 3 ≥ ((d + √ d/400)c 1 )/2. In this situation, the result directly follows:

c 1 + c 2 + c 3 ≥ c 1 + 2c 3 ≥ d + √ d/400 ≥ d + 5 2 d λ because λ > 10 6 .
Step 2.-Hence, we assume c 3 < ((d + √ d/400)c 1 )/2 in what follows. This yields

(c 1 -c 3 )(d -1 -c 1 ) > 1 2 (3c 1 -(d + √ d/400))(d -1 -c 1 ).
The right-hand side is a quadratic polynomial in the variable c 1 ; it vanishes at d/3 + √ d/1200 and d -1, and is positive between these two roots. If c 1 ≥ d/3 + √ d/100 Proposition 5.9 implies that

c 1 ∈ [d/3 + √ d/100, d -d/3].
Both extremities of this interval are between the two roots of the above quadratic polynomial; thus, the infimum of this polynomial function on this interval is equal to its value at d/3 + √ d/100 or at dd/3. One easily estimates these two values from below; the first one is 11 800 18 , and the second one is

√ d • (2/3d - √ d/100 -1) > d √ d 200 because d > 24 • 10
(2d -( √ 3 + 1/400) √ d)/2 • ( √ d/ √ 3 -1) > d 2 • √ d 2 > d √ d 200
for the same reason. This implies Inequality (6.3).

Step 3.-We can then assume that

c 1 < d/3 + √ d/100. (6.4)
In particular, we have

d -1 -c 1 ≥ 2d/3 - √ d/100 -1 > (0.6) • d. If c 1 -c 3 ≥ √ d/100
, we obtain Inequality (6.3); hence, we may add the assumption

c 3 > c 1 - √ d/100. (6.5)
Lemma 6.4 provides the inequality

1 d 2 k ∑ i=1 (a i -b i ) 2 < 2 λd + 2λ d 2 = 2 d 1 λ + λ + 2 .
In particular (a rb r ) 2 < 2d 1 λ + λ + 2 < 2.01 • λd for all indices r. Choosing r such that a r ≥ a j for all j, we know from Noether inequality that a r ≥ d/3 (see Lemma 5.5); this leads to b r > d/3 -1.42

√ λd, c r = (a r + b r )/2 > d/3 - 0.71 √ dλ, and c 3 > c 1 - √ d/100 ≥ c r - √ d/100 > d/3 -(3/4) • √ dλ. Hence, c 3 > d/3 -(3/4) • √ dλ > 3d/10, (6.6)
where the last inequality follows from d > 24λ 3 > 10 12 λ.

For each i ≥ 4, define ε i = min{c 3c i , c i }, and note that

c i • (c 3 -c i ) ≥ ε i • c 3 /2. This gives k ∑ i=4 c i • (c 3 -c i ) ≥ ( k ∑ i=4 ε i ) • c 3 /2. If ∑ k i=4 ε i > √ d/15
, Inequality (6.3) follows from c 3 > 3d/10 (Inequality (6.6)).

Step 4.-We can now add the inequality

k ∑ i=4 ε i ≤ 1 15 √ d (6.7)
to our assumptions. Our goal is to derive a contradiction from these assumptions.

Denote by l the largest index such that c l ≥ c 3 /2. For i = 4, . . . , l, the inequality c i ≥ c 3 /2 corresponds to c i ≤ c 3c i , hence ε i = c 3c i . This yields, together with Inequality (6.7), the following estimates for ∑ l i=4 c i :

(l -3)c 3 - √ d/15 < (l -3)c 3 - l ∑ i=4 ε i = l ∑ i=4 c i ≤ (l -3)c 3 .
Moreover, c 1 < c 3 + √ d/100 (Inequality 6.5), so 3c 3 ≤ c 1 + c 2 + c 3 < 3c 3 + √ d/50. Adding the two estimates yields

lc 3 - √ d/15 < l ∑ i=1 c i < lc 3 + √ d/50. Because ∑ l i=1 c i ≤ ∑ k i=1 c i = 3d -3 (Lemma 6.7
) and c 3 > 3d/10 (Equation 6.6), we have l(3d/10) -√ d/15 < 3d -3, that gives l < 10, hence l ≤ 9.

From ∑ k i=l+1 c i = ∑ k i=l+1 ε i < √ d/15 (Inequality (6.7)), one gets 3d -3 = ∑ k i=1 c i < lc 3 + √ d/50 + √ d/15. Together with c 3 ≤ c 1 < d/3 + √ d/100 (In- equality (6.4)), we get 3d -3 < l(d/3 + √ d/100) + √ d/50 + √ d/15, so l ≥ 9. Since l = 9, Inequality (6.7) yields k ∑ i=10 c i = k ∑ i=10 ε i < √ d/15.
In other words, there is a concentration of the multiplicities on the 9 points p 1 , . . . , p 9 : h behaves like a Halphen element of W ∞ .

Step 5.-To derive a contradiction, we apply Lemma 5.11:

d/3 < (3 + k ∑ i=10 a i ) (3 max{b i } 9 i=1 -d) + k ∑ j=10 b j + 3, (6.8)
because h is loxodromic. Let us estimate ∑ k i=10 c i from below. For i = 1, . . . , 9, write µ i = d/3c i and observe that Inequality (6.4) implies

- √ d/100 < µ 1 ≤ µ 2 ≤ • • • ≤ µ 9 . Lemma 6.7 yields 3d -3 = ∑ k i=1 c i = 3d -∑ 9 i=1 µ i + ∑ k i=10 c i , so that 9 ∑ i=1 µ i = 3 + k ∑ i=10 c i < 3 + √ d/15 < √ d/14, and
µ 9 < √ d/14 - 8 ∑ i=1 µ i < √ d/14 + 8 √ d/100 < √ d/6.
In particular, ∑ 9 i=1 (µ i ) 2 < 9(

√ d/6) 2 = d/4.
We also have

∑ k i=10 (c i ) 2 ≤ (∑ k i=10 c i ) 2 < d/225. We compute k ∑ i=1 (c i ) 2 = 9 ∑ i=1 (d/3 -µ i ) 2 + k ∑ i=10 (c i ) 2 = d 2 - 2 3 d 9 ∑ i=1 µ i + 9 ∑ i=1 (µ i ) 2 + k ∑ i=10 (c i ) 2 < d 2 - 2 3 d(3 + k ∑ i=10 c i ) + 1 4 d + 1 225 d < d 2 - 2 3 d( k ∑ i=10 c i ) - 3 2 d.
On the other hand, Lemma 6.7 yields

k ∑ i=1 (c i ) 2 > d 2 -1 - d 2 λ -1 + λ + 2 > d 2 -(0.501) • (λd),
and we obtain

d 2 -(0.501) • (λd) < d 2 - 2 3 d( k ∑ i=10 c i ) - 3 2 d, hence k ∑ i=10 c i < (0.501) 3 2 λ - 9 4 < 0.7516λ.
In particular, either ∑ k i=10 a i < 0.7516λ or ∑ k i=10 b i < 0.7516λ. We assume the first (otherwise we apply Lemma 5.11 to h -1 instead of h).

For each i, recall that

(a i -b i ) 2 < 2d 1 λ + λ + 2 < 2.01dλ and, conse- quently, |a i -b i | < 1.42 √ λd, hence b i < c i + 0.71 √ λd < d/3 + √ d/100 + 0.71 √ λd < d/3 + 0.72 √ λd. This yields (3 max{b i } 9 i=1 -d) < 2.16 √ dλ. Equation 6.8 implies d < 3(∑ k i=10 a i ) (3 max{b i } 9 i=1 -d) + ∑ k j=10 b j < 3 • (0.7516) • λ(2.16 √ dλ + 2 • 0.7516λ) < 4.88 √ dλ 3/2 .
In particular, √ d < 4.88λ 3/2 , contradicting the hypothesis d > 24λ 3 .

6.3.

From W ∞ to the Cremona group. We can now prove Theorem C, Assertion (1), which we rephrase as follows.

Theorem 6.10. Let f ∈ Bir(P 2 k ) be a loxodromic element of dynamical degree λ > 10 6 . There exists a birational map g ∈ Bir(P 2 k ) such that deg(g f g -1 ) < 4700λ 5 .

To prove Theorem 6.10, we denote by Ax( f • ) the axis of f and by E the projection of the point e 0 onto Ax( f • ) (see Lemma 6.3). We fix a point p 1 ∈ B(P 2 k ) such that e(p 1 ) • E ≥ e(q) • E for each q ∈ B(P 2 k ). We can choose p 1 so that it is a proper point of the plane. 6.3.1. The involutions σ Ω . For each q ∈ B(P 2 k )\{p 1 }, the vector

w q := e 0 -e(p 1 ) 2 -e(q) ∈ Z P 2 k ⊗ Q
has self-intersection -1; we denote by ν q the orthogonal reflection of Z P 2 k ⊗ Q with respect to the hyperplane orthogonal to w q . The Q-linear automorphism ν q is given by

u → u + 2(u • w q )w q = u -2 (u • w q ) (w q • w q ) w q .
The transformations ν q , for q in B(P 2 k )\{p 1 }, constitute a family of commuting involutions because w q is orthogonal to w q if q = q .

For any finite set Ω ⊂ B(P 2 k )\{p 1 } consisting of an even number 2m -2 ≥ 0 of points, we denote by σ Ω the composition of all ν q for q in Ω. By induction, the transformation σ Ω is the automorphism of Z P 2 k given by σ Ω (e 0 ) = me 0 -(m -1)e(p 1 ) -∑ q∈Ω e(q) = e 0 + ∑ q∈Ω ( e 0 -e(p 1 )

2

e(q)); σ Ω (e(p 1 )) = (m -1)e 0 -(m -2)e(p 1 ) -∑ q∈Ω e(q) = e(p 1 ) + ∑ q∈Ω ( e 0 -e(p 1 )

2

e(q)); σ Ω (e q ) = e 0e(p 1 )e(q) if q ∈ Ω;

σ Ω (e q ) = e q if q ∈ B(P 2 k )\({p 1 } ∪ Ω). The following statement is easily proved. It implies that the σ Ω form a subgroup of W ∞ . Lemma 6.11. For all pairs Ω, Ω of subsets of B(P 2 k )\{p 1 } with an even number of elements, and for all pairs (q, q ) of distinct points in B(P 2 k )\{p 1 }, we have:

(i) σ Ω of Z P 2 k
is the product of all ν q , q ∈ Ω, and, as such, is an involution;

(ii) σ Ω • σ Ω = σ Ω∪Ω \Ω∩Ω ; (iii) σ {q,q } is conjugate to the composition of σ 0 ∈ W ∞ with a transposition η ∈ Sym B(P 2 k ) . (iv) σ Ω is an element of W ∞ .
λ 1 λ + λ + 2 = 5 -2 √ 3 √ 2(λ + 1) .
We denote by S the collection of all finite subsets Ω ⊂ B(P 2 k )\{p 1 } consisting of an even number of points, and satisfying the following properties:

(1) Each point q ∈ Ω is either a proper point of P 2 k or is in the first neighbourhood of a point q ∈ Ω.

(2) There is no line of P 2 k passing through p 1 and through two distinct points of Ω (taking here also infinitely near points).

(3) For any three distinct points q i , q j , q k ∈ Ω, the points q j , q k cannot simultaneously belong, as proper or infinitely near points, to the exceptional curve obtained by blowing-up q i (this means that q j , q k are not both "proximate" to q i ). ( 4)

Either Ω = / 0 or cosh (dist(e 0 , σ Ω Ax( f • ))) < cosh (dist(e 0 , Ax( f • ))) -δ. (note that σ Ω (Ax( f • )) = Ax(σ Ω f • σ -1 Ω )))
We put a partial order on S, defined by Ω < Ω if and only if

cosh (dist(e 0 , σ Ω Ax( f • ))) < cosh (dist(e 0 , σ Ω Ax( f • ))) -δ.
The set S is of course not empty, since it contains the set Ω = / 0. The definition of the order implies that there is no infinite decreasing sequence Ω 1 > Ω 2 > . . . in S, so S contains minimal elements. We now prove the following assertion:

( ) Let Ω be a minimal element of S, and let E be the projection of e 0

onto σ Ω Ax( f • ). Either deg(σ Ω f • σ -1 Ω ) ≤ 24λ 3
or there exists q ∈ B(P 2 k ) which satisfies e(q) • E > e(p 1 ) • E.

To prove ( ), we take an element Ω of S, we assume that deg(σ Ω f • σ Ω ) > 24λ 3 and that e(p 1 ) • E ≥ e(q) • E for all q ∈ B(P 2 k ), and we show that Ω is not minimal in S.

• Proposition 6.8 and Lemma 6.6 provide three distinct points q 1 , q 2 , and q 3 such that (e(q 1 ) + e(q 2 ) + e(q 3 )e 0 ) • E > δ. Because e(p 1 ) • E ≥ e(q i ) • E for all indices i, we can assume that q 1 = p 1 . Since (σ {q 2 ,q 3 } ) -1 (e 0 ) = 2e 0e(p 1 )e(q 2 )e(q 3 ),

we obtain E •e 0 -σ {q 2 ,q 3 } (E)•e 0 = E •e 0 -E •(σ {q 2 ,q 3 } ) -1 (e 0 ) > δ; this implies that e 0 • σ {q 2 ,q 3 } (E) < cosh(dist(e 0 , σ Ω Ax( f • ))) -δ.
As a consequence, if we define Ω = Ω ∪ {q 2 , q 3 }\(Ω ∩ {q 2 , q 3 }), then σ Ω = σ {q 2 ,q 3 } • σ Ω , and the point

E = (σ Ω ) -1 (E) ∈ Ax( f • ) satisfies σ {q 2 ,q 3 } (E) =
σ Ω (E ); thus, the inequalities cosh(dist(e 0 , σ Ω Ax( f

• ))) ≤ e 0 • σ Ω (E ) < cosh(dist(e 0 , σ Ω Ax( f • ))) -δ
imply that Ω does not satisfy one of the assertions (1), ( 2), (3) in the definition of S, because Ω is minimal.

• We now replace Ω by a new set Ω such that σ Ω (e 0 )•E does not increase

and Ω satisfies the defining properties of S. From § 6.3.1 we know that e 0 • σ Ω (E ) = σ Ω (e 0 ) • E = (e 0 + ∑ q∈Ω ((e 0e(p 1 ))/2e(q))) • E

If there are two distinct points q, q ∈ Ω such that (e 0e(p 1 )e(q)e(q )) • E ≥ 0, we can replace Ω with Ω\{q, q }, and this does not increase σ Ω (e 0 ) • E . We can thus assume that (e 0e(p 1 )e(q)e(q )) • E < 0 for all pairs of distinct points (q, q ) of Ω . If q ∈ Ω is in the first neighborhood of a point q , the divisor e(q)e(q ) is effective, and intersects E non-negatively, because E is on the axis Ax( f • ) of f ∈ Cr 2 (k) (Lemma 6.2). If q does not belong to Ω , we can thus replace Ω with Ω ∪ {q }\{q}; again, this does not increase σ Ω (e 0 ) • E .

These replacements down, we get a new set Ω . Let us show that Ω belongs to S. Property ( 4) is obviously satisfied. The fact that (e 0e(p 1 )e(q)e(q )) • E < 0 for all pairs of distinct points (q, q ) in Ω implies that p 1 , q, q are not collinear (Assertion (2)). Similarly, the second family of modifications of Ω shows that Ω satisfies Assertion [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]. It remains to show that Assertion (3) holds for Ω . Let q i , q j , and q k be three distinct points of Ω such that q j and q k are proximate to q i ; then, the divisor e(q i )e(q j )e(q k ) is effective and intersects thus E non-negatively (Lemma 6.2). This yields 0 > (e 0e(p 1 )e(q j )e(q k )) • E ≥ (e 0e(p 1 )e(q i )) • E , which is impossible. Indeed, this implies that (e 0e(p 1 )e(q)) • E < 0, where q = p 1 is a point which is either a proper point of P 2 k or in the first neighbourhood of p 1 (choose either q = q i or q such that q i is infinitely near to q), and this is impossible because e 0e(p 1 )e(q) is effective, as it corresponds to a line of P 2 k . This concludes the proof of ( ).

Strategy.

To prove Theorem 6.10, we provide an algorithm which runs as follows. Start with f and choose a minimal configuration Ω ∈ S. If there is an element g in Bir(P 2 k ) such that g • (e 0 ) = σ Ω (e 0 ), we prove that the distance from e 0 to the axis is decreased by a multiplicative factor that depends only on δ; we can thus replace f by g f g -1 . If this is not the case, it is proved that a conjugate of f satisfies deg(g f g -1 ) < 4700λ 5 , and the algorithm stops. 6.3.4. Algorithm: first case. We now take an element Ω ⊂ S, which is minimal in S. By Property ( ), the set Ω is not empty (otherwise the Theorem is proved, with g equal to the identity), hence we have cosh(dist(e 0 , Ax(σ Ω f • σ Ω )) < cosh(dist(e 0 , Ax( f • )))δ.

We write now explicitly Ω = {p 2 , . . . , p 2m-1 }, we denote by E the projection of e 0 on σ Ω Ax( f • ), and we denote by E ∈ Ax( f • ) the element (σ Ω ) -1 (E); in general, this point differs from the projection of e 0 onto Ax( f • ).

Suppose that there exists an element g ∈ Bir(P 2 k ) with

(g • ) -1 (e 0 ) = σ -1 Ω (e 0 ) = σ Ω (e 0 ). The point g • (E ) ∈ Ax((g f g -1 ) • ) satisfies e 0 • g • (E ) = (g • ) -1 (e 0 ) • E = σ Ω (e 0 ) • σ Ω (E) = e 0 • E,
and this implies cosh(dist(e 0 , Ax((g f g -1 ) • ))) < cosh(dist(e 0 , Ax( f • )))δ.

(6.9)

We can thus replace f with g f g -1 and repeat the process (see below § 6.3.6).

6.3.5. Algorithm: second case. Suppose now that such a birational transformation g does not exist. Denote by p i the elements of Ω (including the point p 1 ).

•An inequality.-Recall that p 1 is a proper point of P 2 k . Since Ω is in the family S, properties (1) to (4) of Proposition 5.12 are fulfilled. Thus, if there is no birational transformation g such that g -1 (e 0 ) = σ -1 (e 0 ) = me 0 -(m -1)e(p 1 ) -∑ i e(p i ), one of the two assumptions (5)-( 6) of Proposition 5.12 is not satisfied. We now study these two possibilities. Write

E = α 0 e 0 -∑ α i e(p i ),
where p 1 , . . . , p 2m-1 are as above and the remaining points p k , k ≥ 2m, are elements of B(P 2 k ); the α i 's are real non-negative numbers (apply Lemma 6.2). If Assumption (5) is not fulfilled, the number of points in {p 2 , . . . , p 2m-1 } which belong, as proper or infinitely near points, to the exceptional divisor associated to p 1 is equal to m + l with 0 ≤ l ≤ m -2; we write these points as p i 1 , . . . , p i m+l . The divisor e(p 1 ) -∑ m+l j=1 e(p i j ) is thus effective; hence, it intersects E non-negatively.

Applying σ Ω , we see that E = σ Ω (E f ) intersects non-negatively This gives (-1l)α 0 + (l + 2)α 1 ≥ 0, i.e.

α 1 α 0 ≥ l + 1 l + 2 . ( 6.10)
If Assumption (6) of Proposition 5.12 is not satisfied, we obtain the existence of a curve of degree k ≥ 1 with multiplicity k -1 at p 1 which passes through k + m + l of the points {p 2 , . . . , p 2m-1 }, for some l ≥ 0; note that this curve is unique because it corresponds to the exceptional section of the Hirzebruch surface obtained by blowing-up p 1 and performing elementary links at p 2 , . . . , p 2m-2 . As before, this implies that and this leads again to (-1l)α 0 + (l + 2)α 1 ≥ 0 and to Equation (6.10).

•Upper bound on the degree.-Coming back to the proof of Proposition 5.12 we know that, in both cases, the problem is that the Hirzebruch surface obtained after blowing-up p 1 and performing elementary links at p 2 , . . . , p 2m-2 is equal to a new Hirzebruch surface F 1+2l which does not coincide with F 1 ; so the map σ Ω does not correspond to a geometric Jonquières map.

To recover a well defined birational transformation of the plane, we choose 2l general points of P 2 k , that we call p 2m , . . . , p 2m+2l-1 . Then, we obtain the existence of a Jonquières transformation g ∈ Bir(P 2 k ) such that

(g • ) -1 (e 0 ) = (m + l)e 0 -(m + l -1)e(p 1 ) - 2m+2l-1 ∑ i=2 e(p i ).
We now estimate the degree of g f g -1 from above. The bound comes from the computation of the intersection of g • (E ) ∈ Ax((g f g -1 ) • ) with e 0 , a number which is smaller than, or equal to cosh(dist(e 0 , Ax((g f g -1 ) • ))). First,

g -1
• (e 0 ) = σ Ω (e 0 ) + l(e 0e(p 1 )) - In particular, we obtain

e 0 • g • (E ) = σ Ω (g -1 • (e 0 )) • E ≤ α 0 + l(α 0 -α 1 ).
Inequality (6.10) provides the estimate α 1 ≥ α 0 • l+1 l+2 , and therefore

e 0 • g • (E ) ≤ α 0 • (1 + l -l • l + 1 l + 2 ) = 2α 0 • l + 1 l + 2 < 2α 0 = 2e 0 • E.
Since E is the projection of e 0 on σ Ω Ax( f ), one gets cosh(dist(e 0 , Ax((g f g -1 ) • ))) < 2α 0 = 2 cosh(dist(e 0 , σ Ω Ax( f ))).

Remark 6.12. We have α i + α j ≤ α 0 for all indices i = j. Indeed, one can choose p i , p j proper points and obtain that e 0e(p i )e(p j ) is effective, hence has non-negative intersection with E. This also follows from Lemma 6.2, as e 0e(p i )e(p j ) is in the same W ∞ -orbit as e(p i ).

From Equation (6.10), we get α 1 ≥ α 0 2 ; hence the remark shows that α 1 ≥ α i for all i ≥ 2. By ( ), this implies that deg(σ Ω f σ Ω ) ≤ 24λ 3 .

According to the Inequality (6.1) we have

α 0 = cosh(dist(e 0 , σ Ω Ax( f ))) < 2deg(σ Ω f • σ Ω ) λ -1 λ and 2deg(g f g -1 ) 1 λ + λ + 2 < cosh(dist(e 0 , Ax((g f g -1 ) • ))).
As a consequence, 2deg(g f g -1 )

1 λ + λ + 2 < 4deg(σ Ω hσ Ω ) λ -1 λ and deg(g f g -1 ) < 8deg(σ Ω hσ Ω ) 2 (λ -1 λ ) 2 ( 1 λ + λ + 2) < 4700λ 5 .
Thus, we may stop the algorithm, since deg(g f g -1 ) < 4700λ 5 .

Conclusion. Thus,

• either § 6.3.4 apply, which means that we can find an element g in the Cremona group such that the hyperbolic cosine of the distance from e 0 to the axis of g f g -1 decrease by δ. We can then repeat the process for g f g -1 as long as deg(g f g -1 ) > 24λ 3 ; • or the process stops, which means that § 6.3.4 does not apply, and then § 6.3.5 shows that there exists an element g in Bir(P 2 k ) with deg(g f g -1 ) < 4700λ 5 .

To sum up, the theorem 6.10 is proved in at most cosh(dist(e 0 , Ax( f )))/δ steps. 6.4. Proof of Theorem C. To conclude the proof of Theorem C, we need to prove the second assertion (using the first one, provided by Theorem 6.10). Let f be a loxodromic element of Bir(P 2 k ). By the spectral gap property (see § 2.6), λ( f ) ≥ λ l 1.176280; hence, λ( f 86 ) > 10 6 and the first assertion of Theorem C provides an element g in Bir(P 2 k ) such that deg(g f 86 g -1 ) ≤ 4700 λ( f 86 ) 5 .

Let h be equal to g f g -1 ; we have λ(h) = λ( f ). Both h and h 86 have the same axis, and we denote by L the distance from e 0 to it (see Figure 2). By definition, the distance from e 0 to (h 86 ) and we obtain the inequality mcdeg( f ) ≤ cosh(18 + 345 log(λ( f ))). This concludes the proof of Theorem C.

ALGEBRAIC FAMILIES OF BIRATIONAL TRANSFORMATIONS AND DECREASING SEQUENCES OF DYNAMICAL DEGREES

In this section, we prove the main corollaries of Theorem C. This includes Theorem D, which states that Λ(P 2 k ) is well ordered and that is is closed if k is uncountable and algebraically closed. Proof. We may assume that k is algebraically closed. Let λ n , n ≥ 1, be a sequence of dynamical degrees. Suppose that λ n+1 < λ n for all indices n. Our goal is to prove that the number of terms in this sequence is bounded.

Each element λ n of this sequence is the dynamical degree of some birational transformation f n : X n X n . The subsequence of dynamical degrees λ n for which X n is not geometrically rational is bounded, because the set of dynamical degrees of birational transformations of irrational surfaces is discrete. Thus, in what follows, we assume that X n is equal to the projective plane P 2 k for all n ≥ 1.

If λ n 0 = 1 for some index n 0 , the sequence contains n 0 terms, because all dynamical degrees are larger than or equal to 1. Thus, we assume that λ n > 1 for all n. Let λ ∞ be the limit of the sequence (λ n ); by Corollary 2.7,

λ ∞ + 1 ≥ λ n ≥ λ ∞ ≥ λ L 1.17628 if n is large enough (n ≥ n 0 ). Theorem C provides conjugates g n of f M n , such that deg(g n ) ≤ cosh(18 + 345 log(λ ∞ + 1)).
The set Bir d (P 2 k ) of all birational transformations of degree d is naturally endowed with the structure of an algebraic variety (see [START_REF] Blanc | Topologies and structures on the Cremona group[END_REF]); we denote by F the Zariski closure of the set { f i } i≥0 in Bir d (P 2 k ). The dimension of F is positive because the λ i are pairwise distinct and, extracting a new subsequence, we assume that F is irreducible.

For each positive integer n, consider the set

X n = {g ∈ Bir d (P 2 k ) | deg(g n ) ≤ max i (deg( f n i ))}.
By [START_REF] Blanc | Topologies and structures on the Cremona group[END_REF], this set is Zariski closed 2 ; hence F is contained in X n for all n ≥ 1.

Similarly, the set

Y n = {g ∈ F | deg(g n ) ≤ max i (deg( f n i )) -1}
is a Zariski closed subset of F, and is a strict subset of F because its complement contains at least one f j . Since k is not countable, there is at least one element h in F \ ∪ n≥1 Y n . This birational transformation satisfies

deg(h n ) = max i (deg( f n i ))
for all n ≥ 1. This implies that deg( f n i ) ≤ deg(h n ) for all indices i and n; in particular, λ( f i ) ≤ λ(h) for all i, and

1 < λ ∞ ≤ λ(h).
Thus, h is a loxodromic transformation of degree d. Lemma 7.3. Let k be a field and d ≥ 2 be an integer. There exists a constant ∆(d) such that, for all loxodromic element g ∈ Bir(P

2 k ) of degree d, dist(e 0 , Ax(g)) ≤ ∆(d)/2 (7.1) | log(deg(g m )) -m log(λ(g))| ≤ m log(λ(g m )) + ∆(d) (7.2)
for all m ≥ 1.

Proof. From the spectral gap property λ(g) ≥ λ L = 1.176280. Thus, hyperbolic geometry implies that dist(e 0 , g • (e 0 )) goes to infinity with dist(e 0 , Ax(g)); more precisely, there is a uniform constant ε > 0 such that dist(e 0 , g • (e 0 )) ≥ 2dist(e 0 , Ax(g)) + log(λ(g))ε.

2 With the notation of [START_REF] Blanc | Topologies and structures on the Cremona group[END_REF], consider the set Hd of triples (p, q, r) of homogeneous polyno- Since dist(e 0 , g • e 0 ) is bounded from above by log(2deg(g)), the first upper bound follows.

The triangular inequality implies dist(e 0 , (g m ) • (e 0 )) ≤ 2dist(e 0 , Ax(g • )) + m log(λ(g))

and hyperbolicity implies

dist(e 0 , (g m ) • (e 0 )) ≥ m log(λ(g)) + 2dist(e 0 , Ax(g • )) -δ
where δ is a uniform constant (δ < 100). The result follows.

Apply this lemma to h and to the f i :

m log(λ(h)) -∆(d) ≤ log(deg(h m )) ≤ m log(λ(h)) + ∆(d), m log(λ( f i )) -∆(d) ≤ log(deg( f m i )) ≤ m log(λ( f i )) + ∆(d).
Let ε be a positive real number, and m be a positive integer such that ε ≥ 2∆(d)/m. Then, there exists i such that deg(

f m i ) = deg(h m ), and we get m log(λ(h)) -∆(d) ≤ log(deg(h m )) = log(deg( f m i )) ≤ m log(λ( f i )) + ∆(d).
Hence, log(λ(h)) ≤ log(λ( f ∞ )) + ε. Since this inequality holds for all ε > 0, we obtain λ(h) ≤ λ ∞ , and thus λ(h) = λ ∞ , as desired.

7.4. Increasing approximation by Salem dynamical degrees. The set Pis is contained in the closure of the set Sal. In this section, we show that the same property holds for dynamical degrees:

Theorem 7.4. Let k be an algebraically closed field of characteristic 0. Let β be an element of Λ P (P 2 k ). There exists a strictly increasing sequence (α n ) n≥0 of elements of Λ S (P 2 k ) that converges towards β. Corollary 7.5. Let k be an algebraically closed field of characteristic 0 and let X be a projective surface defined over k. The dynamical spectrum Λ(X) is contained in the closure of the set of dynamical degrees λ( f ) of automorphisms of surfaces which are birationally equivalent to X.

Proof of the corollary. If X is rational, this is a consequence of Theorem 7.4 and Theorem A. If X is not rational, all dynamical degrees are realized by dynamical degrees of automorphisms of surfaces which are birationally equivalent to X (see Section 3).

The proof of Theorem 7.4 is given in §7.4.1 when β is not a reciprocal quadratic integer. The case of reciprocal quadratic integers is dealt with in §7.4.2. 7.4.1. Pisot numbers which are not reciprocal quadratic integers. Let β ∈ Λ P (P 2 k ) be a Pisot number which is not a reciprocal quadratic integer; thus, β is the dynamical degree of a birational transformation of P 2 k , but is not the dynamical degree of an automorphism of rational projective surface.

Choose f ∈ Bir(P 2 k ) with λ( f ) = β, denote by d the degree of f , by p i and q i the base points of f and f -1 (1 ≤ i ≤ m), and write

f • (e 0 ) = de 0 - m ∑ i=1 a i e(q i ) f • (e(p i )) = d i e 0 - m ∑ i=1 c i, j e(q j ).
The multiplicities a i are positive integers; the c i, j are non-negative integers.

Say that q i has an infinite length if f l (q i ) is not a base point of f for all l ≥ 0, and say that q i has a finite length (equal to i ) if f l (q i ) is not a base point of f for 0 ≤ l ≤ i -1 but f i (q i ) is one of the base points p j of f . If all the base points q i , 1 ≤ i ≤ m, have a finite length, one can blow up the points f l (q i ), 1 ≤ l ≤ i to get a new surface on which f is an automorphism. Since f is not conjugate to an automorphism, at least one of the base points q i has an infinite length.

Order the base points q i in such a way that q 1 , ..., q n have infinite length and q n+ j has finite length n+ j for j = 1, . . . , mn. Then, number the p j in such a way that p n+ j = f j (q n+ j ) for all j ≥ 1. We shall now construct a sequence of birational transformations f k such that

• each f k is conjugate to an automorphism ;

• λ( f k ) converges to λ( f ) = β as k goes to +∞. The idea is to transform the points q i into base points of finite length i for i ≤ n, but with length i = k going to +∞.

For this purpose, define

A = {e 0 } ∪ m i=n+1 l i -1 j=0 f j • (e(q i )) B j = n i=1 f j • (e(q i )) for any j ≥ 0 C = n i=1 {e(p i )}.
The elements of these three sets are linearly independent in Z P 2 k . In particular, A is a basis of the sub-module V A spanned by A in Z P 2 k . Similarly, B j (resp. C) is a basis of the sub-module V B j spanned by B j (resp. C) for all j ≥ 0. The map f • restricts to an isomorphism between V C ⊕V A and V A ⊕V B 0 , and also to an isomorphism

V C ⊕V A ⊕V B 0 ⊕ • • • ⊕V B k f * -→ V A ⊕V B 0 ⊕ • • • ⊕V B k+1 . Writing V k = V C ⊕V A ⊕V B 0 ⊕ • • • ⊕V B k , we define a linear transformation F k ∈ Aut(V k ) by F k = π k • f • ,
where π k : V k+1 → V k is the isomorphism defined by π k ( f k+1

• (e(q i ))) = e(p i ) for i = 1, . . . , n and π k (

x) = x for x ∈ V A ⊕V B 0 ⊕ • • • ⊕V B k = V k ∩V k+1 .
Since f • preserves the intersection form of Z P 2 k , F k preserves the intersection form of V k . This latter space is of Minkowski type: An orthonormal basis is given by e 0 (of self-intersection +1) and by the other elements of A, B j ,C (each of self-intersection -1). Since f • satisfies the Noether equalities (5.1), so does F k :

m ∑ i=1 a 2 i = d 2 -1; m ∑ i=1 a i = 3d -3.
For a fixed integer k ≥ 1, denote by r + 1 the dimension of V k . Then, denote by W r the subgroup of W ∞ generated by

• the finite group of permutations of the set e(q), where e(q) runs over the elements of A \ {e 0 }, of B j ( j ≤ k), and of C; • the involution σ 0 (with base points p 1 , p 2 , p 3 chosen among the base points p j ); • the involutions τ p,q for e(p) and e(q) in the sets A \ {e 0 }, B j ( j ≤ k)

and C. This group is isomorphic to the Coxeter group of the Dynkin diagram T 2,3,r-3 introduced in Section 2.5. Since F k satisfies the Noether equalities, F k is an element of the Coxeter group W r (this is a version of Nagata's Theorem mentioned in § 2.5; see [START_REF] Dolgachev | Point sets in projective spaces and theta functions[END_REF] and the proof of Lemma 5.3). By Uehara's theorem (see [START_REF] Uehara | Rational surface automorphisms with positive entropy[END_REF]), there is an element f k ∈ Bir(P 2 k ) for which λ( f k ) is equal to the spectral radius of the linear transformation F k ; moreover, f k is conjugate to an automorphism of a projective rational surface X k . Lemma 7.6. If β is not a reciprocal quadratic integer, the sequence (λ( f k )) converges towards β as k goes to +∞, and it contains a sub-sequence that increases strictly towards β.

This lemma concludes the proof of Theorem 7.4 when β is not a reciprocal quadratic integer. Indeed, λ( f k ) is not a quadratic integer if k is large, because the set of reciprocal quadratic integers is discrete; hence, (λ( f k )) contains a strictly increasing sequence of Salem numbers that converges towards β.

Proof of Lemma 7.6. Let λ k = λ( f k ). This number is the largest real eigenvalue of F k ∈ Aut(V k ⊗ R). V B i . We obtain the system of equations

      0 λ( f ) • v A λ( f ) • v B 1 λ( f ) • v B 2 . . .       = λ( f ) • v = f * (v) =       0 Nv A Qv A v B 1 . . .      
, from which we deduce that v A is an eigenvector of N ∈ GL(V A ), and v B i = Qv A /λ( f ) i for i ≥ 1. Moreover, v A = 0 because v intersects e 0 positively. Thus, β is a root of the characteristic polynomial det(tI -N).

The matrix of F k , acting on The characteristic polynomial of F k is equal to x k P(1/x k , x). Hence,

V k = V C ⊕V A ⊕V B 1 ⊕ • • • ⊕V B k , is M F k          0
P(λ( f k ) -k , λ( f k )) = 0.
Moreover, P(0,t) = t l det(tI -N) for some integer l. Hence, the biggest real root of the polynomial P(0,t) is β, and this root is simple. We choose real numbers β -, β + with 1 < β -< β < β + and define δ k to be

δ k = max t∈[β -,β + ]
|P(0,t) -P(1/t k ,t)|.

By construction, lim k→∞ δ k = 0. Hence, for large k the rational function P(1/t k ,t) has a real root β k near β, and lim k→∞ β k = β. Since F k is an element of the Coxeter group W r , it has at most one real root bigger than 1. Hence, β k = λ( f k ). Thus, λ( f k ) converges towards β. Since β is a Pisot number and is not a reciprocal quadratic integer, it is not equal to the dynamical degree of an automorphism. Thus, λ( f k ) = β for all k, and one can extracts a sub-sequence from (λ( f k )) whose members are pairwise distinct. Theorem D implies that the sequence is strictly increasing. We fix integers m, k ≥ 2 and choose a set ∆ ⊂ B(P 2 ) of 2m -1 + (m -2)k distinct points that we denote by ∆ = {q i } 2m-1 i=1 ∪ {a i, j } i=1,...,m-2, j=1,...,k . We choose 2m -1 from these points, that we write p 1 , . . . , p 2m-1 . These are p i = a i,k for i = 1, . . . , m -2, p i = q i for i = m -1, . . . , 2m -1. Then we construct an element h ∈ W ∞ defined by h(e 0 ) = me 0 -(m -1)e(q 1 ) -∑ 2m-1 i=2 e(q i ); h(e(p 1 )) = (m -1)e 0 -(m -2)e(q 1 ) -∑ 2m-1 i=2 e(q i ); h(e(p i )) = e 0e(q 1 )e(q i ) for i = 2, . . . , 2m -1; h(e(q i )) = e(a i,1 ) for i = 1, . . . , m -1; h(e(a i, j )) = e(a i, j+1 ) for i = 1, . . . , m -1, j = 1, . . . , k -1.

h(e(r)) = e(r) for r ∈ B(P 2 ) \ ∆.

Note that h preserves the Z-module W generated by e 0 and the {e(r)} r∈∆ . It corresponds then to an element of the Coxeter group associated to these points. By Uhehara (see [START_REF] Uehara | Rational surface automorphisms with positive entropy[END_REF]), there is an element f m,k ∈ Bir(P 2 k ) for which λ( f m,k ) is equal to the spectral radius λ m,k of the linear transformation h; moreover, f m,k is conjugate to an automorphism of a projective rational surface. Hence, Theorem 7.4 follows from the following lemma in the case of reciprocal quadratic integers. Then, W is invariant by h, and the matrix of h relative to the above basis is where C is a curve with positive self-intersection on some rational surface.

M h =              
Similarly, the modular group (or mapping class group) Mod(g) of the closed, connected, and orientable surface Σ g of genus g ≥ 2 acts by isometries on several metric spaces, for instance on the Teichmüller space, endowed with its Teichmüller metric. 3The comparison of those two isometric actions provides a fruitful analogy between Bir(P 2 k ) and Mod(g) for g ≥ 2 (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Cantat | Proceedings of the sixth European Congress of Mathematics[END_REF]). In this analogy, loxodromic elements f ∈ Bir(P 2 k ) correspond to pseudo-Anosov classes ϕ ∈ Mod(g). The dynamical degree λ( f ) may be compared to the dilatation factor λ(ϕ) of ϕ; both λ( f ) and λ(ϕ) are algebraic numbers: The degree of λ( f ) is bounded from above by the Picard number of a surface on which f is conjugate to an algebraically stable transformation, while the degree of λ(ϕ) is at most 6g -6.

Theorem A may be compared to Franks and Rykken result, according to which a pseudo-Anosov homeomorphism Φ : Σ g → Σ g with a quadratic dilatation factor and with orientable stable and unstable foliations is semi-conjugate, via a ramified cover, to a linear automorphism of a torus (see [START_REF] Franks | Pseudo-Anosov homeomorphisms with quadratic expansion[END_REF]). As for birational transformations, the infimum of λ(ϕ) when ϕ describes the set of pseudo-Anosov classes that are composition of Dehn-multitwists is the Lehmer number (see [START_REF] Christopher | On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number[END_REF]).

8.2.

Another measure of the complexity of a pseudo-Anosov isotopy class ϕ is obtained as follows. According to Thurston and Mostow, the three-dimensional manifold M ϕ = (Σ g × [0, 1]) /(x, 0) = (Φ(x), 1) (where Φ is a diffeomorphism of Σ g in the isotopy class ϕ) admits a unique hyperbolic metric (a riemannian metric of constant curvature -1). The volume of M ϕ with respect to this riemannian metric is a positive real number vol(ϕ); this volume is, up to a bounded multiplicative error, the translation length of ϕ on the Teichmüller space with respect to the Weil-Petersson metric (see [START_REF] Brock | Weil-Petersson translation distance and volumes of mapping tori[END_REF]). Jorgensen and Thurston proved that the set of all volumes vol(M) of all compact hyperbolic three-manifolds is infinite countable, contains accumulation points, and is well ordered (see [START_REF] Benedetti | Lectures on hyperbolic geometry[END_REF]). Thus, the set {vol(ϕ)} where ϕ describes the set of pseudo-Anosov classes of some higher genus surface is well ordered too; moreover, this set is not discrete (consider sequences vol(φ • τ n ) where τ is a Dehn twist). This parallels Theorem C. Moreover, as shown in Section 7.4, accumulation points in Λ(P 2 k ) are obtained by replacing orbits of base points with an infinite length by orbits with finite length. For volumes of hyperbolic manifolds, one obtains accumulation points by Dehn fillings of cusps. Thus, cusps correspond to base points of infinite length in this dictionary. 8.3. It may also be interesting to compare our results to the description obtained by Thurston of the possible topological entropies of multimodal continuous maps of the interval [0, 1] into itself which are postcritically finite (see [START_REF] Thurston | Entropies in dimension one -contains sarah's edits[END_REF]). Those entropies are logarithms of Perron numbers, and all Perron numbers λ > 1 are realized. Thus, in this setting, there is no gap phenomenon similar to the gaps in the dynamical spectrum Λ(P 2 k ).

  yz : zx : xy].

Example 1 . 4 .

 14 Consider the group GL 2 (Z) acting by (monomial) automorphisms of the multiplicative group k * × k * : If A = a b c d

Example 2 . 3 .

 23 Start with the matrix C defined by

of p 2 and maps p 2 onto p 1 .

 1 The bubble space B(X) is the union of all points of all surfaces above X modulo the equivalence relation generated by these identifications.One can also see a point of B(X) as a divisorial valuation having a center in X a closed point.If p is a point of the bubble space, represented by a point p on a surface Y → X, one denotes by E(p) the exceptional divisor of the blow-up of p, and by e(p) its divisor class, viewed as a point in Z X . These classes satisfy e(p) • e(p ) = 0 if p = p and e(p) • e(p) = -1.

  and v • a > 0 for all ample classes a ∈ NS(X)}.

Example 4 . 5 .

 45 Let f be a birational transformation of the plane P 2 k . Let e 0 be the class of a line, viewed as a point in H P 2 k . Then

Remark 4 . 9 .

 49 The constant 28 is the smallest integer m such that m log(λ L ) ≥ 4 log(3), and the occurrence of log(3) comes from the fact that (H P 2 k , dist) is log(3)-hyperbolic in the sense of Gromov (see Remark 4.1).

Corollary 4 . 11 .

 411 An element f ∈ Bir(P 2 k ) is loxodromic if and only if deg( f 400 ) ≥ 3 19 deg( f 200 ).

5. 1 .

 1 Definition of W ∞ . In what follows, e 0 ∈ Z P 2 k is the class of a line. Let p 1 , p 2 , and p 3 be the three points of the plane defined by p 1 = [1 : 0 : 0], p 2 = [0 : 1 : 0], p 3 = [0 : 0 : 1]. The "infinite Weyl group" W ∞ is the group of Z-linear automorphisms of Z P 2 k generated by: (1) the group Sym B(P 2 k ) of permutations of the set B(P 2 k ), that acts on Z P 2 k by sending e 0 to itself and permuting the e(p). (2) the involution σ 0 that sends e 0 onto 2e 0e(p 1 )e(p 2 )e(p 3 ), sends e(p i ) onto e 0e(p 1 )e(p 2 )e(p 3 ) + e(p i ) for i = 1, 2, 3 and fixes e(p) for all p in B(P 2 k )\{p 1 , p 2 , p 3 }.

k.Lemma 5 . 1 . 2 k→ Z P 2 k

 5122 Let k be an algebraically closed field. If f is an element of Bir(P 2 k ), the linear transformation f • : Z P is an element of W ∞ . Proof. If f has degree 1, it is an element of the group Aut(P 2 k ), acts by permutation on B(P 2 k ) and fixes the class e 0 . In other words, the map f → f • provides an embedding of Aut(P 2 k ) into Sym B(P 2 k ) ⊂ W ∞ . If f is the standard quadratic transformation [x : y : z] [yz : xz : xy] it has three base-points, namely p 1 = [1 : 0 : 0], p 2 = [0 : 1 : 0], and p 3 = [0 : 0 : 1]. Moreover, f • acts as σ 0 on e 0 and the e(p i ) for i = 1, 2, 3, and transforms

5. 2 .

 2 Degrees, multiplicity, base-points. Let h be an element of W ∞ . We define the degree deg(h) by deg(h) = e 0 • h(e 0 ); the degree is a positive integer because all elements of W ∞ preserve H P 2 k . Writing h(e 0 ) = deg(h)e 0 -∑ p a(p)e(p), where p runs over B(P 2 k ), we say that p is a base-point of h -1 if a(p) = 0; the integer a(p) is the multiplicity of the base-point. Since h is an isometry of H P 2 k

5. 4 .

 4 Jonquières elements. An element of W ∞ is called a Jonquières element with respect to e 0 -e(p) (or to the point p ∈ B(P 2 k )) if h(e 0 -e(p)) = e 0 -e(p). Jonquières twists f in Bir(P 2 k ) are conjugate to Jonquières elements within Bir(P 2 k ) if k is algebraically closed.

2 k

 2 (q j )where the multiplicities a k and b k are non-negative integers (1 ≤ k ≤ m).Denote by K ∈ Z P the element 3e 0 -∑ 9 i=1 e i , and writeh(K) = ne 0 -m ∑ i=1 c k e k ;to obtain such a formula, we may have to allow new base points, and thus increase the number m. By Lemma 5.3, n is a positive integer, and the c k are non-negative integers.The canonical form Ω vanishes on K. The invariance of Ω gives 3n = K) = e 0e k for all k (because Ω(e 0e k ) = 2 = 0). From Lemma 5.3 we get K • (e 0e k ) = h(K) • h(e 0e k ) > 0 for all indices k:

Lemma 6 . 2 . 2 k

 622 If u ∈ Z P is equal to one of the following: e 0 , e(p 1 ), e 0e(p 1 ), 3e 0 -

Lemma 6 . 3 . 2 k

 632 Let h be a loxodromic element of W ∞ of degree d and dynamical degree λ. Denote by v + and v -the eigenvectors of h in Z P for the eigenvaluesλ and λ -1 , such that v + • e 0 = v -• e 0 = 1. Then (i) d = h(e 0 )•e 0 = e 0 •h -1 (e 0) and this degree is equal to cosh(dist(h(e 0 ), e 0 )) and cosh(dist(e 0 , h -1 (e 0 ))); (ii) log(λ(h)) is the translation length of h, i.e. to the minimum of dist(x, h(x))for x in H P 2 k . (iii) The set of points of H P 2 k that realize the translation length is the axis of h; it coincides with the geodesic line

6. 3 . 2 .

 32 Minimal sets. To simplify the notation, we define

  i j )) = (m -1)e 0 -(m -2)e(p 1 ) -2m-1 ∑ i=2 e(p i ) -m+l ∑ j=1(e 0e(p 1 )e(p i j )).

ke 0 -

 0 (k -1)e(p 1 ) -k+m+l ∑ j=1 e(p i j ) = k(e 0e(p 1 )) + e(p 1 ) -k+m+l ∑ j=1 e(p i j ) intersects non-negatively E . Applying σ Ω , we see that E intersects non-negatively k(e 0 -e(p 1 ))+(m-1)e 0 -(m-2)e(p 1 )--e(p 1 )-e(p i j )),

- 1 •

 1 (e 0 )) = e 0 + l(e 0e(p 1 )) -

7. 1 . 7 . 1 . 1 .Example 7 . 1 .FIGURE 2 .Theorem 7 . 2 .

 171171272 FIGURE 2. Axis and distances.-The blue points are the base point e 0 and its image by h = g f g -1 ; the green points are the projections of these blue points onto the axis of h.

2 k.

 2 The map f• preserves V C ⊕V A ∞ i=1V B i and its matrix can be written as a matrix by blocks as follows:Let v be an eigenvector of f • with eigenvalue λ( f ); such a vector exists (andis isotropic) in Z P Decompose v as v = 0 + v A + v B 1 + v B 2 + . . . with respect to the direct sum V C ⊕V A ∞ i=1

.

  Its characteristic polynomial det(xI -M F k ) is equal to det xI -N 0 -P + x k+1 I -Q x k I  Let P(s,t) be the polynomial function in two variables that is defined by P(s,t) = det

7. 4 . 2 .

 42 Reciprocal quadratic integers. It remains to prove Theorem 7.4 for reciprocal quadratic integers.

Lemma 7 . 7 .

 77 For integers m ≥ 2, the sequence (λ m,k ) k≥1 converges towards the largest root λ m,∞ of P m (x) = x 2 -(m + 1)x + 1, and λ m,k is a Salem number if k is large enough. Proof. Denote by W ⊂ W the sub-Z-module whose basis is e(p 1 ), i,1 ), . . . , e(a 1,k-1 ), m-2 ∑ i=2 e(a i,k-1 ).

8 . 1 .

 81 -2) -(m -3) -(m -1) -(m + 1) 0 . . . to the one done in the proof ofLemma 7.6 shows that its characteristic polynomial det(xI-M h ) is equal to det -1) -(m -3) xm -(m + 1equal to x 2k+2 (x 2 -x(m-1)+1)+x k+1 ((m-1)x 2 -4x+(m-1))+(x 2 -(m-1)x+1)Fixing m, we see that the sequence (λ m,k ) k converges towards λ m,∞ , and that λ m,k = λ m,∞ for k large. Each λ m,k being the spectral radius of an element in a Coxeter group W r k , it is either equal to 1, to a quadratic integer or a Salem number. The set of quadratic integers being discrete, and λ m,k being different from λ m,∞ , the λ m,k are all Salem numbers for k large enough. 8. APPENDIX: MODULAR GROUPS, DILATATIONS, AND VOLUMES The Cremona group Bir(P 2 k ) acts faithfully on the hyperbolic space H P 2

  1.4.2. Well ordered sets. The set Λ(P 2 k ) is a subset of R + and, as such, is totally ordered. The following statement, which follows from Theorem C, asserts that Λ(P 2 k ) is well ordered: Every non-empty subset of Λ(P 2 k ) has a minimum; equivalently, it satisfies the descending chain condition (if ( f n ) n≥0 is a sequence of birational transformations of P 2

  .1.3. Approximation of the points of the axis. Denote by • the Euclidean norm on Z P 2

	k

  • (e 0 ) is at most equal to log(2deg(h 86 )) and, by hyperbolicity of H P 2 + log(9400) + 4 • 86 log(λ( f )) ≤ 18 + 344 log(λ( f )).

	, it is bounded from below by 86 log(λ( f ))+2L-8 log(3). From the last inequality, we get k
	log(2deg(h 86 ) ≤ log(9400) + 5 • 86 log(λ( f ));
	hence,
	2L ≤ 8 log(3) With this upper bound in hands, we estimate
	dist(e 0 , (h)

• e 0 ) ≤ 2L + log(λ( f )) ≤ 18 + 345 log(λ( f ))

  is a birational map; let H d be the quotient of Hd by the equivalence relation for which two triples are equivalent if they are multiple of each other by a non-zero constant. Then (p, q, r) → f is a map from H d to the set of birational transformations of degree ≤ d. Denote by H d,d the subset of H d made of triples of (p, q, r) that give rise to a birational map of degree d exactly; this set is a Zariski open subset of H d , and the projection π

mials of degree d such that f : [x : y : z] → [p : q : r] d : H d,d → Bir d (P 2 k ) is an isomorphism. The map H d → H d n that applies f to f n is a morphism. Since Bir ≤ (P 2 k ) is closed in Bir(P 2 k ) (see Corollary 2.8 of

[START_REF] Blanc | Topologies and structures on the Cremona group[END_REF]

), one deduces that X n is closed.

In this article, log denotes the neperian logarithm.

It also acts on the complex of curves of the surface, a metric space which is Gromov hyperbolic (see[START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF]).

The first author acknowledge support by the Swiss National Science Foundation Grant "Birational Geometry" PP00P2_128422 /1 and both authors acknowledge support by the French National Research Agency Grant "BirPol", ANR-11-JS01-004-01.

Hence, the degree of g n is bounded and, extracting a subsequence, we may assume that the degree of g n does not depend on n: There exists a degree d such that g n is contained in the algebraic variety Bir d (P 2 k ) of elements of Bir d (P 2 ) of degree d, for n ≥ n 0 . Junyi Xie proved in [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF] that the dynamical degree

+∞[ is lower semi-continous for the Zariski topology. In other words, the level subsets

. As the sequence of dynamical degrees (λ M n ) is strictly decreasing, we deduce that the sequence of Zariski closed sets L(λ M n ) ⊂ Bir d (P 2 k ) decreases strictly. Since the Zariski topology is Noetherian, the sequence (λ n ) contains only finitely many terms. 7.2. Small Pisot numbers and spectral gaps. Theorem 7.2 implies that there are gaps in the dynamical spectrum on the right of every dynamical degree; the first gap occurs after the Lehmer number λ L , the first gap on the right of a Pisot number occurs on the right of the plastic number λ P . If one restricts the study to dynamical degrees in the Pisot family, one can prove the following properties as a corollary of our previous results, known facts on Pisot numbers (see [START_REF]Pisot and Salem numbers[END_REF]), and a systematic study of quadratic birational transformations of the plane (see [START_REF] Diller | Cremona transformations, surface automorphisms, and plane cubics[END_REF] and [START_REF] Blanc | Small dynamical degrees and pisot numbers below the golden mean[END_REF] for assertion (b)):

(a) The Golden mean λ G is the smallest accumulation point in the set of Pisot numbers; it is an accumulation point from below, and from above. (b) All Pisot numbers below the Golden mean are realized as dynamical degrees of quadratic birational transformations of the plane. (c) There is an ε > 0 such that ]λ G , λ G + ε[ does not contain any dynamical degree; hence, the infimum of the set {λ ∈ Pis | λ is not a dynamical degree} is equal to λ G .

7.3.

The dynamical spectrum is closed. We now prove that the dynamical spectrum Λ(P 2 k ) is closed if k is uncountable and algebraically closed. Let ( f i ) i≥0 be a sequence of birational transformations of the projective plane such that λ( f i ) converges towards a real number λ ∞ . Our goal is to construct an element h in Bir(P 2 k ) such that λ(h) = λ ∞ . Thus, we may (and do) assume that the numbers λ i := λ( f i ) form a strictly increasing sequence converging to λ ∞ > 1. Theorem C applies: Changing each f i into a conjugate element of Bir(P 2 k ), we assume that deg(

where D depends on sup(λ i ) but does not depend on i. Replacing ( f i ) by a subsequence, we assume that the f i have the same degree d (with 2 ≤ d ≤ D).