We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the wellknown subgroup of the Cremona group made up of the polynomial automorphisms of C 2 . Then we deal with the classification of birational maps and some applications (Tits alternative, nonsimplicity...) Since any birational map can be written as a composition of quadratic birational maps up to an automorphism of the complex projective plane, we spend time on these special maps. Some questions of group theory are evoked: the classification of the finite subgroups of the Cremona group and related problems, the description of the automorphisms of the Cremona group and the representations of some lattices in the Cremona group. The description of the centralizers of discrete dynamical systems is an important problem in real and complex dynamic, we make a state of art of this problem in the Cremona group.

Let Z be a compact complex surface which carries an automorphism f of positive topological entropy. Either the Kodaira dimension of Z is zero and f is conjugate to an automorphism on the unique minimal model of Z which is either a torus, or a K3 surface, or an Enriques surface, or Z is a non-minimal rational surface and f is conjugate to a birational map of the complex projective plane. We deal with results obtained in this last case: construction of such automorphisms, dynamical properties (rotation domains...) are touched on. vi vii Dear Pat, You came upon me carving some kind of little figure out of wood and you said: "Why don't you make something for me ?" I asked you what you wanted, and you said, "A box." "What for ?" "To put things in." "What things ?" "Whatever you have," you said.

Well, here's your box. Nearly everything I have is in it, and it is not full. Pain and excitement are in it, and feeling good or bad and evil thoughts and good thoughts -the pleasure of design and some despair and the indescribable joy of creation.

INTRODUCTION

The study of the Cremona group Bir(P 2 ), i.e. the group of birational maps from P 2 (C) into itself, started in the XIXth century. The subject has known a lot of developments since the beginning of the XXIth century; we will deal with these most recent results. Unfortunately we will not be exhaustive.

We introduce a special subgroup of the Cremona group: the group Aut(C 2 ) of polynomial automorphisms of the plane. This subgroup has been the object of many studies along the XXth century. It is more rigid and so, in some sense, easier to understand. Indeed Aut(C 2 ) has a structure of amalgamated product so acts non trivially on a tree (Bass-Serre theory); this allows to give properties satisfied by polynomial automorphisms. There are a lot of different proofs of the structure of amalgamated product. We present one of them due to Lamy in Chapter 2; this one is particularly interesting for us because Lamy considers Aut(C 2 ) as a subgroup of the Cremona group and works in Bir(P 2 ) (see [START_REF] Lamy | Une preuve géométrique du théorème de Jung[END_REF]).

A lot of dynamical aspects of a birational map are controlled by its action on the cohomology H 2 (X , R) of a "good" birational model X of P 2 (C). The construction of such a model is not canonical; so Manin has introduced the space of infinite dimension of all cohomological classes of all birational models of P 2 (C). Its completion for the bilinear form induced by the cup product defines a real Hilbert space Z(P 2 ) endowed with an intersection form. One of the two sheets of the hyperboloid {[D] ∈ Z(P 2 ) | [D] 2 = 1} owns a metric which yields a hyperbolic space (Gromov sense); let us denote it by H Z . We get a faithful representation of Bir(P 2 ) into Isom(H Z ). The classification of isometries into three types has an algrebraic-geometric meaning and induces a classification of birational maps ( [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]); it is strongly related to the classification of Diller and Favre ([77]) built on the degree growth of the sequence {deg f n } n∈N . Such a sequence either is bounded (elliptic maps), or grows linearly (de Jonquières twists), or grows quadratically (Halphen twists), or grows exponentially (hyperbolic maps). We give some applications of this construction: Bir(P 2 ) satisfies the Tits alternative ( [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]) and is not simple ( [START_REF] Cantat | Normal subgroups in the cremona group[END_REF]).

One of the oldest results about the Cremona group is that any birational map of the complex projective plane is a product of quadratic birational maps up to an automorphism of the complex projective plane. It is thus natural to study the quadratic birational maps and also the cubic ones in order to make in evidence some direct differences ( [START_REF] Cerveau | Transformations birationnelles de petit degré[END_REF]). In Chapter 4 we present a stratification of the set of quadratic birational maps. We recall that this set is smooth. We also INTRODUCTION give a geometric description of the quadratic birational maps and a criterion of birationality for quadratic rational maps. We then deal with cubic birational maps; the set of such maps is not smooth anymore.

While Noether was interested in the decomposition of the birational maps, some people studied finite subgroups of the Cremona group ( [START_REF] Bertini | Ricerche sulle trasformazioni univoche involutorie nel piano[END_REF][START_REF] Kantor | Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene[END_REF][START_REF] Wiman | Zur theorie der endlichen gruppen von birazionalen transformationen in der ebene[END_REF]). A strongly related problem is the characterization of the birational maps that preserve curves of positive genus. In Chapter 5 we give some statements and ideas of proof on this subject; people recently went back to this domain [START_REF] Bayle | Birational involutions of P 2[END_REF][START_REF] Beauville | p-elementary subgroups of the Cremona group[END_REF][START_REF] Beauville | On Cremona transformations of prime order[END_REF][START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF][START_REF] De Fernex | On planar Cremona maps of prime order[END_REF][START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF][START_REF] Blanc | Sur un théorème de Castelnuovo[END_REF][START_REF] Pan | Sur le sous-groupe de décomposition d'une courbe irrationnelle dans le groupe de Cremona du plan[END_REF][START_REF] Diller | Invariant curves for birational surface maps[END_REF], providing new results about the number of conjugacy classes in Bir(P 2 ) of birational maps of order n for example ( [START_REF] De Fernex | On planar Cremona maps of prime order[END_REF][START_REF] Blanc | The number of conjugacy classes of elements of the Cremona group of some given finite order[END_REF]). We also present another construction of birational involutions related to holomorphic foliations of degree 2 on P 2 (C) (see [START_REF] Cerveau | Feuilletages et transformations périodiques[END_REF]).

A classical question in group theory is the following: let G be a group, what is the automorphisms group Aut(G) of G ? For example, the automorphisms of PGL n (C) are, for n ≥ 3, obtained from the inner automorphisms, the involution u → t u -1 and the automorphisms of the field C. A similar result holds for the affine group of the complex line C; we give a proof of it in Chapter 6. We also give an idea of the description of the automorphisms group of Aut(C 2 ), resp. Bir(P 2 ) (see [START_REF] Déserti | Sur le groupe des automorphismes polynomiaux du plan affine[END_REF][START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF]).

Margulis studies linear representations of the lattices of simple, real Lie groups of real rank strictly greater than 1; Zimmer suggests to generalize it to non-linear ones. In that spirit we expose the representations of the classical lattices SL n (Z) into the Cremona group ( [START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF]). We see, in Chapter 7, that there is a description of embeddings of SL 3 (Z) into Bir(P 2 ) (up to conjugation such an embedding is the canonical embedding or the involution u → t u -1 ); therefore SL n (Z) cannot be embedded as soon as n ≥ 4.

The description of the centralizers of discrete dynamical systems is an important problem in dynamic; it allows to measure algebraically the chaos of such a system. In Chapter 8 we describe the centralizer of birational maps. Methods are different for elliptic maps of infinite order, de Jonquières twists, Halphen twists and hyperbolic maps. In the first case, we can give explicit formulas ( [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]); in particular the centralizer is uncountable. In the second case, we do not always have explicit formulas ( [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF])... When f is an Halphen twist, the situation is different: the centralizer contains a subgroup of finite index which is abelian, free and of rank ≤ 8 (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Gizatullin | Rational G-surfaces[END_REF]). Finally for a hyperbolic map f the centralizer is an extension of a cyclic group by a finite group ( [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]).

The study of automorphisms of compact complex surfaces with positive entropy is strongly related with birational maps of the complex projective plane. Let f be an automorphism of a compact complex surface S with positive entropy; then either f is birationally conjugate to a birational map of the complex projective plane, or the Kodaira dimension of S is zero and then f is conjugate to an automorphism of the unique minimal model of S which has to be a torus, a K3 surface or an Enriques surface ( [START_REF] Cantat | Dynamique des automorphismes des surfaces projectives complexes[END_REF]). The case of K3 surfaces has been studied in [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF][START_REF] Mcmullen | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF][START_REF] Oguiso | The third smallest Salem number in automorphisms of K3 surfaces. In Algebraic geometry in East Asia-Seoul[END_REF][START_REF] Silverman | Rational points on K3 surfaces: a new canonical height[END_REF][START_REF] Wang | Rational points and canonical heights on K3-surfaces in P 1 × P 1 × P 1[END_REF]. One of the first example given in the context of rational surfaces is due to Coble ([61]). Let us mention another well-known example: let us consider Λ = Z[i] and E = C/Λ. The group SL 2 (Λ) acts linearly on C 2 and preserves the lattice Λ × Λ; then any element A of SL 2 (Λ) induces an automorphism f A on E × E which commutes with ι(x, y) = (ix, iy). The automorphism f A lifts to an automorphism f A on the desingularization of INTRODUCTION Enfin merci à l'Université de Bâle, à l'Université Paris 7 et à l'IMPA pour leur accueil.

CHAPTER 1 FIRST STEPS 1.1. Divisors and intersection theory

Let X be an algebraic variety. A prime divisor on X is an irreducible closed subset of X of codimension 1.

Examples 1.1.1. -

• If X is a surface, the prime divisors of X are the irreducible curves that lie on it.

• If X = P n (C) then prime divisors are given by the zero locus of irreducible homogeneous polynomials.

A Weil divisor on X is a formal finite sum of prime divisors with integer coefficients m ∑ i=1 a i D i , m ∈ N, a i ∈ Z, D i prime divisor of X .

Let us denote by Div(X ) the set of all Weil divisors on X . If f ∈ C(X ) * is a rational function and D a prime divisor we can define the multiplicity ν f (D) of f at D as follows:

• ν f (D) = k > 0 if f vanishes on D at the order k;

• ν f (D) = -k if f has a pole of order k on D;

• and ν f (D) = 0 otherwise. To any rational function f ∈ C(X ) * we associate a divisor div( f ) ∈ Div(X ) defined by

div( f ) = ∑ D prime divisor ν f (D) D.
Note that div( f ) ∈ Div(X ) since ν f (D) is zero for all but finitely many D. Divisors obtained like that are called principal divisors. As div( f g) = div( f )+ div(g) the set of principal divisors is a subgroup of Div(X ).

Two divisors D, D ′ on an algebraic variety are linearly equivalent if D -D ′ is a principal divisor. The set of equivalence classes corresponds to the quotient of Div(X ) by the subgroup of principal divisors; when X is smooth this quotient is isomorphic to the Picard group Pic(X ). (1) Example 1.1.2. -Let us see that Pic(P n ) = ZH where H is the divisor of an hyperplane.

Consider the homorphism of groups given by

Θ : Div(P n ) → Z, D of degree d → d.
Let us first remark that its kernel is the subgroup of principal divisors. Let D = ∑ a i D i be a divisor in the kernel, where each D i is a prime divisor given by an homogeneous polynomial f i ∈ C[x 0 , . . . , x n ] of some degree d i . Since ∑ a i d i = 0, f = ∏ f a i i belongs to C(P n ) * . We have by construction D = div( f ) so D is a principal divisor. Conversely any principal divisor is equal to div( f ) where f = g/h for some homogeneous polynomials g, h of the same degree. Thus any principal divisor belongs to the kernel.

Since Pic(P n ) is the quotient of Div(P n ) by the subgroup of principal divisors, we get, by restricting Θ to the quotient, an isomorphism Pic(P n ) → Z. We conclude by noting that an hyperplane is sent on 1.

We can define the notion of intersection. Given a point p in a smooth algebraic variety X of dimension n we say that π : Y → X is a blow-up of p ∈ X if Y is a smooth variety, if

π |Y \{π -1 (p)} : Y \ {π -1 (p)} → X \ {p}
is an isomorphism and if π -1 (p) ≃ P n-1 (C). Set E = π -1 (p); E is called the exceptional divisor.

If π : Y → X and π ′ : Y ′ → X are two blow-ups of the same point p then there exists an isomorphism ϕ : Y → Y ′ such that π = π ′ ϕ. So we can speak about the blow-up of p ∈ X . Remark 1.1.4. -When n = 1, π is an isomorphism but when n ≥ 2 it is not: it contracts E = π -1 (p) ≃ P n-1 (C) onto the point p.

Example 1.1.5. -We now describe the blow-up of (0 : 0 : 1) in P 2 (C). Let us work in the affine chart z = 1, i.e. in C 2 with coordinates (x, y). Set Bl (0,0) P 2 = (x, y), (u : v) ∈ C 2 × P 1 xv = yu .

The morphism π : Bl (0,0) P 2 → C 2 given by the first projection is the blow-up of (0, 0):

• First we can note that π -1 (0, 0) = (0, 0), (u : v) (u : v) ∈ P 1 so E = π -1 (0, 0) is isomorphic to P 1 ;

• Let q = (x, y) be a point of C 2 \ {(0, 0)}. We have π -1 (q) = (x, y), (x : y) ∈ Bl (0,0) P 2 \ E so π |Bl (0,0) P 2 \E is an isomorphism, the inverse being (x, y) → (x, y), (x : y) .

How to compute ? In affine charts: let U (resp. V ) be the open subset of Bl (0,0) P 2 where v = 0 (resp. u = 0). The open subset U is isomorphic to C 2 via the map

C 2 → U,
(y, u) → (yu, y), (u : 1) ;

we can see that V is also isomorphic to C 2 . In local coordinates we can define the blow-up by

C 2 → C 2 , (y, u) → (yu, y), E is described by {y = 0} C 2 → C 2 , (x, v) → (x, xv), E is described by {x = 0}
Let π : Bl p S → S be the blow-up of the point p ∈ S. The morphism π induces a map π * from Pic(S) to Pic(Bl p S) which sends a curve C on π -1 (C). If C ⊂ S is irreducible, the strict transform C of C is C = π -1 (C \ {p}).

We now recall what is the multiplicity of a curve at a point. If C ⊂ S is a curve and p is a point of S, we can define the multiplicity m p (C) of C at p. Let m be the maximal ideal of the ring of functions O p,S (2) . Let f be a local equation of C; then m p (C) can be defined as the integer k such that f ∈ m k \ m k+1 . For example if S is rational, we can find a neighborhood U of p in S with U ⊂ C 2 , we can assume that p = (0, 0) in this affine neighborhood, and C is described by the equation n ∑ i=1 P i (x, y) = 0, P i homogeneous polynomials of degree i in two variables.

The multiplicity m p (C) is equal to the lowest i such that P i is not equal to 0. We have • m p (C) ≥ 0;

• m p (C) = 0 if and only if p ∈ C;

• m p (C) = 1 if and only if p is a smooth point of C. Assume that C and D are distinct curves with no common component then we define an integer (C • D) p which counts the intersection of C and D at p:

• it is equal to 0 if either C or D does not pass through p;

• otherwise let f , resp. g be some local equations of C, resp. D in a neighborhood of p and define (C • D) p to be the dimension of O p,S /( f , g).

This number is related to C • D by the following statement. Let C be a curve in S, p = (0, 0) ∈ S. Let us take local coordinates x, y at p and let us set k = m p (C); the curve C is thus given by P k (x, y) + P k+1 (x, y) + . . . + P r (x, y) = 0, where P i denotes a homogeneous polynomial of degree i. The blow-up of p can be viewed as (u, v) → (uv, v); the pull-back of C is given by v k p k (u, 1) + vp k+1 (u, 1) + . . . + v r-k p r (x, y) = 0, i.e. it decomposes into k times the exceptional divisor E = π -1 (0, 0) = (v = 0) and the strict transform. So we have the following statement: We also have the following statement. Proposition 1. 1.8 ([115], Chapter V, Proposition 3.2). -Let S be a smooth surface, let p be a point of S and let π : Bl p S → S be the blow-up of p. We denote by E ⊂ Bl p S the curve π -1 (p) ≃ P 1 . We have Pic(Bl p S) = π * Pic(S) + ZE. If X is an algebraic variety, the nef cone Nef(X ) is the cone of divisors D such that D •C ≥ 0 for any curve C in X .

The intersection form on Bl

Birational maps

A rational map from P 2 (C) into itself is a map of the following type f : P 2 (C) P 2 (C), (x : y : z) ( f 0 (x, y, z) : f 1 (x, y, z) : f 2 (x, y, z))

where the f i 's are homogeneous polynomials of the same degree without common factor.

A birational map from P 2 (C) into itself is a rational map

f : P 2 (C) P 2 (C)
such that there exists a rational map ψ from P 2 (C) into itself satisfying f

• ψ = ψ • f = id.
The Cremona group Bir(P 2 ) is the group of birational maps from P 2 (C) into itself. The elements of the Cremona group are also called Cremona transformations. An element f of Bir(P 2 ) is equivalently given by (x, y) → ( f 1 (x, y), f 2 (x, y)) where C( f 1 , f 2 ) = C(x 1 , x 2 ), i.e.

Bir(P 2 ) ≃ Aut C (C(x, y)).

The degree of f : (x : y : x) ( f 0 (x, y, z) : f 1 (x, y, z) : f 2 (x, y, z)) ∈ Bir(P 2 ) is equal to the degree of the f i 's: deg f = deg f i .

Examples 1.2.1. -

• Every automorphism f : (x : y : z) (a 0 x + a 1 y + a 2 z : a 3 x + a 4 y + a 5 z : a 6 x + a 7 y + a 8 z), det(a i ) = 0 of the complex projective plane is a birational map. The degree of f is equal to 1. In other words Aut(P 2 ) = PGL 3 (C) ⊂ Bir(P 2 ). • The map σ : (x : y : z) (yz : xz : xy) is rational; we can verify that σ • σ = id, i.e. σ is an involution so σ is birational. We have: deg σ = 2.

Definitions. -Let f : (x : y : z) ( f 0 (x, y, z) : f 1 (x, y, z) : f 2 (x, y, z)) be a birational map of P 2 (C); then:

• the indeterminacy locus of f , denoted by Ind f , is the set

m ∈ P 2 (C) f 0 (m) = f 1 (m) = f 2 (m) = 0
• and the exceptional locus Exc f of f is given by • Let us denote by σ the map defined by σ : (x : y : z) (yz : xz : xy); we note that Exc σ = x = 0, y = 0, z = 0 , Ind σ = (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) .

m ∈ P 2 (C) det jac( f ) (m) = 0 .
• If ρ is the following map ρ : (x : y : z) (xy : z 2 : yz), then Exc ρ = y = 0, z = 0 & Ind ρ = (1 : 0 : 0), (0 : 1 : 0) .

Definition. -Let us recall that if X is an irreducible variety and Y a variety, a rational map f : X Y is a morphism from a non-empty open subset U of X to Y .

Let f : P 2 (C) P 2 (C) be the birational map given by (x : y : z) ( f 0 (x, y, z) : f 1 (x : y : z) : f 2 (x, y, z))

where the f i 's are homogeneous polynomials of the same degree ν, and without common factor.

The linear system Λ f of f is the pre-image of the linear system of lines of P 2 (C); it is the system of curves given by ∑ a i f i = 0 for (a 0 : a 1 : a 2 ) in P 2 (C). Let us remark that if A is an automorphism of P 2 (C), then Λ f = Λ A f . The degree of the curves of Λ f is ν, i.e. it coincides with the degree of f . If f has one point of indeterminacy p 1 , let us denote by π 1 : Bl p 1 P 2 → P 2 (C) the blow-up of p 1 and E 1 the exceptional divisor. The map ϕ 1 = f • π 1 is a birational map from Bl p 1 P 2 into P 2 (C). If ϕ 1 is not defined at one point p 2 then we blow it up via π 2 : Bl p 1 ,p 2 P 2 → P 2 (C); set E 2 = π -1 2 (p 2 ). Again the map ϕ 2 = ϕ 1 • π 1 : Bl p 1 ,p 2 P 2 P 2 (C) is a birational map. We continue the same processus until ϕ r becomes a morphism. The p i 's are called base-points of f or base-points of Λ f . Let us describe Pic(Bl p 1 ,...,p r P 2 ). First Pic(P 2 ) = ZL where L is the divisor of a line (Example 1. 1.2). Set E i = (π i+1 . . . π r ) * E i and ℓ = (π 1 . . . π r ) * (L). Applying r times Proposition 1.1.8 we get Pic(Bl p 1 ,...,p r P 2 ) = Zℓ ⊕ ZE 1 ⊕ . . . ⊕ ZE r . Moreover all elements of the basis (ℓ, E 1 , . . . , E r ) satisfy the following relations

ℓ 2 = ℓ • ℓ = 1, E 2 i = -1, E i • E j = 0 ∀ 1 ≤ i = j ≤ r, E i • ℓ = 0 ∀1 ≤ i ≤ r.
The linear system Λ f consists of curves of degree ν all passing through the p i 's with multiplicity m i . Set E i = (π i+1 . . . π r ) * E i . Applying r times Lemma 1.1.7 the elements of Λ ϕ r are equivalent to νL -∑ r i=1 m i E i where L is a generic line. Remark that these curves have selfintersection

ν 2 - r ∑ i=1 m 2 i .
All members of a linear system are linearly equivalent and the dimension of Λ ϕ r is 2 so the selfintersection has to be non-negative. This implies that the number r exists, i.e. the number of base-points of f is finite. Let us note that by construction the map ϕ r is a birational morphism from Bl p 1 ,...,p r P 2 to P 2 (C) which is the blow-up of the points of f -1 ; we have the following diagram

S ′ π r •...•π 1 Ð Ð ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ϕ r 0 0 ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ S f G G ❴ ❴ ❴ ❴ ❴ ❴ ❴

S

The linear system Λ f of f corresponds to the strict pull-back of the system O P 2 (1) of lines of P 2 (C) by ϕ. The system Λ ϕ r which is its image on Bl p 1 ,...,p r P 2 is the strict pull-back of the system O P 2 (1). Let us consider a general line L of P 2 (C) which does not pass through the p i 's; its pull-back ϕ -1 r (L) corresponds to a smooth curve on Bl p 1 ,...,p r P 2 which has selfintersection -1 and genus 0. We thus have (ϕ -1 r (L)) 2 = 1 and by adjunction formula ϕ -1 r (L) • K Bl p 1 ,...,pr P 2 = -3. Since the elements of Λ ϕ r are equivalent to νLr ∑ i=1 m i E i and since K Bl p 1 ,...,pr P 2 = -3L + ∑ r i=1 E i we have

r ∑ i=1 m i = 3(ν -1), r ∑ i=1 m 2 i = ν 2 -1.
In particular if ν = 1 the map f has no base-points. If ν = 2 then r = 3 and m 1 = m 2 = m 3 = 1. As we will see later (Chapter 4) it doesn't mean that "there is one quadratic birational map".

So there are three standard ways to describe a Cremona map • the explicit formula (x : y : z) ( f 0 (x, y, z) : f 1 (x, y, z) : f 2 (x, yz)) where the f i 's are homogeneous polynomials of the same degree and without common factor; • the data of the degree of the map, the base-points of the map and their multiplicity (it defines a map up to an automorphism); • the base-points of π and the curves contracted by η with the notations of Theorem 1.3.1 (it defines a map up to an automorphism).

Zariski's theorem

Let us recall the following statement.

Theorem 1.3.1 (Zariski, 1944

). -Let S, S be two smooth projective surfaces and let f : S S be a birational map. There exists a smooth projective surface S ′ and two sequences of blow-ups π 1 : S ′ → S, π 2 : S ′ → S such that f = π 2 π -1 is the composition of two sequences of blow-ups

1 S ′ π 1 Ð Ð ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ π 2 0 0 ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ S f G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ S Example 1.3.
P 2 (C) L AB E B E A E C L AC L BC L BC L AC E A E B L AB P 2 (C) C L AC L AB A B σ L BC π 1 π 2 E C
with A = (1 : 0 : 0), B = (0 : 1 : 0), C = (0 : 0 : 1), L AB (resp. L AC , resp. L BC ) the line passing through A and B (resp. A and C, resp. B and C) E A (resp. E B , resp. E C ) the exceptional divisor obtained by blowing up A (resp. B, resp. C) and L AB (resp. L AC , resp. L BC ) the strict transform of L AB (resp. L AC , resp. L BC ).

There are two steps in the proof of Theorem 1.3.1. The first one is to compose f with a sequence of blow-ups in order to remove all the points of indeterminacy (remark that this step is also possible with a rational map and can be adapted in higher dimension); we thus have

S ′ π 1 Ð Ð ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ f 0 0 ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ S f G G ❴ ❴ ❴ ❴ ❴ ❴ ❴

S

The second step is specific to the case of birational map between two surfaces and can be stated as follows.

Proposition 1.3.3 ([136]

). -Let f : S → S ′ be a birational morphism between two surfaces S and S ′ . Assume that f -1 is not defined at a point p of S ′ ; then f can be written πφ where π : Bl p S ′ → S ′ is the blow-up of p ∈ S ′ and φ a birational morphism from S to Bl p S ′

Bl p S ′ π 3 3 ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ S φ a a ④ ④ ④ ④ ④ ④ ④ ④ f G G S ′
Before giving the proof of this result let us give a useful Lemma.

Lemma 1.3.4 ([15]

). -Let f : S S ′ be a birational map between two surfaces S and S ′ .

If there exists a point p ∈ S such that f is not defined at p there exists a curve C on S ′ such that f -1 (C ) = p.

Proof of the Proposition 1. 3.3. -Assume that φ = π -1 f is not a morphism. Let m be a point of S such that φ is not defined at m. On the one hand f (m) = p and f is not locally invertible at m, on the other hand there exists a curve in Bl p S ′ contracted on m by φ -1 (Lemma 1.3.4). This curve is necessarily the exceptional divisor E obtained by blowing up.

Let q 1 , q 2 be two different points of E at which φ -1 is well defined and let C 1 , C 2 be two germs of smooth curves transverse to E. Then π(C 1 ) and π(C 2 ) are two germs of smooth curve transverse at p which are the image by f of two germs of curves at m. The differential of f at m is thus of rank 2: contradiction with the fact that f is not locally invertible at m.

φ -1 (C 2 ) f π φ q 2 E C 1 C 2 π(C 2 ) m φ -1 (C 1 ) p = f (m) π(C 1 ) q 1 S S ′

S

We say that f : S P 2 (C) is induced by a polynomial automorphism (3) -Let S be a surface. Let f be a birational map from S to P 2 (C) induced by a polynomial automorphism of C 2 . Assume that f is not a morphism. Then

of C 2 if • S = C 2 ∪ D where D is a union of irreducible curves, D is called divisor at infinity; • P 2 (C) = C 2 ∪ L
• f has a unique point of indeterminacy p 1 on the divisor at infinity;

• f has base-points p 2 , . . ., p s and for all i = 2, . . . , s the point p i is on the exceptional divisor obtained by blowing up p i-1 ; • each irreducible curve contained in the divisor at infinity is contracted on a point by f ;

• the first curve contracted by π 2 is the strict transform of a curve contained in the divisor at infinity; • in particular if S = P 2 (C) the first curve contracted by π 2 is the transform of the line at infinity (in the domain).

Proof. -According to Lemma 1.3.4 if p is a point of indeterminacy of f there exists a curve contracted by f -1 on p. As f is induced by an automorphism of C 2 the unique curve on P 2 (C) which can be blown down is the line at infinity so f has at most one point of indeterminacy. As f is not a morphism, it has exactly one.

3. A polynomial automorphism of C 2 is a bijective application of the following type

f : C 2 → C 2 , (x, y) → ( f 1 (x, y), f 2 (x, y)), f i ∈ C[x, y].
The second assertion is obtained by induction. Each irreducible curve contained in the divisor at infinity is either contracted on a point, or sent on the line at infinity in P 2 (C). Since f -1 contracts the line at infinity on a point the second eventuality is excluded.

According to Theorem 1.3.1 we have

S ′ π 1 Ð Ð ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ π 2 4 4 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ S f G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
where S ′ is a smooth projective surface and π 1 : S ′ → S, π 2 : S ′ → P 2 (C) are two sequences of blow-ups. The divisor at infinity in S ′ is the union of • a divisor of self-intersection -1 obtained by blowing-up p s ,

• the other divisors, all of self-intersection ≤ -2, produced in the sequence of blow-ups,

• and the strict transform of the divisor at infinity in S ′ . The first curve contracted by π 2 is of self-intersection -1 and cannot be the last curve produced by π 1 (otherwise p s is not a point of indeterminacy); so the first curve contracted by π 2 is the strict transform of a curve contained in the divisor at infinity.

The last assertion follows from the previous one.

CHAPTER 2 SOME SUBGROUPS OF THE CREMONA GROUP

A special subgroup: the group of polynomial automorphisms of the plane

A polynomial automorphism of C 2 is a bijective application of the following type

f : C 2 → C 2 , (x, y) → ( f 1 (x, y), f 2 (x, y)), f i ∈ C[x, y]. The degree of f = ( f 1 , f 2 ) is defined by deg f = max(deg f 1 , deg f 2 ). Note that deg ψ f ψ -1 = deg f in general so we define the first dynamical degree of f d( f ) = lim(deg f n ) 1/n
which is invariant under conjugacy (1) . The set of the polynomial automorphisms is a group denoted by Aut(C

C 2 → C 2 , (x, y) → (a 1 x + b 1 y + c 1 , a 2 x + b 2 y + c 2 ), a i , b i , c i ∈ C, a 1 b 2 -a 2 b 1 = 0 is an automorphism of C 2 .
The set of all these maps is the affine group A. • The map

C 2 → C 2 , (x, y) → (αx + P(y), βy + γ), α, β, γ ∈ C, αβ = 0, P ∈ C[y]
is an automorphism of C 2 . The set of all these maps is a group, the elementary group E.

• Of course

S = A ∩ E = (a 1 x + b 1 y + c 1 , b 2 y + c 2 ) a i , b i , c i ∈ C, a 1 b 2 = 0 is a subgroup of Aut(C 2 ).
The group Aut(C 2 ) has a very special structure. In other words A and E generate Aut(C 2 ) and each element f in Aut(C 2 ) \ S can be written as follows

f = (a 1 )e 1 . . . a n (e n ), e i ∈ E \ A, a i ∈ A \ E.
Moreover this decomposition is unique modulo the following relations There are many proofs of Theorem 2.1.2; you can find a "historical review" in [START_REF] Lamy | Une preuve géométrique du théorème de Jung[END_REF]. We will now give an idea of the proof done in [START_REF] Lamy | Une preuve géométrique du théorème de Jung[END_REF] and give details in §2.2. Let

a i e i = (a i s)(s -1 e i ), e i-1 a i = (e i-1 s ′ )(s ′-1 a i ), s, s ′ ∈ S.
f : (x, y) → ( f 1 (x, y), f 2 (x, y))
be a polynomial automorphism of C 2 of degree ν. We can view f as a birational map:

f : P 2 (C) P 2 (C), (x : y : z) z ν f 1 x z , y z : z ν f 2 x z , y z : z ν .
Lamy proved there exists ϕ ∈ Bir(P 2 ) induced by a polynomial automorphism of C 2 such that # Ind f ϕ -1 < # Ind f ; more precisely "ϕ comes from an elementary automorphism". Proceeding recursively we obtain a map g such that #Ind f = 0, in other words an automorphism of P 2 (C) which gives an affine automorphism.

According to Bass-Serre theory ( [START_REF] Serre | Arbres, amalgames, SL 2[END_REF]) we can canonically associate a tree to any amalgamated product. Let T be the tree associated to Aut(C 2 ):

• the disjoint union of Aut(C 2 )/E and Aut(C 2 )/A is the set of vertices,

• Aut(C 2 )/S is the set of edges. All these quotients must be understood as being left cosets; the cosets of f ∈ Aut(C 2 ) are noted respectively f E, f A, and f S. By definition the edge hS links the vertices f A and gE if hS ⊂ f A and hS ⊂ gE (and so f A = hA and gE = hE). In this way we obtain a graph; the fact that A and E are amalgamated along S is equivalent to the fact that T is a tree ( [START_REF] Serre | Arbres, amalgames, SL 2[END_REF]). This tree is uniquely characterized (up to isomorphism) by the following property: there exists an action of Aut(C 2 ) on T , such that the fundamental domain of this action is a segment, i.e. an edge and two vertices, with E and A equal to the stabilizers of the vertices of this segment (and so S is the stabilizer of the entire segment). This action is simply the left translation: From a dynamical point of view affine automorphisms and elementary automorphisms are simple. Nevertheless there exist some elements in Aut(C 2 ) with a rich dynamic; this is the case of Hénon automorphisms, automorphisms of the type ϕg 1 . . .

g(hS) = (g • h)S.
g p ϕ -1 with ϕ ∈ Aut(C 2 ), g i = (y, P i (y) -δ i x), P i ∈ C[y], deg P i ≥ 2, δ i ∈ C * . Note that g i = ∈ A\E (y, x) ∈ E\A
(-δ i x + P i (y), y) . Using Jung's theorem, Friedland and Milnor proved the following statement.

Proposition 2.1.5 ([97]

). -Let f be an element of Aut(C 2 ).

Either f is conjugate to an element of E, or f is a Hénon automorphism.

If f belongs to E, then d( f ) = 1. If f = g 1 . . . g p with g i = (y, P i (y) -δ i x), then d( f ) = p ∏ i=1 deg g i ≥ 2. Then we have • d( f ) = 1 if and only if f is conjugate to an element of E; • d( f ) > 1 if and only if f is a Hénon automorphism.
Hénon automorphisms and elementary automorphisms are very different:

• Hénon automorphisms: no invariant rational fibration ( [START_REF] Brunella | Minimal models of foliated algebraic surfaces[END_REF]), countable centralizer ( [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]), infinite number of hyperbolic periodic points; • Elementary automorphisms: invariant rational fibration, uncountable centralizer.

Proof of JUNG'S theorem

Assume that Φ is a polynomial automorphism of C 2 of degree n

Φ : (x, y) → (Φ 1 (x, y), Φ 2 (x, y)), Φ i ∈ C[x, y];
we can extend Φ to a birational map still denoted by Φ Φ :

(x : y : z) z n Φ 1 x z , y z : z n Φ 2 x z , y z : z n .
The line at infinity in P 2 (C) is z = 0. The map Φ : P 2 (C) P 2 (C) has a unique point of indeterminacy which is on the line at infinity (Lemma 1.3.5). We can assume, up to conjugation by an affine automorphism, that this point is (1 : 0 : 0) (of course this conjugacy doesn't change the number of base-points of Φ). We will show that there exists ϕ : P 2 (C) P 2 (C) a birational map induced by a polynomial automorphism of C 2 such that

P 2 (C) Φ•ϕ -1 5 5 ❍ ❍ ❍ ❍ ❍ P 2 (C) ϕ Y Y ✈ ✈ ✈ ✈ ✈ Φ G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
and # base-points of Φϕ -1 < # base-points of Φ. To do this we will rearrange the blow-ups of the sequences π 1 and π 2 appearing when we apply Zariski's Theorem: the map ϕ is constructed by realising some blow-ups of π 1 and some blow-ups of π 2 .

Hirzebruch surfaces.

-Let us consider the surface F 1 obtained by blowing-up (1 : 0 : 0) ∈ P 2 (C). This surface is a compactification of C 2 which has a natural rational fibration corresponding to the lines y = constant. The divisor at infinity is the union of two rational curves (i.e. curves isomorphic to P 1 (C)) which intersect in one point. One of them is the strict transform of the line at infinity in P 2 (C), it is a fiber denoted by f 1 ; the other one, denoted by s 1 is the exceptional divisor which is a section for the fibration. We have: f 2 1 = 0 and s 2 1 = -1 (Proposition 1.1.8). More generally for any n we denote by F n a compactification of C 2 with a rational fibration and such that the divisor at infinity is the union of two transversal rational curves: a fiber f ∞ and a section s ∞ of self-intersection -n. These surfaces are called Hirzebruch surfaces:

P P 1 (C) O P 1 (C) ⊕ O P 1 (C) (n) .
Let us consider the surface F n . Let p be the intersection of s n and f n , where f n is a fiber. Let π 1 be the blow-up of p ∈ F n and let π 2 be the contraction of the strict transform f n of f n . We can go from F n to F n+1 via π 2 π -1 1 :

s n+1 -(n + 1) 0 π 2 -(n + 1) s n -1 -1 f π 1 s n -n p 0 f F n+1 F n
We can also go from F n+1 to F n via π 2 π -1 1 where

• π 1 is the blow-up of a point p ∈ F n+1 which belongs to the fiber f n and not to the section s n+1 , • π 2 the contraction of the strict transform f n of f n :

s n s n+1 -(n + 1) π 2 f 0 -1 -1 π 1 -(n + 1)
s n+1 p 0 f F n F n+1 -n
2.2.2. First step: blow-up of (1 : 0 : 0). -The point (1 : 0 : 0) is the first blown-up point in the sequence π 1 . Let us denote by ϕ 1 the blow-up of (1 : 0 : 0) ∈ P 2 (C), we have

F 1 ϕ 1 | | ② ② ② ② g 1 4 4 ❊ ❊ ❊ ❊ P 2 (C) Φ G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
Note that # base-points of g 1 = # base-points of Φ -1. Let us come back to the diagram given by Zariski's theorem. The first curve contracted by π 2 which is a curve of self-intersection -1 is the strict transform of the line at infinity (Lemma 1.3.5, last assertion); it corresponds to the fiber f 1 in F 1 . But in F 1 we have f 2 1 = 0; the self-intersection of this curve has thus to decrease so the point of indeterminacy p of g 1 has to belong to f 1 . But p also belongs to the curve produced by the blow-up (Lemma 1.3.5, second assertion); in other words p = f 1 ∩ s 1 .

Second step

p = f n ∩ s n . Let ϕ : F n F n+1

be the birational map which is the blow-up of p composed with the contraction of the strict transform of f n . Let us consider the birational map h

′ = h • ϕ -1 : F n+1 h ′ 5 5 • • • • F n ϕ a a ④ ④ ④ ④ h G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C) Then • # base-points of h ′ = # base-points of h -1;
• the point of indeterminacy of h ′ belongs to f n+1 .

Proof. -Let us apply Zariski Theorem to h; we obtain

S π 1 π 2 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ F n h G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
where S is a smooth projective surface and π 1 , π 2 are two sequences of blow-ups.

Since s n 2 ≤ -2 (where s n is the strict transform of s n ) the first curve contracted by π 2 is the transform of f n (Lemma 1.3.5). So the transform of f n in S is of self-intersection -1; we also have f 2 n = 0 in F n . This implies that after the blow-up of p the points appearing in π 1 are not on f n . Instead of realising these blow-ups and then contracting the transform of f n we first contract and then realise the blow-ups. In other words we have the following diagram

S 6⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ η 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ η 1 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ π Ñ Ñ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ S ′ 4 4 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ F n h R R ❚ ❯ ❲ ( ) ❭ ( ❴ ❛ ❜ ❞ ❡ ❣ ✐ F n+1 h ′ G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
where π is the blow-up of p and η the contraction of f n . The map ηπ -1 is exactly the first link mentioned in §2.2.1. We can see that to blow-up p allows us to decrease the number of points of indeterminacy and to contract f n does not create some point of indeterminacy. So # base-points of h ′ = # base-points of h -1

Moreover the point of indeterminacy of h ′ is on the curve obtained by the blow-up of p, i.e. f n .

After the first step we are under the assumptions of the Lemma 2.2.1 with n = 1. The Lemma gives an application h ′ : F 2 P 2 (C) such that the point of indeterminacy is on f 2 . If this point also belongs to s 2 we can apply the Lemma again. Repeating this as long as the assumptions of the Lemma 2.2.1 are satisfied, we obtain the following diagram

F n g 2 4 4 ❊ ❊ ❊ ❊ F 1 ϕ 2 c c ⑧ ⑧ ⑧ ⑧ g 1 G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
where ϕ 2 is obtained by applying n -1 times Lemma 2.2.1. Moreover # base-points of g 2 = # base-points of g 1n + 1 and the point of indeterminacy of g 2 is on f n but not on s n (remark: as, for n ≥ 2, there is no morphism from F n to P 2 (C), the map g 2 has a point of indeterminacy).

′ = h • ϕ -1 : F n-1 h ′ 5 5 • • • • F n ϕ a a ④ ④ ④ ④ h G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C) Then • # base-points of h ′ = # base-points of h -1;
• if h ′ has a point of indeterminacy, it belongs to f n-1 and not to s n-1 .

Proof. -Let us consider the Zariski decomposition of h

S π 1 π 2 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ F n h G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
Since s n 2 = -n with n ≥ 2, the first curve blown down by π 2 is the transform of f n (Lemma 1.3.5). Like in the proof of Lemma 2.2.1 we obtain the following commutative diagram

S 6⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ η 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ η 1 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ π Ñ Ñ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ S ′ 4 4 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ F n h R R ❚ ❯ ❲ ( ) ❭ ( ❴ ❛ ❜ ❞ ❡ ❣ ✐ F n-1 h ′ G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
where π is the blow-up of p and η the contraction of f n . We immediately have:

# base-points of h ′ = # base-points of h -1.
Let F ′ be the exceptional divisor associated to π; the map h has a base-point on F ′ . Assume that this point is

F ′ ∩ f n , then (π -1 1 ( f n )) 2 ≤ -2:
contradiction with the fact that it is the first curve blown down by π 2 . So the base-point of h is not F ′ ∩ f n and so it is the point of indeterminacy of h ′ that is on f n-1 but not on s n-1 .

After the second step the assumptions in Lemma 2.2.2 are satisfied. Let us remark that if n ≥ 3 then the map h ′ given by Lemma 2.2.2 still satisfies the assumptions in this Lemma.

After applying n -1 times Lemma 2.2.2 we have the following diagram

F 1 g 3 4 4 ❊ ❊ ❊ ❊ F n ϕ 3 c c ⑧ ⑧ ⑧ ⑧ g 2 G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C) 2.2.5. Last contraction. -Applying Zariski's theorem to g 3 we obtain S ϕ π 2 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ F 1 g 3 G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
The fourth assertion of the Lemma 1.3.5 implies that the first curve contracted by π 2 is either the strict transform of f 1 by π 1 , or the strict transform of s 1 by π 1 . Assume that we are in the first case; then after realising the sequence of blow-ups π 1 and contracting this curve the transform of s 1 is of self-intersection 0 and so cannot be contracted: contradiction with the third assertion of Lemma 1.3.5. So the first curve contracted is the strict transform of s 1 which can be done and we obtain P 2 (C)

g 4 5 5 ❍ ❍ ❍ ❍ ❍ F 1 ϕ 4 a a ③ ③ ③ ③ ③ ③ ③ ③ ③ g 3 G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
The morphism ϕ 4 is the blow-up of a point and the exceptional divisor associated to its blow-up is s 1 ; up to an automorphism we can assume that s 1 is contracted on (1 : 0 : 0). Moreover # base-points of g 3 = # base-points of g 4 .

Conclusion.

-After all these steps we have

P 2 (C) g 4 5 5 ❍ ❍ ❍ ❍ ❍ P 2 (C) ϕ 4 •ϕ 3 •ϕ 2 •ϕ 1 Y Y ✈ ✈ ✈ ✈ ✈ Φ G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C) where # base-points of g 4 = # base-points of Φ -2n + 1 (with n ≥ 2). Let us check that ϕ = ϕ 4 • ϕ 3 • ϕ 2 • ϕ 1 is induced by an element of E.
It is sufficient to prove that ϕ preserves the fibration y = constant, i.e. the pencil of curves through (1 : 0 : 0); indeed

• the blow-up ϕ 1 sends lines through (1 : 0 : 0) on the fibers of F 1 ;

• ϕ 2 and ϕ 3 preserve the fibrations associated to F 1 and F n ;

• the morphism ϕ 4 sends fibers of F 1 on lines through (1 : 0 : 0).

Finally g 4 is obtained by composing Φ with a birational map induced by an affine automorphism and a birational map induced by an element of E so g 4 is induced by a polynomial automorphism; morevoer # base-points of g 4 < # base-points of Φ.

2.2.7.

Example. -Let us consider the polynomial automorphism Φ of C 2 given by

Φ = y + (y + x 2 ) 2 + (y + x 2 ) 3 , y + x 2 .
Let us now apply to φ the method just explained above. The point of indeterminacy of Φ is (0 : 1 : 0). Let us compose Φ with (y, x) to deal with an automorphism whose point of indeterminacy is (1 : 0 : 0). Let us blow up this point

F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ P 2 (C)
Then we apply Lemma 2.2.1

⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ F 2 P 2 (C)
On F 2 the point of indeterminacy is on the fiber, we thus apply Lemma 2.2.2

⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ F 2 F 1 P 2 (C)
and contracts s 1

⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ F 2 F 1 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ P 2 (C) (x+y 2 ,y)(y,x) G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
We get the decomposition Φ = Φ ′ (x + y 2 , y)(y, x) with

Φ ′ = (y + x 2 + x 3 , x) = (x + y 2 + y 3 , y)(y, x).
We can check that Φ ′ has a unique point of indeterminacy (0 : 1 : 0). Let us blow up the point (1 : 0 : 0)

F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ P 2 (C)
and then apply two times Lemma 2.2.1

⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ F 2 F 3 P 2 (C) then two times Lemma 2.2.2 ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ F 2 F 3 F 2 F 1 P 2 (C)
Finally we contract the section s 1

⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 1 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ F 1 } } ③ ③ ③ ③ ③ ③ ③ ③ F 2 F 3 F 2 F 1 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ P 2 (C) Φ ′ =(x+y 2 +y 3 ,y)(y,x) G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
and obtain Φ ′ = (x + y 2 + y 3 , y)(y, x).

The de Jonquières group

The de Jonquières maps are, up to birational conjugacy, of the following type

a(y)x + b(y) c(y)x + d(y) , αy + β γy + δ , a(y) b(y) c(y) d(y) ∈ PGL 2 (C(y)), α β γ δ ∈ PGL 2 (C);
let us remark that the family of lines y = constant is preserved by such a Cremona transformation. De Jonquières maps are exactly the Cremona maps which preserve a rational fibration (2) . The de Jonquières maps form a group, called de Jonquières group and denoted by dJ. Remark that the exceptional set of φ is reduced to a finite number of fibers y = cte and possibly the line at infinity. In some sense dJ ⊂ Bir(P 2 ) is the analogue of E ⊂ Aut(C 2 ). In the 80's Gizatullin and Iskovskikh give a presentation of Bir(P 2 ) (see [START_REF] Kh | Defining relations for the Cremona group of the plane[END_REF][START_REF] Iskovskikh | Proof of a theorem on relations in the two-dimensional Cremona group[END_REF]); let us state the result of Iskovskikh presented in P 1 (C) × P 1 (C) which is birationally isomorphic to P 2 (C).

Theorem 2.3.1 ([124]

). -The group of birational maps of P 1 (C) × P 1 (C) is generated by dJ and Aut(P 1 (C) × P 1 (C)) (3) .

Moreover the relations in Bir(P 1 (C)×P 1 (C)) are the relations of dJ, of Aut(P 1 (C)×P 1 (C)) and the relation

(ηe) 3 = 1 x , 1 y where η : (x, y) → (y, x) & e : (x, y) → x, x y .
Let f be a birational map of P 2 (C) of degree ν. Assume that f has a base-point p 1 of multiplicity m 1 = ν -1. Then we have

ν 2 -(ν -1) 2 - r ∑ i=2 m 2 i = 1, 3ν -(ν -1) - r ∑ i=2 m i = 3
where p 2 , . . ., p r are the other base-points of f and m i the multiplicity of p i . This implies that ∑ r i=2 m i (m i -1) = 0, hence m 2 = . . . = m r = 1 and r = 2ν -1. For simplicity let us assume that the p i 's are in P 2 (C). The homaloidal system Λ f consists of curves of degree ν with singular point p 1 of multiplicity ν -1 passing simply to 2ν -2 points p 2 , . . ., p 2ν-1 . The corresponding Cremona transformation is a de Jonquières transformation. Indeed let Γ be an element of Λ f . Let Ξ be the pencil of curves of Λ f that have in common with Γ a point m distinct from p 1 , . . ., p 2ν-1 . The number of intersections of Γ with a generic curve of Ξ that are absorbed by the p i 's is at least

(ν -1)(ν -2) + 2ν -2 + 1 = ν(ν -1) + 1 2.
Here a rational fibration is a rational application from P 2 (C) into P 1 (C) whose fibers are rational curves. 3. The de uières group is birationally isomorphic to the subgroup of Bir(P 1 (C) × P 1 (C)) which preserves the first projection p :

P 1 (C) × P 1 (C) → P 1 (C).
one more than the number given by Bezout's theorem. The curves of Ξ are thus all split into Γ and a line of the pencil centered in p 1 . Let us assume that p 1 = (1 : 0 : 0); then Γ is given by

xψ ν-2 (y, z) + ψ ν-1 (y, z), deg ψ i = i.
To describe Λ f we need an arbitrary curve taken from Λ f and outside Ξ which gives

(xψ ν-2 + ψ ν-1 )(a 0 y + a 1 z) + xϕ ν-1 (y, z) + ϕ ν (y, z), deg ϕ i = i.
Therefore f can be represented by

(x : y : z) xϕ ν-1 + ϕ ν : (xψ ν-2 + ψ ν-1 )(ay + bz) : (xψ ν-2 + ψ ν-1 )(cy + dz)
with adbc = 0. We can easily check that f is invertible and that Λ f and Λ f -1 have the same type. At last we have in the affine chart z = 1

xϕ ν-1 (y) + ϕ ν (y) xψ ν-2 (y) + ψ ν-1 (y) , ay + b cy + d .

No dichotomy in the Cremona group

There is a strong dichotomy in Aut(C 2 ) (see §2.1); we will see that there is no such dichotomy in Bir(P 2 ). Let us consider the family of birational maps ( f α,β ) given by -The first dynamical degree (4) of f α,β is equal to 1; more precisely deg f n α,β ∼ n. Assume that α and β are generic and have modulus 1. If g commutes with f α,β , then g coincides with an iterate of f α,β ; in particular the centralizer of f α,β is countable.

The elements f 2 α,β have two fixed points m 1 , m 2 and

• there exists a neighborhood V 1 of m 1 on which f α,β is conjugate to (αx, βy); in particular the closure of the orbit of a point of V 1 (under f α,β ) is a torus of dimension 2; • there exists a neighborhood V 2 of m 2 such that f 2 α,β is locally linearizable on V 2 ; the closure of a generic orbit of a point of V 2 (under f 2 α,β ) is a circle.

In the affine chart (x, y) the maps f α,β preserve the 3-manifolds |y| = cte. The orbits presented below are bounded in a copy of R 2 × S 1 . The dynamic happens essentially in dimension 3; different projections allow us to have a good representation of the orbit of a point. In the affine chart z = 1 let us denote by p 1 and p 2 the two standard projections. The given pictures are representations (in perspective) of the following projections.

4. For a birational map f of P 2 (C) the first dynamical degree is given by λ

( f ) = lim n→+∞ (deg f n ) 1/n .
• Let us first consider the set

Ω 1 (m, α, β) = (p 1 ( f n α,β (m)), Im(p 2 ( f n α,β (m)))) n = 1..30000 ;
this set is contained in the product of R 2 with an interval. The orbit of a point under the action of f α,β is compressed by the double covering (x, ρe iθ ) → (x, ρ sin θ).

• Let us introduce Ω 2 (m, α, β) = (Re(p 1 ( f n α,β (m))), p 2 ( f n α,β (m))) n = 1..30000
which is contained in a cylinder R × S 1 ; this second projection shows how to "decompress" Ω 1 to have the picture of the orbit.

Let us assume that α = exp(2i √ 3) and β = exp(2i √ 2); let us denote by Ω i (m) instead of Ω i (m, α, β).

The following pictures illustrate Theorem 2.4.1.

Ω 1 (10 -4 i, 10 -4 i) Ω 2 (10 -4 i, 10 -4 i)

It is "the orbit" of a point in the linearization domain of (0 : 0 : 1); we note that the closure of an orbit is a torus. Ω 1 (10000 + 10 -4 i, 10000 + 10 -4 i) Ω 2 (10000 + 10 -4 i, 10000 + 10 -4 i)

It is "the orbit" under f 2 α,β of a point in the linearization domain of (0 : 1 : 0); the closure of an "orbit" is a topological circle. The singularities are artifacts of projection.

Remark 2.4.2. -The line z = 0 is contracted by f α,β on (0 : 1 : 0) which is blow up on z = 0 : the map f α,β is not algebraically stable (see Chapter 3) that's why we consider f 2 α,β instead of f α,β .

The theory does not explain what happens outside the linearization domains. Between V 1 and V 2 the experiences suggest a chaotic dynamic as we can see below. We note a deformation of the invariant tori.

Ω 1 (0.9 + 10 -4 i, 0.9 + 10 -4 i) Ω 2 (0.9 + 10 -4 i, 0.9 + 10 -4 i)

Ω 1 (1 + 10 -4 i, 1 + 10 -4 i) Ω 2 (1 + 10 -4 i, 1 + 10 -4 i) Ω 1 (1.08 + 10 -4 i, 1.08 + 10 -4 i) Ω 2 (1.08 + 10 -4 i, 1.08 + 10 -4 i)
The invariant tori finally disappear; nevertheless the pictures seem to organize themselves around a closed curve.

So if there is no equivalence between first dynamical degree strictly greater than 1 and countable centraliser we have an implication; more precisely we have the following statement.

Theorem 2.4.3 ([47]

). -Let f be a birational map of the complex projective plane with first dynamical degree λ( f ) strictly greater than 1. If ψ is an element of Bir(P 2 ) which commutes with f , there exist two integers m in N * and n in Z such that ψ m = f n .

CHAPTER 3 CLASSIFICATION AND APPLICATIONS

Notions of stability and dynamical degree

Let X , Y be two compact complex surfaces and let f : X Y be a dominant meromorphic map. Let Γ f be the graph of f and let π 1 : Γ f → X , π 2 : Γ f → Y be the natural projections. If Γ f is a singular submanifold of X ×Y , we consider a desingularization of Γ f without changing the notation. If β is a differential form of bidegree (1, 1) on Y , then π * 2 β determines a form of bidegree (1, 1) on Γ f which can be pushed forward as a current f * β := π 1 * π * 2 β on X thanks to the first projection. Let us note that f * induces an operator between H 1,1 (Y, R) and H 1,1 (X , R) : if β and γ are homologous, then f * β and f * γ are homologous. In a similar way we can define the push-forward f * := π 2 * π * 1 : H p,q (X ) → H p,q (Y ). Note that when f is bimeromorphic f * = ( f -1 ) * .

Assume that X = Y . The map f is algebraically stable if there exists no curve V in X such that f k (V ) belongs to Ind f for some integer k ≥ 0.

Theorem-Definition 3.1.1 ([77]

). -Let f : S → S be a dominating meromorphic map on a Kähler surface and let ω be a Kähler form. Then f is algebraically stable if and only if any of the following holds:

• for any α ∈ H 1,1 (S) and any k in N, we have

( f * ) k α = ( f k ) * α;
• there is no curve C in S such that f k (C ) ⊂ Ind f for some integer k ≥ 0;

• for all k ≥ 0 we have

( f k ) * ω = ( f * ) k ω.
In other words for an algebraically stable map the following does not happen . . ... . .

f f f f f C i.
e. the positive orbit (1) of p 1 ∈ Ind f -1 intersects Ind f . Let us say that the coefficients of an automorphism A of P 2 (C) are algebraically independent if A has a representative in GL 3 (C) whose coefficients are algebraically independent over Q. We can deduce the following: let A be an automorphism of the projective plane whose coefficients are algebraically independent over Q, then Aσ and (Aσ) -1 are algebraically stable.

Diller and Favre prove the following statement. -Let S be a rational surface and let f : S S be a birational map. There exists a birational morphism ε : S → S such that ε f ε -1 is algebraically stable.

Idea of the proof. -Let us assume that f is not algebraically stable; hence there exists a curve C and an integer k such that C is blown down onto p 1 and p k = f k-1 (p 1 ) is an indeterminacy point of f . The idea of Diller and Favre is the following: after blowing up the points p i the image of C is, for i = 1, . . . , k, a curve. Doing this for any element of Exc f whose an iterate belongs to Ind f we get the statement.

Remark 3.1.6.

-There is no similar result in higher dimension. Let us recall the following statement due to Lin ([137, Theorem 5.7]): suppose that A = (a i j ) ∈ M n (Z) is an integer matrix with det A = 1. If λ and λ are the only eigenvalues of A of maximal modulus, also with algebraic multiplicity one, and if λ = |λ|e 2iπϑ with ϑ ∈ Q; then there is no toric birational model which makes the corresponding monomial map

f A : C n → C n , (x 1 , . . . , x n ) → ∏ j x a 1 j j , . . . , ∏ j x a n j j algebraically stable. A 3 × 3 example is ([116]) A =   -1 1 0 -1 0 1 1 0 0   ;
in higher dimension A 0 0 Id where 0 is the zero matrix and Id is the identity matrix works.

The first dynamical degree of f is defined by

λ( f ) = lim sup n→+∞ |( f n ) * | 1/n
where | . | denotes a norm on End(H 1,1 (X , R)) ; this number is greater or equal to 1 (see [START_REF] Russakovskii | Value distribution for sequences of rational mappings and complex dynamics[END_REF][START_REF] Friedland | Entropy of algebraic maps[END_REF]). Let us remark that for all birational maps f we have the inequality

λ( f ) n ≤ deg f n
where deg f is the algebraic degree of f (the algebraic degree of f = ( f 0 :

f 1 : f 2 )
is the degree of the homogeneous polynomials f i ).

Examples 3.1.7. -• The first dynamical degree of a birational map of the complex projective plane of finite order is equal to 1.

• The first dynamical degree of an automorphism of P 2 (C) is equal to 1.

• The first dynamical degree of an elementary automorphism (resp. a de Jonquières map) is equal to 1. • The first dynamical degree of a Hénon automorphism of degree d is equal to d.

• The first dynamical degree of the monomial map Let us give some properties about the first dynamical degree. Let us recall that a Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real algebraic integer is a Salem number if all its conjugate roots have absolute value no greater than 1, and at least one has absolute value exactly 1.

f B : (x, y) → (x a y b , x c y d ) is the largest eigenvalue of B = a b c d . • Let us set E = C/Z[i], Y = E × E = C 2 /Z[i] × Z[i]
Theorem 3.1.8 ([77]). -The set λ( f ) | f ∈ Bir(P 2 )
is contained in {1} ∪ P ∪ S where P (resp. S) denotes the set of Pisot (resp. Salem) numbers.

In particular it is a subset of algebraic numbers. quadratically) then f preserves a pencil of rational curves (resp. elliptic curves); up to birational conjugacy f preserves a pencil of lines, i.e. belongs to the de Jonquières group (resp. preserves an Halphen pencil, i.e. a pencil of (elliptic) curves of degree 3n passing through 9 points with multiplicity n).

Classification of birational maps

Picard-Manin space

Manin describes in [140, Chapter 5] the inductive limit of the Picard group of any surface obtained by blowing up any point of a surface S. Then he shows that the group Bir(S) linearly acts on this limit group.

• Let S be a Kähler compact complex surface. Let Pic(S) be the Picard group of S and let NS(S) be its Néron-Severi group (2) . Let us consider the morphism from Pic(S) into NS(S) which associates to any line bundle L its Chern class c 1 (L); its kernel is denoted by Pic 0 (S). The dimension of NS(R) ⊗ R is called the Picard number of S and is denoted by ρ(S). There is an intersection form on the Picard group, there is also one on the Néron-Severi group; when S is projective, its signature is (1, ρ(S) -1). The nef cone is denoted by NS + (S) or Pic + (S) when NS(S) = Pic(S). Let S and S ′ be two surfaces and let π : S → S ′ be a birational morphism. The morphism π * is injective and preserves the nef cone: π * (NS + (S ′ )) ⊂ NS + (S). Moreover for any ℓ, ℓ ′ in Pic(S), we have (π * ℓ, π * ℓ ′ ) = (ℓ, ℓ ′ ).

• Let S be a Kähler compact complex surface. Let B(S) be the category which objects are the birational morphisms π ′ : S ′ → S. A morphism between two objects π 1 : S ′ 1 → S and π 2 : S ′ 2 → S of this category is a birational morphism ε : S ′ 1 → S ′ 2 such that π 2 ε = π 1 . In particular the set of morphisms between two objects in either empty, or reduced to a unique element.

This set of objects is ordered as follows: π 1 ≥ π 2 if and only if there exists a morphism from π 1 to π 2 ; we thus say that π 1 (resp. S ′ 1 ) dominates π 2 (resp. S ′ 2 ). Geometrically this means that the set of base-points of π -1 1 contains the set of base-points of π -1 2 . If π 1 and π 2 are two objects of B(S) there always exists another one which simultaneously dominates π 1 and π 2 .

Let us set

Z(S) = lim → NS(S ′ )
the inductive limit is taken following the injective morphism π * .

The group Z(S) is called Picard-Manin space space of S. The invariant structures of π * induce invariant structures for Z(S):

• an intersection form (, ) : Z(S) × Z(S) → Z;

• a nef cone Z + (S) = lim → NS + (S);

• a canonical class, viewed as a linear form Ω : Z(S) → Z.

Note that NS(S ′ ) embeds into Z(S) so we can identify NS(S ′ ) and its image in Z(S).

Let us now describe the action of birational maps of S on Z(S). Let S 1 and S 2 be two surfaces and let f be a birational map from S 1 to S 2 . According to Zariski Theorem we can remove the indeterminacy of f thanks to two birational morphisms π 1 : S ′ → S 1 and π 2 :

S ′ → S 2 such that f = π 2 π -1
1 . The map π 1 (resp. π 2 ) embeds B(S ′ ) into B(S 1 ) (resp. B(S 2 )). Since any object of B(S 1 ) (resp. B(S 2 )) is dominated by an object of π 1 * (B(S)) (resp. π 2 * (B(S))) we get two isomorphisms

π 1 * : Z(S ′ ) → Z(S 1 ), π 2 * : Z(S ′ ) → Z(S 2 ).
Then we set f * = π 2 * π -1 1 * .

Theorem 3.3.1 ([140], page 192

). -The map f → f * induces an injective morphism from Bir(S) into GL(Z(S)).

If f belongs to Bir(S), the linear map f * preserves the intersection form and the nef cone.

Let us denote by Eclat(S) the union of the surfaces endowed with a birational morphism π : S ′ → S modulo the following equivalence relation: S ∋ p 1 ∼ p 2 ∈ S if and only if ε -1 2 ε 1 sends p 1 onto p 2 and is a local isomorphism between a neighborhood of p 1 and a neighborhood of p 2 . A point of Eclat(S) corresponds either to a point of S, or to a point on an exceptional divisor of a blow-up of S etc. Any surface S ′ which dominates S embeds into Eclat(S). Let us consider the free abelian group Ec(S) generated by the points of Eclat(S); we have a scalar product on Ec(S)

(p, p) E = -1, (p, q) = 0 if p = q.
The group Ec(S) can be embedded in Z(S) (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). If p is a point of Eclat(S) let us denote by e p the point of Z(S) associated to p, i.e. e p is the class of the exceptional divisor obtained by blowing up p. This determines the image of the basis of Ec(S) in Z(S) so we have the morphism defined by

Ec(S) → Z(S), ∑ a(p)p → ∑ a(p)e p .
Using this morphism and the canonical embedding from NS(S) into Z(S) we can consider the morphism NS(S) × Ec(S) → Z(S). Let us now consider q ∈ Bl p P 2 and as previously we define Bl p,q P 2 and E q . The elements e p and e q belong to the image of NS(Bl p,q P 2 ) in Z(P 2 ).

If E p is the strict transform of E p in Bl p,q P 2 the element e p (resp. e q ) corresponds to E p + E q (resp. E q ). We can check that (e p , e q ) = 0 and (e p , e p ) = 1.

• The completed Picard-Manin space Z(S) of S is the L 2 -completion of Z(S); in other words

Z(S) = [D] + ∑ a p [E p ] [D] ∈ NS(S), a p ∈ R, ∑ a 2 p < ∞ .
Note that Z(S) corresponds to the case where the a p vanishes for all but a finite number of p ∈ Eclat(S).

a 0 [H] + ∑ p∈Eclat(S) a p [E p ],
with ∑ a 2 p < ∞.

The group Bir(S) acts on Z(S); let us give details when S = P 2 (C). Let f be a birational map from P 2 (C) into itself. According to Zariski Theorem there exist two morphisms

π 1 , π 2 : S → P 2 (C) such that f = π 2 π -1 1 . Defining f * by f * = (π * 1 ) -1 π * 2 and f * by f * = ( f * ) -1
we get the representation f → f * of the Cremona group in the orthogonal group of Z(P 2 ) (resp. Z(P 2 )) with respect to the intersection form. Since for any p in P 2 (C) such that f is defined at p we have f * (e p ) = e f (p) this representation is faithful; it also preserves the integral structure of Z(P 2 ) and the nef cone.

• Only one of the two sheets of the hyperboloid [D] ∈ Z(P 2 ) [D] 2 = 1 intersects the nef cone Z(P 2 ); let us denote it by H Z . In other words

H Z = [D] ∈ Z(P 2 ) [D] 2 = 1, [H] • [D] > 0 .
We can define a distance on

H Z : cosh(dist([D 1 ], [D 2 ])) = [D 1 ] • [D 2 ].
The space H Z is a model of the "hyperbolic space of infinite dimension"; its isometry group is denoted by Isom(H Z ) (see [109], §6). As the action of Bir(P 2 ) on Z(P 2 ) preserves the two-sheeted hyperboloid and as the action also preserves the nef cone we get a faithful representation of Bir(P 2 ) into Isom(H Z ). In the context of the Cremona group we will see that the classification of isometries into three types has an algebraic-geometric meaning.

• As H Z is a complete cat(-1) metric space, its isometries are either elliptic, or parabolic, or hyperbolic (see [START_REF] Ghys | Sur les groupes hyperboliques d'après Mikhael Gromov[END_REF]). In the case of hyperbolic case we can characterize these isometries as follows:

-elliptic isometry: there exists an element ℓ in Z(S) such that f * (ℓ) = ℓ and (ℓ, ℓ) > 0 then f * is a rotation around ℓ and the orbit of any p in Z(S) (resp. any p in H Z ) is bounded;

-parabolic isometry: there exists a non zero element ℓ in Z + (S) such that f * (ℓ) = ℓ. Moreover (ℓ, ℓ) = 0 and Rℓ is the unique invariant line by f * which intersects Z + (S). If p belongs to Z + (S), then lim

n→∞ f n * (Rp) = Rℓ.
-hyperbolic isometry: there exists a real number λ > 1 and two elements ℓ + and ℓ -in Z(S)

such that f * (ℓ + ) = λℓ + and f * (ℓ -) = (1/λ)ℓ -. If p is a point of Z + (S) \ Rℓ -, then lim n→∞ 1 λ n f n * (p) = v ∈ Rℓ + \ {0},
We have a similar property for ℓ -and f -1 . This classification and Diller-Favre classification (Theorem 3.2.1) are related by the following statement.

Theorem 3.3.5 ([47]

). -Let f be a birational map of a compact complex surface S. Let f * be the action induced by f on Z(S).

• f * is elliptic if and only if f is an elliptic map: there exists an element ℓ in Z + (S) such that f (ℓ) = ℓ and (ℓ, ℓ) > 0, then f * is a rotation around ℓ and the orbit of any p in Z(S) • or Γ contains a solvable (3) subgroup of finite index.

(resp. any p in H Z ) is bounded. • f * is parabolic if
Let us mention that the group of diffeomorphisms of a real manifold of dimension ≥ 1 does not satisfy Tits alternative (see [START_REF] Ghys | Groups acting on the circle[END_REF] and references therein). Nevertheless the group of polynomial automorphisms of C 2 satisfies Tits alternative ( [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]); Lamy obtains this property from the classification of subgroups of Aut(C 2 ), classification established by using the action of this group on T :

Theorem 3.4.2 ([135]

). -Let G be a subgroup of Aut(C 2 ). Exactly one of the followings holds:

• any element of G is conjugate to an element of E, then -either G is conjugate to a subgroup of E;

-or G is conjugate to a subgroup of A; -or G is abelian, G = i∈N G i with G i ⊂ G i+1
and any G i is conjugate to a finite cyclic group of the form (αx, βy) with α, β roots of unicity of the same order. Any element of G has a unique fixe point (4) and this fixe point is the same for any element of G.

Finally the action of G fixes a piece of the tree T .

• G contains Hénon automorphisms, all having the same geodesic, in this case G is solvable and contains a subgroup of finite index isomorphic to Z. 1) for k ≥ 1. The group G is solvable if there exists an integer k such that G (k) = {id}.

(0) = G et G (k) = [G (k-1) , G (k-1) ] = aba -1 b -1 | a, b ∈ G (k-
4. as polynomial automorphism of C 2

• there exist X 1 and X 2 two non empty subsets of X such that

X 2 X 1 ; ∀ α ∈ Γ 1 \ {id}, α(X 2 ) ⊂ X 1 ; ∀ β ∈ Γ 2 \ {id}, β(X 1 ) ⊂ X 2 .
Then Γ is isomorphic to the free product

Γ 1 * Γ 2 of
Γ 1 = 1 2 0 1 n n ∈ Z , Γ 2 = 1 0 2 1 n n ∈ Z , X 1 = (x, y) ∈ R 2 |x| > |y| & X 2 = (x, y) ∈ R 2 |x| < |y| .
Let us consider an element γ of Γ 1 \ {id} and (x, y) an element of X 2 , we note that γ(x, y) is of the form (x + my, y), with |m| ≥ 2; therefore γ(x, y) belongs to X 1 . If γ belongs to Γ 2 \ {id} and if (x, y) belongs to X 1 , the image of (x, y) by γ belongs to X 2 . According to Lemma 3.4.

3 we have 1 2 0 1 , 1 0 2 1 ≃ F 2 = Z * Z = Γ 1 * Γ 2 .
We also obtain that 1 k 0 1 and 1 0 k 1 generate a free group of rank 2 in SL 2 (Z) for any k ≥ 2. Nevertheless it is not true for k = 1, the matrices 1 1 0 1 and 1 0 1 1 generate SL 2 (Z).

Example 3.4.5. -Two generic matrices in SL ν (C), with ν ≥ 2, generate a free group isomorphic to F 2 .

In [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF] Cantat characterizes the finitely generated subgroups of Bir(P 2 ); Favre reformulates, in [START_REF] Favre | Le groupe de Cremona et ses sous-groupes de type fini[END_REF], this classification:

Theorem 3.4.6 ([47]

). -Let G be a finitely generated subgroup of the Cremona group. Exactly one of the following holds:

• Any element of G is elliptic thus -either G is, up to finite index and up to birational conjugacy, contained in the connected component of Aut(S) where S denotes a minimal rational surface; -or G preserves a rational fibration.

• G contains a (de Jonquières or Halphen) twist and does not contain hyperbolic map, thus

G preserves a rational or elliptic fibration. • G contains two hyperbolic maps f and g such that f , g is free.

• G contains a hyperbolic map and for any pair ( f , g) of hyperbolic maps, f , g is not a free group, then

1 -→ ker ρ -→ G ρ -→ Z -→ 1
and ker ρ contains only elliptic maps.

One consequence is the following statement.

Theorem 3.4.7 ([47]

). -The Cremona group Bir(P 2 ) satisfies Tits alternative.

3.4.2.

Simplicity. -Let us recall that a simple group has no non trivial normal subgroup. We first remark that Aut(C 2 ) is not simple; let φ be the morphism defined by

Aut(C 2 ) → C * , f → det jac f .
The kernel of φ is a proper normal subgroup of Aut(C 2 ). In the seventies Danilov has established that ker φ is not simple ( [START_REF] Danilov | Non-simplicity of the group of unimodular automorphisms of an affine plane[END_REF]). Thanks to some results of Schupp ([168]) he proved that the normal subgroup (5) generated by

(ea) 13 , a = (y, -x), e = (x, y + 3x 5 -5x 4 ) is strictly contained in Aut(C 2 ).
More recently Furter and Lamy gave a more precise statement. Before giving it let us introduce a length ℓ(.) for the elements of Aut(C 2 ).

• If f belongs to A ∩ E, then ℓ( f ) = 0;

• otherwise ℓ( f ) is the minimal integer n such that f = g 1 . . . g n with g i in A or E. The length of the element given by Danilov is 26.

We note that ℓ(.) is invariant by inner conjugacy, we can thus assume that f has minimal length in its conjugacy class.

Theorem 3.4.8 ([99]

). -Let f be an element of Aut(C 2 ). Assume that det jac f = 1 and that f has minimal length in its conjugacy class.

• If f is non trivial and if ℓ( f ) ≤ 8, the normal subgroup generated by f coincides with the group of polynomial automorphisms f of C 2 with det jac f = 1; • if f is generic (6) and if ℓ( f ) ≥ 14, the normal subgroup generated by f is strictly contained in the subgroup f

∈ Aut(C 2 ) det jac f = 1 of Aut(C 2 ).
Is the Cremona group simple ? Cantat and Lamy study the general situation of a group G acting by isometries on a δhyperbolic space and apply it to the particular case of the Cremona group acting by isometries on the hyperbolic space H Z . Let us recall that a birational map f induces a hyperbolic isometry f * ∈ H Z if and only if {deg f k } k∈N grows exponentially (Theorem 3.3.5). Another characterization given in [START_REF] Cantat | Normal subgroups in the cremona group[END_REF] is the following: f induces a hyperbolic isometry f * ∈ H Z if and only if 5. Let G be a group and let f be an element of G; the normal subgroup generated by [START_REF] Furter | Normal subgroup generated by a plane polynomial automorphism[END_REF] for more details.

f in G is h f h -1 | h ∈ G . 6. See
there is a f * -invariant plane in the Picard-Manin space that intersects H Z on a curve Ax( f * ) (a geodesic line) on which f * acts by a translation:

dist(x, f * (x)) = log λ( f ), ∀x ∈ Ax( f * ).
The curve Ax( f * ) is uniquely determined and is called the axis of

f * . A birational map f is tight if • f * ∈ Isom(H Z ) is hyperbolic;
• there exists a positive number ε such that: if g is a birational map and if g * (Ax( f * )) contains two points at distance ε which are at distance at most

1 from Ax( f * ) then g * (Ax( f * )) = Ax( f * ); • if g is a birational map and g * (Ax( f * )) = Ax( f * ) then g f g -1 = f or f -1 .
Applying their results on group acting by isometries on δ-hyperbolic space to the Cremona group, Cantat and Lamy obtain the following statement.

Theorem 3.4.9 ([50]

). -Let f be a birational map of the complex projective plane. If f is tight, then f k generates a non trivial normal subgroup of Bir(P 2 ) for some positive interger k.

They exhibit tight elements by working with the unique irreducible component of maximal dimension

V d = φψϕ -1 | φ, ϕ ∈ Aut(P 2 ), ψ ∈ dJ, deg ψ = d of Bir d .

Corollary 3.4.10 ([50]

). -The Cremona group contains an uncountable number of normal subgroups.

In particular Bir(P 2 ) is not simple.

Representations of cocompact lattices of SU(n, 1) in the Cremona group. -In [68]

Delzant and Py study actions of Kähler groups on infinite dimensional real hyperbolic spaces, describe some exotic actions of PSL 2 (R) on these spaces, and give an application to the study of the Cremona group. In particular they give a partial answer to a question of Cantat ([47]):

Theorem 3.4.11 ([68]). -Let Γ be a cocompact lattice in the group SU(n, 1) with n ≥ 2. If ρ : Γ → Bir(P 2
) is an injective homomorphism, then one of the following two possibilities holds:

• the group ρ(Γ) fixes a point in the Picard-Manin space;

• the group ρ(Γ) fixes a unique point in the boundary of the Picard-Manin space.

CHAPTER 4 QUADRATIC AND CUBIC BIRATIONAL MAPS

Some definitions and notations

Let Rat k be the projectivization of the space of triplets of homogeneous polynomials of degree k in 3 variables:

Rat k = P ( f 0 , f 1 , f 2 ) f i ∈ C[x, y, z] k .
An element of Rat k has degree ≤ k. We associate to f = ( f 0 :

f 1 : f 2 ) ∈ Rat k the rational map f • : (x : y : z) δ( f 0 (x, y, z) : f 1 (x, y, z) : f 2 (x, y, z)), where δ = 1 pgcd( f 0 , f 1 , f 2 ) . Let f be in Rat k ; we say that f = ( f 0 : f 1 : f 2 )
is purely of degree k if the f i 's have no common factor. Let us denote by Rat k the set of rational maps purely of degree k. Whereas Rat k can be identified to a projective space, Rat k is an open Zariski subset of it. An element of Rat k \ Rat k can be written ψ f = (ψ f 0 : ψ f 1 : ψ f 2 ) where f belongs to Rat ℓ , where ℓ < k, and ψ is a homogeneous polynomial of degree k -ℓ. Let us denote by Rat the set of all rational maps from P

2 (C) into itself: it is k≥1 Rat k . It's also the injective limite of the Rat • k 's where Rat • k = f • f ∈ Rat k . Note that if f ∈ Rat k is purely of degree k then f can be identified to f • . This means that the application Rat k → Rat • k is injective.
Henceforth when there is no ambiguity we use the notation f for the elements of Rat k and for those of Rat • k . We will also say that an element of Rat k "is" a rational map. The space Rat contains the group of birational maps of P 2 (C). Let Bir k ⊂ Rat k be the set of birational maps f of Rat k such that f • is invertible, and let us denote by Bir k ⊂ Bir k the set of birational maps purely of degree k. Set

Bir • k = f • f ∈ Bir k .
The Cremona group can be identified to k≥1 Bir k . Note that Bir 1 ≃ PGL 3 (C) is the group of automorphisms of P 2 (C); we have Bir 1 ≃ Bir • 1 = Bir 1 . The set Rat 1 can be identified to P 8 (C) and Rat 1 is the projectivization of the space of matrices of rank greater than 2.

For k = 2 the inclusion Bir 2 ⊂ Bir 2 is strict. Indeed if A is in PGL 3 (C) and if ℓ is a linear form, ℓA is in Bir 2 but not in Bir 2 .

There are two "natural" actions on Rat k . The first one is the action of PGL 3 (C) by dynamic conjugation

PGL 3 (C) × Rat k → Rat k , (A, Q) → AQA -1
and the second one is the action of PGL 3 (C) 2 by left-right composition (l.r.) -Let σ be the birational map given by

PGL 3 (C) × Rat k × PGL 3 (C) → Rat k , (A, Q, B) → AQB -1 .

Remark that Rat

P 2 (C) P 2 (C), (x : y : z) (yz : xz : xy).
The map σ is an involution whose indeterminacy and exceptional sets are given by: Ind σ = (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) , Exc σ = x = 0, y = 0, z = 0 .

The Cremona transformation ρ : (x : y : z) (xy : z 2 : yz) has two points of indeterminacy which are (0 : 1 : 0) and (1 : 0 : 0); the curves contracted by ρ are z = 0, resp. y = 0. Let τ be the map defined by (x : y : z) (x 2 : xy : y 2xz); we have Ind τ = (0 : 0 : 1) ,

Exc τ = x = 0 .
Notice that ρ and τ are also involutions. The Cremona transformations f and ψ are birationally conjugate if there exists a birational map η such that f = ψηψ -1 . The three maps σ, ρ and τ are birationally conjugate to some involutions of PGL 3 (C) (see for example [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]).

Let us continue with quadratic rational maps. Let C[x, y, z] ν be the set of homogeneous polynomials of degree ν in C 3 . Let us consider the rational map det jac defined by

det jac : Rat 2 P(C[x, y, z] 3 ) ≃ curves of degree 3 [Q] [det jac Q = 0]. Remark 4.1.2.
-The map det jac is not defined for maps [Q] such that det jac Q ≡ 0; such a map is up to l.r. conjugacy (Q 0 : Q 1 : 0) or (x 2 : y 2 : xy).

Proposition 4.1.3 ([56]

). -The map det jac is surjective.

Proof. -For the map σ we obtain three lines in general position, for ρ the union of a "double line" and a line, for τ one "triple line" and for (x 2 : y 2 : (xy)z) the union of three concurrent lines. With det jac -

1 α x 2 + z 2 : - α 2 xz + 1 + α 4 x 2 - 1 4 y 2 : xy = [y 2 z = x(x -z)(x -αz)]
we get all cubics having a Weierstrass normal form.

If Q : (x : y : z) (xy : xz : x 2 + yz), then det jac Q = [x(x 2 -yz) = 0]
is the union of a conic and a line in generic position.

We have det jac (y 2 : x 2 + 2xz :

x 2 + xy + yz) = [y(2x 2 -yz) = 0]
which is the union of a conic and a line tangent to this conic.

We use an argument of dimension to show that the cuspidal cubic belongs to the image of det jac.

Up to conjugation we obtain all plane cubics, we conclude by using the l.r. action.

Criterion of birationality

We will give a presentation of the classification of the quadratic birational maps. Let us recall that if φ is a rational map and P a homogeneous polynomial in three variables we say that φ contracts P if the image by φ of the curve [P = 0] \ Ind φ is a finite set.

Remark 4.2.1.

-In general a rational map doesn't contract det jac f (it is the case for f : (x :

y : z) (x 2 : y 2 : z 2 )). Buts if f is a birational map of P 2 (C) into itself, then det jac f is contracted by f . Let A and B be two elements of PGL 3 (C). Set Q = AσB (resp. Q = AρB, resp. Q = AτB).
Then det jac Q is the union of three lines in general position (resp. the union of a "double" line and a "simple" line, resp. a triple line). We will give a criterion which allows us to determine if a quadratic rational map is birational or not.

Theorem 4.2.2 ([56]

). -Let Q be a rational map; assume that Q is purely quadratic and non degenerate (i.e. det jac Q ≡ 0). Assume that Q contracts det jac Q; then det jac Q is the union of three lines (non-concurrent when they are distincts) and Q is birational.

Moreover:

• if det jac Q is the union of three lines in general position, Q is, up to l.r. equivalence, the involution σ; • if det jac Q is the union of a "double" line and a "simple" line, Q = ρ up to l.r. conjugation.

• if det jac Q is a "triple" line, Q belongs to O l.r. (τ).

Corollary 4.2.3 ([56]

). -A quadratic rational map from P

: z) (Q 0 : Q 1 : Q 2 )
. Up to l.r. conjugacy we can assume that det jac Q is contracted on (1 : 0 : 0); then detjac Q divides Q 1 and Q 2 which is impossible.

In the same way if det jac Q = Lq where L is linear and q non degenerate and quadratic, we can assume that q = 0 is contracted on (1 : 0 : 0); then Q : (x : y : z) (q 1 : q : αq) and so is degenerate.

Therefore det jac Q is the product of three linear forms.

First of all let us consider the case where, up to conjugacy, det jac Q = xyz. If the lines x = 0 and y = 0 are contracted on the same point, for example (1 : 0 : 0), then Q : (x : y : z) (q : xy : αxy) which is degenerate. The lines x = 0, y = 0 and z = 0 are thus contracted on three distinct points. A computation shows that they cannot be aligned. We can assume that x = 0 (resp. y = 0, resp. z = 0) is contracted on (1 : 0 : 0) (resp. (0 : 1 : 0), resp. (0 : 0 : 1)); let us note that Q is the involution (x : y : z) (yz : xz : xy) up to l.r. conjugacy. Now let us consider the case when det jac Q has two branches x = 0 and z = 0. As we just see, the lines x = 0 and z = 0 are contracted on two distinct points, for example (1 : 0 : 0) and (0 : 1 : 0). The map Q is up to l.r. conjugacy Q : (x : y : z) (z(αy + βz) : x(γx + δy) : xz). A direct computation shows that Q is birational as soon as βδαγ = 0 and in fact l.r. equivalent to ρ.

Then assume that det(jac Q) = z 3 . We can suppose that z = 0 is contracted on (1 : 0 : 0); then Q : (x : y : z) (q : zℓ 1 : zℓ 2 ) where q is a quadratic form and the ℓ i 's are linear forms. • If (z, ℓ 1 , ℓ 2 ) is a system of coordinates we can write up to conjugacy

Q : (x : y : z) (q : xz : yz), q = ax 2 + by 2 + cz 2 + dxy. The explicit computation of det(jac Q) implies: a = b = d = 0, i.e. either Q is degenerate, or Q represents a linear map which is impossible. • Assume that (z, ℓ 1 , ℓ 2 ) is not a system of coordinates, i.e. ℓ 1 = az + ℓ(x, y), ℓ 2 = bz + εℓ(x, y).
Let us remark that ℓ is nonzero (otherwise Q is degenerate), thus we can assume that ℓ = x. Up to l.r. equivalence Q : (x : y : z) (q : xz : z 2 ).

An explicit computation implies the following equality: detjac Q = -2z 2 ∂q ∂y ; thus z divides ∂q ∂y . In other words q = αz 2 + βxz + γx 2 + δyz. Up to l.r. equivalence, we obtain Q = τ.

Finally let us consider the case: det(jac Q) = xy(xy). As we just see the lines x = 0 and y = 0 are contracted on two distinct points, for example (1 : 0 : 0) and (0 : 1 : 0). So

Q : (x : y : z) (y(ax + by + cz) : x(αx + βy + γz) : xy) with a, b, c, α, β, γ ∈ C. Let us note that the image of the line x = y by Q is ((a + b)x + cz : (α + β)x + γz : x);
it is a point if and only if c and γ are zero, then Q does not depend on z.

Set

Σ 3 := O l.r. (σ), Σ 2 := O l.r. (ρ), Σ 1 := O l.r. (τ).
Let us consider a birational map represented by

Q : (x : y : z) ℓ(ℓ 0 : ℓ 1 : ℓ 2 )
where ℓ and the ℓ i 's are linear forms, the ℓ i 's being independent. The line given by ℓ = 0 is an apparent contracted line; indeed the action of Q on P 2 (C) is obviously the action of the automorphism (ℓ 0 :

ℓ 1 : ℓ 2 ) of P 2 (C).
Let us denote by Σ 0 the set of these maps Σ 0 = ℓ(ℓ 0 : ℓ 1 : ℓ 2 ) ℓ, ℓ i linear forms, the ℓ i 's being independent .

We will abusively call the elements of Σ 0 linear elements; in fact the set

(Σ 0 ) • = f • f ∈ Σ 0
can be identified to PGL 3 (C). We have Σ 0 = O l.r. (x(x : y : z)): up to l.r. conjugacy a map ℓA can be written xA ′ then xid. This approach allows us to see degenerations of quadratic maps on linear maps. Let us remark that an element of Σ i has i points of indeterminacy and i contracted curves. An element of Σ i cannot be linearly conjugate to an element of Σ j where j = i; nevertheless they can be birationally conjugate: the involutions σ, ρ and τ are birationally conjugate to involutions of PGL 3 (C). Let us mention that a generic element of Σ i , i ≥ 1, is not birationally conjugate to a linear map.

Corollary 4.2.7 ([56]). -We have

Bir 2 = Σ 1 ∪ Σ 2 ∪ Σ 3 , Bir 2 = Σ 0 ∪ Σ 1 ∪ Σ 2 ∪ Σ 3 . Remarks 4.2.8. -i. A Noether decomposition of ρ is (z -y : y -x : y)σ(y + z : z : x)σ(x + z : y -z : z).
We recover the classic fact already mentioned in [START_REF] Hudson | Cremona Transformations in Plane and Space[END_REF][START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]: for any birational quadratic map Q with two points of indeterminacy there exist ℓ 1 , ℓ 2 and

ℓ 3 in PGL 3 (C) such that Q = ℓ 1 σℓ 2 σℓ 3 .
ii. The map τ = (x 2 : xy : y 2xz) of Σ 1 can be written ℓ 1 σℓ 2 σℓ 3 σℓ 4 σℓ 5 where

ℓ 1 = (y -x : 2y -x : z -y + x), ℓ 2 = (x + z : x : y), ℓ 3 = (-y : x + z -3y : x), ℓ 4 = (x + z : x : y), ℓ 5 = (y -x : -2x + z : 2x -y).
Therefore each element of Σ 1 is of the following type ℓ 1 σℓ 2 σℓ 3 σℓ 4 σℓ 5 where ℓ i is in PGL 3 (C) (see [START_REF] Hudson | Cremona Transformations in Plane and Space[END_REF][START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]). The converse is false: if the ℓ i 's are generic then ℓ 1 σℓ 2 σℓ 3 σℓ 4 σℓ 5 is of degree 16.

Some orbits under the left-right action

As we saw Bir 2 is a finite union of l.r. orbits but it is not a closed algebraic subset of Rat 2 :

the "constant" map (yz : 0 : 0) is in the closure of O l.r. (σ) but not in Bir 2 . To precise the nature of Bir 2 we will study the orbits of σ, ρ, τ and x(x : y : z).

Proposition 4.3.1 ([56]

).

-The dimension of

Σ 3 = O l.r. (σ) is 14.
Proof. -Let us denote by Isot σ the isotropy group of σ. Let (A, B) be an element of (SL 3 (C)) 2 such that Aσ = σB; a computation shows that (A, B) belongs to Notice that the computation of Isot ρ shows that we have the following relations

(γδx + βδz : α 2 y : αδz)ρ = ρ(γx + βz : δy : αz), α, γ, δ ∈ C * , β ∈ C.
We can compute the isotropy group of τ and show that:

Proposition 4.3.3 ([56]). -The dimension of Σ 1 is 12.
In particular we obtain the following relations: Aτ = τB when

A =   αε 0 βε εγ + 2αβ α 2 (εδ + β 2 ) 0 0 ε 2   , B =   α β 0 0 ε 0 γ δ α/ε   , where β, γ, δ ∈ C, α, ε ∈ C * .
A similar computation allows us to state the following result.

Proposition 4.3.4 ([56]

). -The dimension of Σ 0 = O l.r. (x(x : y : z)) is 10.

Incidence conditions; smoothness of Bir 2 and non-smoothness of Bir 2

Let us study the incidence conditions between the Σ i 's and the smoothness of Bir 2 :

Proposition 4.4.1 ([56]). -We have Σ 0 ⊂ Σ 1 , Σ 1 ⊂ Σ 2 , Σ 2 ⊂ Σ 3
(the closures are taken in Bir 2 ); in particular Σ 3 is dense in Bir 2 .

Proof.

-By composing σ with (z : y : εx + z) we obtain

σ ε 1 = y(εx + z) : z(εx + z) : yz which is for ε = 0 in O l.r. (σ). But σ ε 1 is l.r. conjugate to σ ε 2 = xy : (εx + z)z : yz . Let us note that lim ε→0 σ ε 2 = (xy : z 2 : yz) = ρ; so Σ 2 ⊂ Σ 3 .
If we compose ρ with (z : x + y : x), we have up to l.r. equivalence (yz + xz : x 2 : xy). Composing with (x : y : y + z), we obtain up to l.r. conjugation the map f = (yz + y 2 + xz : x 2 : xy). Set g ε := f (x/ε : y : -εz); up to l.r. conjugation g ε can be written (-εyz + y 2xz : x 2 : xy). For ε = 0 we have the map τ.

Therefore Σ 1 is contained in Σ 2 .
If ε is nonzero, then τ can be written up to l.r. conjugation:

(x 2 : xy : ε 2 y 2 + xz); for ε = 0 we obtain x(x : y : z) which is in Σ 0 . Hence Σ 0 ⊂ Σ 1 .
Thus we can state the following result.

Theorem 4.4.2 ([56]

). -The closures being taken in Bir 2 we have

Σ 0 = Σ 0 , Σ 1 = Σ 0 ∪ Σ 1 , Σ 2 = Σ 0 ∪ Σ 1 ∪ Σ 2 , Bir 2 = Σ 1 ∪ Σ 2 ∪ Σ 3 , Bir 2 = Σ 3 = Σ 0 ∪ Σ 1 ∪ Σ 2 ∪ Σ 3 with dim Σ 0 = 10, dim Σ 1 = 12, dim Σ 2 = 13 and dim Σ 3 = 14.

Theorem 4.4.3 ([56]

). -The set of quadratic birational maps is smooth in the set of rational maps.

Proof. -Because any Σ i is one orbit and because of the incidence conditions it is sufficient to prove that the closure of Σ 3 is smooth along Σ 0 . The tangent space to Σ 0 in x(x : y : z) is given by:

T x(x:y:z) Σ 0 = (α 1 x 2 + α 4 xy + α 5 xz : β 1 x 2 + β 2 y 2 + β 4 xy + β 5 xz + β 6 yz : γ 1 x 2 + β 6 z 2 + γ 4 xy + γ 5 xz + β 2 yz) α i , β i , γ i ∈ C .
The vector space S generated by (y 2 : 0 : 0), (z 2 : 0 : 0), (yz : 0 : 0), (0 : z 2 : 0),

(0 : 0 : y 2 ), (0 : 0 : z 2 ), (0 : 0 : yz) is a supplementary of T x(x:y:z) Σ 0 in Rat 2 .
Let f be an element of Σ 3 ∩ x(x : y : z) + S , it can be written

(x 2 + Ay 2 + Bz 2 +Cyz : xy + az 2 : xz + αy 2 + βz 2 + γyz).
Necessarily f has three points of indeterminacy.

Assume that a = 0; let us remark that the second component of a point of indeterminacy of f is nonzero. If (x : y : z) belongs to Ind f , then x = -az 2 /y. We have f (-az 2 /y : y : z) = (a 2 z 4 + Ay 4 + By 2 z 2 + Cy 3 z : 0 : -az 3 + αy 3 + βyz 2 + γy 2 z)

= (P : 0 : Q).
As f has three points of indeterminacy, the polynomials P and Q have to vanish on three distinct lines. In particular Q divides P:

a 2 z 4 + Ay 4 + By 2 z 2 +Cy 3 z = (My + Nz)(-az 3 + αy 3 + βyz 2 + γy 2 z). Thus (4.4.1) B = -β 2 -aγ, C = -βγ -aα, A = -αβ.
These three equations define a smooth graph through f and x(x : y : z), of codimension 3 as Σ 3 .

Assume now that a is zero; a point of indeterminacy (x :

y : z) of f satisfies xy = 0. If x = 0 we have f (0 : y : z) = (Ay 2 + Bz 2 +Cyz : 0 : αy 2 + βx 2 + γyz)
and if y = 0 we have f (x : 0 : z) = (x 2 + Bz 2 : 0 : xz + βz 2 ). The map f has a point of indeterminacy of the form (x : 0 : z) if and only if B = -β 2 . If it happens, f has only one such point of indeterminacy. Since f has three points of indeterminacy, two of them are of the form (0 : y : z) and the polynomials Ay 2 + Bz 2 +Cyz and αy 2 + βz 2 + γyz are C-colinear. We obtain the conditions

• a = 0, B = -β 2 , A = -αβ and C = -βγ if β is nonzero; • a = B = β = Aγ -αC = 0 otherwise.
Let us remark that in this last case f cannot have three points of indeterminacy. Finally we note that Σ 3 ∩ x(x : y : z) + S is contained in the graph defined by the equations (4.4.1). The same holds for the closure Σ 3 ∩ x(x : y : z) + S which, for some reason of dimension, coincides thus with this graph. Then Σ 3 is smooth along Σ 0 . Remark 4.4.4. -Since Σ 3 is smooth along Σ 0 and since we have incidence conditions, Σ 3 is smooth along Σ 2 and Σ 1 . Nevertheless we can show these two statements by constructing linear families of birational maps (see [START_REF] Cerveau | Transformations birationnelles de petit degré[END_REF]).

Proposition 4.4.5 ([56]

). -The closure of Bir 2 in P 17 ≃ Rat 2 is not smooth.

Proof. -Let φ be a degenerate birational map given by z(x : y : 0). The tangent space to O l.r. (φ) in φ is given by

T φ O l.r. (φ) = (α 1 x 2 + α 3 z 2 + α 4 xy + α 5 xz + α 6 yz : α 4 y 2 + β 3 z 2 +α 1 xy + β 5 xz + β 6 yz : γ 5 xz + γ 6 yz) α i , β i , γ i ∈ C .
A supplementary S of T φ O l.r. (φ) is the space of dimension 8 generated by (y 2 : 0 : 0), (0 : x 2 : 0), (0 : y 2 : 0), (0 : xy : 0), (0 : 0 : x 2 ), (0 : 0 : y 2 ), (0 : 0 : z 2 ), (0 : 0 : xy).

We will prove that φ + S ∩ Σ 3 contains a singular analytic subset of codimension 3. Since Σ 3 is also of codimension 3 we will obtain, using the l.r. action, the non-smoothness of Σ 3 along the orbit of φ. An element Q of φ + S can be writen

(xz + ay 2 : yz + bx 2 + cy 2 + dxy : ex 2 + f y 2 + gz 2 + hxy).
The points of indeterminacy are given by the three following equations

xz + ay 2 = 0, yz + bx 2 + cy 2 + dxy = 0, ex 2 + f y 2 + hxy = 0;
after eliminating z this yields to P 1 = P 2 = 0 where

P 1 = -ay 3 + bx 3 + cxy 2 + dx 2 y, P 2 = ex 4 + f x 2 y 2 + a 2 gy 4 + hx 3 y.
Let us remark that if, for some values of the parameters, P 1 vanishes on three distinct lines and divides P 2 , then the corresponding map Q has three points of indeterminacy and is birational, more precisely Q is in Σ 3 . The fact that P 1 divides P 2 gives (4.4.2)

P 2 = (Ax + By)P 1 ⇔              e = bA f = cA + dB a 2 g = -aB h = dA + bB aA = cB
Let us note that the set Λ of parameters such that a = 0, b fce = 0, bhde = 0 satisfies the system (4.4.2) (with A = e/b and B = 0). The set Λ is of codimension 3 and is not smooth. The intersection Λ ′ of quadrics b fce = 0 and bhde = 0 is not smooth. Indeed Λ ′ contains the linear space E given by b = e = 0 but is not reduced to E: for example the space defined by b = c = d = e = f = h is contained in Λ ′ and not in E. Since codim E = codim Λ ′ the set Λ ′ is thus reducible and then not smooth; it is the same for Λ.

If a = b = e = 0 (resp. b = c = d = e = f = h = 1, a = 0) the polynomial P 1 is equal to cxy 2 + dx 2 y (resp. x 3 + xy 2 + x 2 y)
and in general vanishes on three distinct lines. So we have constructed in Σ 3 ∩ φ + S a singular analytic set of codimension 3. i.e. a 2-dimensional linear system Λ of conics. Such a system is a homaloidal net if it possesses three base-points, that is three points through which all the elements of Λ pass. There are three different such nets • the nets Λ 3 of conics with three distinct base-points;

• the nets Λ 2 of conics passing through two points, all having at one of them the same tangent; • the nets Λ 1 of conics mutually osculating at a point. In order to have three conics that generate a homaloidal net Λ it suffices to annihilate the minors of a matrix

ℓ 0 ℓ 1 ℓ 2 ℓ ′ 0 ℓ ′ 1 ℓ ′ 2
whose elements are linear forms in the indeterminates x, y and z. Indeed the two conics described by (4.5.1)

ℓ 0 ℓ ′ 1 -ℓ ′ 0 ℓ 1 = 0, ℓ 0 ℓ ′ 2 -ℓ 2 ℓ ′ 0 = 0 have four points in common. One of them ((ℓ 0 = 0) ∩ (ℓ ′ 0 = 0)) doesn't belong to the third conic ℓ 1 ℓ ′ 2 -ℓ ′ 1 ℓ 2 = 0 obtained from (4.5.1) by eliminating ℓ 0 /ℓ ′ 0 . So Λ is given by a 0 (ℓ 0 ℓ ′ 1 -ℓ ′ 0 ℓ 1 ) + a 1 (ℓ 0 ℓ ′ 2 -ℓ 2 ℓ ′ 0 ) + a 2 (ℓ 1 ℓ ′ 2 -ℓ ′ 1 ℓ 2 ) = 0 with (a 0 : a 1 : a 2 ) ∈ P 2 (C).
Let x, y, z be some projective coordinates in P and let u, v, w be some projective coordinates in P ′ , another plane which coincides with P . Let f be the algebraic correspondance between these two planes; it is defined by

ϕ(x, y, z; u, v, w) = 0 ψ(x, y, z; u, v, w) = 0
As f is a birational isomorphism we can write ϕ and ψ as follows

ϕ(x, y, z; u, v, w) = uℓ 0 (x, y, z) + vℓ 1 (x, y, z) + wℓ 2 (x, y, z), ψ(x, y, z; u, v, w) = uℓ ′ 0 (x, y, z) + vℓ ′ 1 (x, y, z) + wℓ ′ 2 (x, y, z) and also ϕ(x, y, z; u, v, w) = xL 0 (u, v, w) + yL 1 (u, v, w) + zL 2 (u, v, w), ψ(x, y, z; u, v, w) = xL ′ 0 (u, v, w) + yL ′ 1 (u, v, w) + zL ′ 2 (u, v, w) where ℓ i , ℓ ′
i , L i and L ′ i are some linear forms. This implies in particular that (4.5.2)

(u : v : w) = (ℓ 1 ℓ ′ 2 -ℓ 2 ℓ ′ 1 : ℓ 2 ℓ ′ 0 -ℓ 0 ℓ ′ 2 : ℓ 0 ℓ ′ 1 -ℓ 1 ℓ ′ 0 ) i.e. u (resp. v, resp. w) is a quadratic form in x, y, z.
On can note that if m = (u : v : w) ∈ P ′ belongs to the line D given by a 0 u + a 1 v + a 2 w = 0 the point (x : y : z) corresponding to it via (4.5.2) belongs to the conic given by

a 0 (ℓ 1 ℓ ′ 2 -ℓ 2 ℓ ′ 1 ) + a 1 (ℓ 2 ℓ ′ 0 -ℓ 0 ℓ ′ 2 ) + a 2 (ℓ 0 ℓ ′ 1 -ℓ 1 ℓ ′ 0 ) = 0.
So the lines of a plane thus correspond to the conics of a homaloidal net of the other plane.

Conversely we can associate a quadratic map between two planes to a homaloidal net of conics in one of them. Let Λ be an arbitrary homaloidal net of conics in P and let us consider a projectivity θ between Λ and the net of lines in P ′ . Let m be a point of P and let us assume that m is not a base-point of Λ. The elements of Λ passing through m is a pencil of conics with four base-points: the three base-points of Λ and m. To this pencil corresponds a pencil of lines whose base-point m is determined by m. To a point m ′ ∈ P ′ corresponds a pencil of conics in P , the image of the pencil of lines centered in m. Therefore the map which sends m to m gives rise to a Cremona map from P into P ′ which sends the conics of P into the lines of P ′ .

So we have the following statement. • If Λ has three distinct base-points we can assume that these points are p 0 = (1 : 0 : 0), p 1 = (0 : 1 : 0), p 2 = (0 : 0 : 1) and Λ is thus given by

a 0 yz + a 1 xz + a 2 xy = 0, (a 0 : a 1 : a 2 ) ∈ P 2 (C).
The map f is defined by (x : y : z) (yz : xz : xy) and can easily be inverted ( f is an involution).

• If Λ has two distinct base-points, we can assume that the conics of Λ are tangent at p 2 = (0 : 0 : 1) to the line x = 0 and also pass through p 0 = (1 : 0 : 0). Then Λ is given by

a 0 xz + a 1 xy + a 2 y 2 = 0, (a 0 : a 1 : a 2 ) ∈ P 2 (C).
The map f is defined by (x : y : z) (xz : xy : y 2 ) and its inverse is (u : v : w) (v 2 : vw : uw).

• If the conics of Λ are mutually osculating at p 2 = (0 : 0 : 1), we can assume that Λ contains the two degenerated conics x 2 = 0 and xy = 0. Let C be an irreducible conic in Λ; assume that C ∩ (y = 0) = p 0 and that p 1 = (0 : 1 : 0) is the pole of y = 0 with respect to C . Assume finally that (1 : 1 : 1) belongs to C then C is given by xz + y 2 = 0 and Λ is defined by

a 0 (xz + y 2 ) + a 1 x 2 + a 2 xy = 0, (a 0 : a 1 : a 2 ) ∈ P 2 (C).
The map f is (x : y : z) (xzy 2 : x 2 : xy) and its inverse is (u

: v : w) (v 2 : vw : uv + w 2 ).
Remark 4.5.3. -We can see that f and f -1 have the same type. So the homaloidal nets associated to f and f -1 have the same type.

Cubic birational maps

The space of birational maps which are purely of degree 2 is smooth and connected. Is it the case in any degree ? Let us see what happens in degree 3. In the old texts we can find a description of cubic birational maps which is based on enumerative geometry. In [START_REF] Cerveau | Transformations birationnelles de petit degré[END_REF]Chapter 6] we give a list of normal forms up to l.r. conjugation, the connectedness appearing as a consequence of this classification. The methods are classical: topology of the complement of some plane curves, contraction of the jacobian determinant... Unfortunately, as soon as the degree is greater than 3 we have no criterion as in degree 2: if f is the map (x 2 y : xz 2 : y 2 z), the zeroes of det jac f are contracted but f is not invertible. Nevertheless if f is birational, the curve det jac f = 0 is contracted and it helps in a lot of cases. We show that in degree 3 the possible configurations of contracted curves are the following unions of lines and conics:

{5} {4} {3} {2} {1} {10} {9} {8} {7} {6} {15} {14} {13} {12} {11}
The following table gives the classification of cubic birational maps up to conjugation:

(xz 2 + y 3 : yz 2 : z 3 ) {1} {1} 13 (xz 2 : x 2 y : z 3 ) {2} {2} 15 (xz 2 : x 3 + xyz : z 3 ) {2} {2} 15 (x 2 z : x 3 + z 3 + xyz : xz 2 ) {2} {2} 14 (x 2 z : x 2 y + z 3 : xz 2 ) {2} {2} 15 (xyz : yz 2 : z 3 -x 2 y) {2} {8} 14 (x 3 : y 2 z : xyz) {3} {3} 15 (x 2 (y -z) : xy(y -z) : y 2 z) {3} {10} 15 (x 2 z : xyz : y 2 (x -z)) {3} {10} 15
(xyz : y 2 z : x(y 2 -xz)) {3} {10} 15 (x 3 : x 2 y : (x -y)yz) {4} {4} 15 (x 2 (x -y) : xy(x -y) : xyz + y 3 ) {4} {4} 16 (xz(x + y) : yz(x + y) : xy 2 ) {5} {5} 16 (x(x + y)(y + z) : y(x + y)(y + z) : xyz) {5} {12} 16 (x(x + y + z)(x + y) : y(x + y + z)(x + y) : xyz) {5} {12} 16 (x(x 2 + y 2 + γxy) : y(x 2 + y 2 + γxy) : xyz), γ 2 = 4 {6} {6} 15 1 parameter (xz(y + x) : yz(y + x) : xy(x -y)) {7} {7} 16 (x(x 2 + y 2 + γxy + γ + xz + yz) : y(x 2 + y 2 + γxy + γ + xz + yz) : xyz) {7} {14} 16 1 parameter (y(x -y)(x + z) : x(x -y)(z -y) : yz(x + y)) {7} {14} 16
+ yz + xz) : xyz), γ = 0, 1 {13} {13} 16 1 parameter (x(x 2 + y 2 + γxy + xz) : y(x 2 + y 2 + γxy + xz) : xyz), γ 2 = 4, {14} {7} 16 1 parameter (x(x 2 + yz + xz) : y(x 2 + yz + xz) : xy(x -y)) {14} {7} 16 (x(x 2 + y 2 + γxy + δxz + yz) : y(x 2 + y 2 + γxy + δxz + yz) : xyz), γ 2 = 4, δ = γ ± {15} {15} 16 2 parameters
where γ denotes a complex number and where

γ + := γ + γ 2 -4 2 γ -:= γ -γ 2 -4 2 .
For any model we mention the configuration of contracted curves of the map (second column), the configuration of the curves contracted by the inverse (third column), the dimension of its orbit under the l.r. action (fourth column) and the parameters (fifth column). Any cubic birational map can be written, up to dynamical conjugation, A f where A denotes an element of PGL 3 (C) and f an element of the previous table. This classification allows us to prove that the "generic" element has the last configuration and allows us to establish that the dimension of the space Bir 3 of birational maps purely of degree 3 is 18. Up to l.r. conjugation the elements having the generic configuration {15} form a family of 2 parameters: in degree 2 there are 3 l.r. orbits, in degree 3 an infinite number.

Let us note that the configurations obtained by degenerescence from picture {15} do not all appear. In degree 2 there is a similar situation: the configuration of three concurrent lines is not realised as the exceptional set of a quadratic birational map.

Let us denote by X the set of birational maps purely of degree 3 having configuration {15}. We establish that the closure of X in Bir 3 is Bir 3 . We can show that Bir 3 is irreducible, in fact rationally connected ([56, Chapter 6]); but if Bir 2 is smooth and irreducible Bir 3 , viewed in P 29 (C) ≃ Rat 3 , doesn't have the same properties ([56, Theorem 6.38]).

Let us mention another result. Let dJ d be the subset of dJ made of birational maps of degree d and let V d be the subset of Bir(P 2 ) defined by

V d = A f B A, B ∈ PGL 3 (C), f ∈ dJ d .
The dimension of Bir d is equal to 4d + 6 and V d its unique irreducible component of maximal dimension ( [START_REF] Dang Nguyen | Composantes irréductibles des transformations de Cremona de degré d[END_REF]).

CHAPTER 5 FINITE SUBGROUPS OF THE CREMONA GROUP

The study of the finite subgroups of Bir(P 2 ) began in the 1870 ′ s with Bertini, Kantor and Wiman ([27, 130, 184]). Since then, many mathematicians have been interested in the subject, let us for example mention [START_REF] Bayle | Birational involutions of P 2[END_REF][START_REF] Beauville | p-elementary subgroups of the Cremona group[END_REF][START_REF] Beauville | On Cremona transformations of prime order[END_REF][START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF][START_REF] De Fernex | On planar Cremona maps of prime order[END_REF][START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]. In 2006 Dolgachev and Iskovkikh improve the results of Kantor and Wiman and give the description of finite subgroups of Bir(P 2 ) up to conjugacy. Before stating one of the key result let us introduce some notions.

Let S be a smooth projective surface. A conic bundle η : S → P 1 (C) is a morphism whose generic fibers have genus 0 and singular fibers are the union of two lines. A surface endowed with conic bundles is isomorphic either to F n , or to F n blown up in a finite number of points, all belonging to different fibers (the number of blow-ups is exactly the number of singular fibers).

A surface S is called a del Pezzo surface if -K S is ample, which means that -K S • C > 0 for any irreducible curve C ⊂ S. Any del Pezzo surface except P 1 (C) × P 1 (C) is obtained by blowing up r points p 1 , . . ., p r of P 2 (C) with r ≤ 8 and no 3 of p i are collinear, no 6 are on the same conic and no 8 lie on a cubic having a singular point at one of them. The degree of S is 9r.

Theorem 5.0.1 ([139, 123]

). -Let G be a finite subgroup of the Cremona group. There exists a smooth projective surface S and a birational map φ : P 2 (C)

S such that φGφ -1 is a subgroup of Aut(S). Moreover one can assume that

• either S is a del Pezzo surface;

• or there exists a conic bundle S → P 1 (C) invariant by φGφ -1 .

Remark 5.0.2. -The alternative is not exclusive: there are conic bundles on del Pezzo surfaces.

Dolgachev and Iskovskikh give a characterization of pairs (G, S) satisfying one of the possibilities of Theorem 5.0.1. Then they use Mori theory to determine when two pairs are birationally conjugate. Let us note that the first point was partially solved by Wiman and Kantor but not the second. There are still some open questions ([83] §9), for example the description of the algebraic varieties that parametrize the conjugacy classes of the finite subgroups of Bir(P 2 ). Blanc gives an answer to this question for finite abelian subgroups of Bir(P 2 ) with no elements with an invariant curve of positive genus, also for elements of finite order (resp. cyclic subgroups of finite order) of the Cremona group ( [START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF][START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]). Let L be the linear system of cubics through the p i 's. A cubic is given by a homogeneous polynomial of degree 3 in three variables. The dimension of the space of homogeneous polynomials of degree 3 in 3 variables is 10 thus dim{C |C cubic} = 10 -1 = 9; cubics have to pass through p 1 , . . ., p 7 so dim L = 2. Let p be a generic point of P 2 (C); let us consider the pencil L p containing elements of L through p. A pencil of generic cubics We can check that such an involution is birational, of degree 8; its fixed points form an hyperelliptic curve of genus 3, degree 6 with 7 ordinary double points which are the p i 's. The exceptional locus of a Geiser involution is the union of seven cubics passing through the seven points of indeterminacy of I G and singular in one of these seven points (cubics with double point).

a 0 C 0 + a 1 C 1 , C 0 , C 1 two cubics (a 0 : a 1 ) ∈ P 1 (C)
The involution I G can be realized as an automorphism of a del Pezzo surface of degree 2.

I B = I B (p 1 , . . . , p 8 ).
Its fixed points form a non hyperelliptic curve of genus 4, degree 9 with triple points in the p i 's and such that the normalisation is isomorphic to a singular intersection of a cubic surface and a quadratic cone in P 3 (C).

The involution I B can be realized as an automorphism of a del Pezzo surface of degree 1.

de Jonquières involutions.

-Let C be an irreductible curve of degree ν ≥ 3. Assume that C has a unique singular point p and that p is an ordinary multiple point with multiplicity ν -2. To (C , p) we associate a birational involution I J which fixes pointwise C and which preserves lines through p. Let m be a generic point of P 2 (C) \ C ; let r m , q m and p be the intersections of the line (mp) and C ; the point I J (m) is defined by the following property: the cross ratio of m, I J (m), q m and r m is equal to -1. The map I J is a de Jonquières involution of degree ν centered in p and preserving C . More precisely its fixed points are the curve C of genus ν -2 for ν ≥ 3.

For ν = 2 the curve C is a smooth conic; we can do the same construction by choosing a point p not on C .

ω = (x, y, z) ∈ C 3 u(x, y, z) = v(x, y, z) = w(x, y, z) = 0 .
Let us give a geometric interpretation of the degree. Let F be a foliation of degree ν on P 2 (C), let D be a generic line, and let p a point of D \ Sing F . We say that F is transversal to D if the leaf L p of F in p is transversal to D in p, otherwise we say that p is a point of tangency between F and D. The degree ν of F is exactly the number of points of tangency between F and D. Indeed, if ω be a 1-form of degree ν + 1 on C 3 defining F , it is of the following type ω = P 0 dx + P 1 dy + P 2 dz, P i homogeneous polynomial of degree ν + 1.

Let us denote by ω 0 the restriction of ω to the affine chart x = 1

ω 0 = ω |x=1 = P 1 (1, y, z)dy + P 2 (1, y, z)dz.
Assume that the line D = z = 0 is a generic line. In the affine chart x = 1 the fact that the radial vector field vanishes on D implies that

P 0 (1, y, 0) + yP 1 (1, y, 0) = 0.
Generically (on the choice of D) the polynomial P 0 (1, y, 0) is of degree ν + 1 so P 1 (1, y, 0) is of degree ν. Since ω 0|D = P 1 (1, y, 0)dy, the restriction of ω 0 to D vanishes into ν points: the number of tangencies between F and D is ν.

The classification of foliations of degree 0 and 1 on P 2 (C) is known since the XIXth century. A foliation of degree 0 on P 2 (C) is a pencil of lines, i.e. is given by xdyydx = x 2 d y x , the pencil of lines being y x = cte. Each foliation of degree 1 on the complex projective plane has 3 singularities (counting with multiplicity), has, at least, one invariant line and is given by a rational closed 1-form (in other words there exists a homogeneous polynomial P such that ω/P is closed); the leaves are the connected components of the "levels" of a primitive of this 1-form. The possible 1-forms are

x λ 0 y λ 1 z λ 2 , λ i ∈ C, ∑ i λ i = 0, x y exp z y , Q x 2
where Q is a quadratic form of maximal rank. More generally a foliation of degree 0 on P n (C) is associated to a pencil of hyperplanes, i.e. is given by the levels of ℓ 1 /ℓ 2 where ℓ 1 , ℓ 2 are two independent linear forms. Let F be a foliation of degree 1 on P n (C). Then

• either there exists a projection τ : P n (C) P 2 (C) and a foliation of degree 1 on P 2 (C)

such that F = τ * F 1 ,
• or the foliation is given by the levels of Q/L 2 where Q (resp. L) is of degree 2 (resp. 1). For ν ≥ 2 almost nothing is known except the generic nonexistence of an invariant curve ( [START_REF] Jouanolou | Équations de Pfaff algébriques[END_REF][START_REF] Cerveau | Holomorphic foliations in CP(2) having an invariant algebraic curve[END_REF]). Let us mention that

• there exists a description of the space of foliations of degree 2 in P 3 (C) (see [START_REF] Cerveau | Irreducible components of the space of holomorphic foliations of degree two in CP(n), n ≥ 3[END_REF]);

• any foliation of degree 2 is birationally conjugate to another (not necessary of degree 2) given by a linear differential equation dy dx = P(x, y) where P is in C(x)[y] (see [START_REF] Cerveau | Complex codimension one singular foliations and Godbillon-Vey sequences[END_REF]).

A regular point m of F is an inflection point for F if L m has an inflection point in m. Let us denote by Flex F the closure of these points. A way to find this set has been given by Pereira in [START_REF] Pereira | Vector fields, invariant varieties and linear systems[END_REF]: let

Z = E ∂ ∂x + F ∂ ∂y + G ∂ ∂z
be a homogeneous vector field on C 3 non colinear to the radial vector field

R = x ∂ ∂x + y ∂ ∂y + z ∂ ∂z describing F (i.e. ω = i R i Z dx ∧ dy ∧ dz). Let us consider H = x E Z(E) y F Z(F) z G Z(G)
; the zeroes of H is the union of FlexF and the lines invariant by F .

5.2.2.

Foliations of degree 2 and involutions. -To any foliation F of degree 2 on P 2 (C) we can associate a birational involution I F : let us consider a generic point m of F , since F is of degree 2, the tangent T m L m to the leaf through m is tangent to F at a second point p, the involution I F is the map which swaps these two points. More precisely let us assume that F is given by the vector field χ. The image by I F of a generic point m is the point m + sχ(m) where s is the unique nonzero parameter for which χ(m) and χ(m + sχ(m)) are colinear.

Let q be a singular point of F and let P (q) be the pencil of lines through q. The curve of points of tangency Tang(F , P (q)) between F and P (q) is blown down by I F on q. We can verify that all contracted curves are of this type.

Jouanolou example.

-The foliation F J is described in the affine chart z = 1 by

(x 2 y -1)dx -(x 3 -y 2 )dy;
this example is due to Jouanolou and is the first known foliation without invariant algebraic curve.

We can compute I F J :

(xy 7 + 3x 5 y 2 z -x 8 -5x 2 y 4 z 2 + 2y 3 z 5 + x 3 yz 4 -xz 7 : 3xy 5 z 2 + 2x 5 z 3 -x 7 y -5x 2 y 2 z 4 + x 4 y 3 z + yz 7 -y 8 :
xy 4 z 3 -5x 4 y 2 z 2 -y 7 z + 2x 3 y 5 + 3x 2 yz 5 -z 8 + x 7 z).
its degree is 8 and Ind I F J = Sing F J = (ξ j : ξ -2 j : 1) j = 0, . . . , 6, ξ 7 = 1 .

As there is no invariant algebraic curve for F J we have

Flex F J = Fix I F J = 2(3x 2 y 2 z 2 -xy 5 -x 5 z -yz 5 );
this curve is irreducible.

The subgroup of Aut(P 2 ) which preserves a foliation F of P 2 (C) is called the isotropy group of F ; it is an algebraic subgroup of Aut(P 2 ) denoted by

Iso F = ϕ ∈ Aut(P 2 ) ϕ * F = F .
The point (1 : 1 : 1) is a singular point of Flex F J , it is an ordinary double point. If we let Iso F J act, we note that each singular point of F J is an ordinary double point of Flex F J and that Flex F J has no other singular point. Therefore Flex F J has genus (6-1)(6-2)

2 -7 = 3.
The singular points of Sing F J are in general position so I F J is a Geiser involution. The group I F J , Iso F J is a finite subgroup of Bir(P 2 ); it cannot be conjugate to a subgroup of Aut(P 2 ) because Fix I J is of genus 3. This group of order 42 appears in the classification of finite subgroups of Bir(P 2 ) (see [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]).

The generic case.

-Let us recall that if F is of degree ν, then # Sing F = ν 2 + ν + 1 (let us precise that points are counted with multiplicity). Thus a quadratic foliation has seven singular points counted with multiplicity; moreover if we choose seven points p 1 , . . . , p 7 in general position, there exists one and only one foliation F such that Sing F = p 1 , . . . , p 7 (see [START_REF] Gómez-Mont | Stability of meromorphic vector fields in projective spaces[END_REF]).

Theorem 5.2.1 ([54]

). -Let p 1 , . . . , p 7 be seven points of P 2 (C) in general position. Let F be the quadratic foliation such that Sing F = p 1 , . . . , p 7 and let I G be the Geiser involution associated to the p i 's. Then I G and I F coincide.

Corollary 5.2.2 ([54]

). -The involution associated to a generic quadratic foliation of P 2 (C) is a Geiser involution. This allows us to give explicit examples of Geiser involutions. Indeed we can explicitely write a generic foliation of degree 2 of P 2 (C) : we can assume that (0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0) and (1 : 1 : 1) are singular for F and that the line at infinity is not preserved by F so the foliation F is given in the affine chart z = 1 by the vector field -Let us consider a foliation F of degree 3 on P 2 (C). Every generic line of P 2 (C) is tangent to F in three points. The "application" which switches these three points is in general multivalued; we give a criterion which says when this application is birational. This allows us to give explicit examples of trivolutions and finite subgroups of Bir(P 2 ) (see [START_REF] Cerveau | Feuilletages et transformations périodiques[END_REF]).

Number of conjugacy classes of birational maps of finite order

The number of conjugacy classes of birational involutions in Bir(P 2 ) is infinite (Theorem 5.1.2). Let n be a positive integer; what is the number ν(n) of conjugacy classes of birational maps of order n in Bir(P 2 ) ? De Fernex gives an answer for n prime ( [START_REF] De Fernex | On planar Cremona maps of prime order[END_REF]); there is a complete answer in [START_REF] Blanc | The number of conjugacy classes of elements of the Cremona group of some given finite order[END_REF].

Theorem 5.3.1 ([29]

). -For n even, ν(n) is infinite; this is also true for n = 3, 5.

For any odd integer n = 3, 5 the number of conjugacy classes ν(n) of elements of order n in Bir(P 2 ) is finite. Furthermore

• ν(9) = 3; • ν(15) = 9; • ν(n) = 1 otherwise.
Let us give an idea of the proof. Assume that n is even. Let us consider an element P of C[x n ] without multiple root. Blanc proves that there exists a birational map f of order 2n such that f n is the involution (x, P(x)/y) that fixes the hyperelliptic curve y 2 = P(x). So the case n = 2 allows to conclude for any even n ≥ 4.

To any elliptic curve C we can associate a birational map f C of the complex projective plane whose set of fixed points is C . Indeed let us consider the smooth cubic plane curve C = {(x : y : z) ∈ P 2 (C) | P(x, y, z) = 0} where P is a non-singular form of degree 3 in 3 variables.

The surface S = {(w : x : y : z) ∈ P 3 (C) | w 3 = P(x, y, z)} is a del Pezzo surface of degree 3 (see for example [START_REF] Kollár | Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). The map f C : w → exp( 2iπ3 )w gives rise to an automorphism of S whose set of fixed points is isomorphic to C . Since the number of isomorphism classes of ellitpic curves is infinite the number of conjugacy classes in Bir(P 2 ) of elements of order 3 is thus also infinite. A similar construction holds for birational maps of order 5.

To show the last part of the statement Blanc applies Theorem 5.0.1 to the subgroup generated by a birational map of odd order n ≥ 7.

Birational maps and invariant curves

Examining to Theorem 5.1.2, it is not surprising that simultaneously Castelnuovo was interested in birational maps that preserve curves of positive genus. Let C be an irreducible curve of P 2 (C); the inertia group of C , denoted by Ine(C ), is the subgroup of Bir(P 2 ) that fixes pointwise C . Let C ⊂ P 2 (C) be a curve of genus > 1, then an element of Ine(C ) is either a de Jonquières map, or a birational map of order 2, 3 or 4 (see [START_REF] Castelnuovo | Sulle transformazioni cremoniane del piano, che ammettono una curva fiera[END_REF]). This result has been recently precised as follows.

Theorem 5.4.1 ([35]

). -Let C ⊂ P 2 (C) be an irreducible curve of genus > 1. Any f of Ine(C ) is either a de Jonquières map, or a birational map of order 2 or 3. In the first case, if f is of finite order, it is an involution.

To prove this statement Blanc, Pan and Vust follow Castelnuovo's idea; they construct the adjoint linear system of C : let π : Y → P 2 (C) be an embedded resolution of singularities of C and let C be the strict transform of C . Let ∆ be the fixed part of the linear system

| C + K Y |. If | C + K Y | is neither empty, nor reduced to a divisor, π * | C + K Y |\∆
is the adjoint linear system. By iteration they obtain that any element f of Ine(C ) preserves a fibration F that is rational or elliptic. If F is rational, f is a de Jonquières map. Let us assume that F is elliptic. Since C is of genus > 1 the restriction of f to a generic fiber is an automorphism with at most two fixed points: f is thus of order 2, 3 or 4. Applying some classic results about automorphisms of elliptic curves Blanc, Pan and Vust show that f is of genus 2 or 3. Finally they note that this result cannot be extended to curves of genus ≤ 1; this eventuality has been dealt with in [START_REF] Pan | Sur le sous-groupe de décomposition d'une courbe irrationnelle dans le groupe de Cremona du plan[END_REF][START_REF] Blanc | On the inertia group of elliptic curves in the Cremona group of the plane[END_REF] with different technics.

Let us also mention results due to Diller, Jackson and Sommese that are obtained from a more dynamical point of view.

Theorem 5.4.2 ([78]

). -Let S be a projective complex surface and f be a birational map on S. Assume that f is algebraically stable and hyperbolic. Let C be a connected invariant curve of f . Then C is of genus 0 or 1.

If C is of genus 1, then, after contracting some curves in S, there exists a meromorphic

1-form such that • f * ω = αω with α ∈ C,
• and -C is the divisor of poles of ω.

The constant α is determined solely by C and f |C .

They are also interested in the number of irreducible components of an invariant curve of a birational map f ∈ Bir(S) where S denotes a rational surface. They prove that except in a particular case, this number is bounded by a quantity that only depends on S.

Theorem 5.4.3 ([78]

). -Let S be a rational surface and let f be a birational map on S.

Assume that f is algebraically stable and hyperbolic. Let C ⊂ S be a curve invariant by f .

If one of the connected components of C is of genus 1 the number of irreducible components of C is bounded by dim Pic(S) + 2.

If every connected component of C has genus 0 then • either C has at most dim Pic(S) + 1 irreducible components;

• or there exists an holomorphic map π : S → P 1 (C), unique up to automorphisms of P 1 (C), Diff q (N) are isomorphic as abstract groups then p = q and the isomorphism is induced by a C p -diffeomorphism from M to N.

such

Theorem 6.1.1 ([91]

). -Let M and N be two smooth manifolds without boundary. Let ϕ be an isomorphism between Diff p (M) and Diff q (N). Then p is equal to q and there exists

ψ : M → N of class C p such that ϕ( f ) = ψ f ψ -1 , ∀ f ∈ Diff p (M).
There are similar statements for diffeomorphisms which preserve a volume form, a symplectic form ( [START_REF] Banyaga | On isomorphic classical diffeomorphism groups[END_REF][START_REF] Banyaga | The structure of classical diffeomorphism groups, volume 400 of Mathematics and its Applications[END_REF])... If M is a Riemann surface of genus larger than 2, then the group of diffeomorphisms which preserve the complex structure is finite. Thus there is no hope to obtain a similar result as Theorem 6.1.1: we can find two distinct curves of genus 3 whose group of automorphisms is trivial. More generally if M is a complex compact manifold of general type, then Aut(M) is finite and often trivial. On the contrary let us mention two examples of homogeneous manifolds:

• any automorphism of Aut(P 2 ) is the composition of an inner automorphism, the action of an automorphism of the field C and the involution u → t u -1 (see for example [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF]); • the automorphisms group of the torus

C/Γ is the semi-direct product C/Γ ⋊ Z/2Z ≃ R 2 /Z 2 ⋊ Z/2Z for all lattices Γ = Z[i], Z[j].
In the first part of the Chapter we deal with the structure of the automorphisms group of the affine group Aff(C) of the complex line (Theorem 6.2.1). Let us say a few words about it. Let φ be an automorphism of Aff(C) and let G be a maximal (for the inclusion) abelian subgroup of Aff(C); then φ(G) is still a maximal abelian subgroup of Aff(C). We get the nature of φ from the precise description of the maximal abelian subgroups of Aff(C).

In the second part of the Chapter we are focused on the automorphisms group of polynomial automorphisms of C 2 . Let φ be an automorphism of Aut(C 2 ). Using the structure of amalgamated product of Aut(C 2 ) (Theorem 2.1.2) Lamy determines the centralisers of the elements of Aut(C 2 ) (see [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]); we thus obtain that the set of Hénon automorphisms is preserved by φ (Proposition 6.3.5). Since the elementary group E is maximal among the solvable subgroups of length 3 of Aut(C 2 ) (Proposition 6.3.7) we establish a property of rigidity for E: up to conjugation by a polynomial automorphism of the plane φ(E) = E (see Proposition 6.3.8). This rigidity allows us to characterize φ.

We finish Chapter 6 with the description of Aut(Bir(P 2 )). Let φ be an automorphism of Bir(P 2 ). The study of the uncountable maximal abelian subgroups G of Bir(P 2 ) leads to the following alternative: either G owns an element of finite order, or G preserves a rational fibration (that is G is, up to conjugation, a subgroup of dJ = PGL 2 (C(y)) ⋊ PGL 2 (C)). This allows us to prove that PGL 3 (C) is pointwise invariant by φ up to conjugacy and up to the action of an automorphism of the field C. The last step is to establish that ϕ(σ) = σ; we then conclude with Theorem 2.1.4.

The affine group of the complex line

Let Aff(C) = z → az + b a ∈ C * , b ∈ C
( f ) = τ(ψ f ψ -1 ), ∀ f ∈ Aff(C).
Proof. -If G is a maximal abelian subgroup of Aff(C) then φ(G) too. The maximal abelian subgroups of Aff(C) are

T = z → z + α α ∈ C and D z 0 = z → α(z -z 0 ) + z 0 α ∈ C * .
Note that T has no element of finite order so φ(T) = T and φ(D z 0 ) = D z ′ 0 . Up to a conjugacy by an element of T one can suppose that φ(D 0 ) = D 0 . In other words one has

• an additive morphism τ 1 :

C → C such that φ(z + α) = z + τ 1 (α), ∀ α ∈ C; • a multiplicative one τ 2 : C * → C * such that φ(αz) = τ 2 (α)z, ∀ α ∈ C * .
On the one hand we have φ(αz

+ α) = φ(αz)φ(z + 1) = τ 2 (α)z + τ 2 (α)τ 1 (1)
and on the other hand φ(αz

+ α) = φ(z + α)φ(αz) = τ 2 (α)z + τ 1 (α).
Therefore τ 1 (α) = τ 2 (α)κ where κ = τ 1 (1). In particular τ 1 is multiplicative and additive, i.e. τ 1 is an automorphism of the field C (and τ 2 too).

Then

φ(αz + β) = τ 2 (α)z + τ 1 (β) = τ 2 (α)z + τ 2 (β)κ = τ 2 (αz + τ -1 2 (κ)β) = τ 2 (τ -1 2 (κ)z • αz + β • τ 2 (κ)z).
Let us denote by Aut(C n ) the group of polynomial automorphisms of C n . Ahern and Rudin show that the group of holomorphic automorphisms of C n and the group of holomorphic automorphisms of C m have different finite subgroups when n = m (see [START_REF] Ahern | Periodic automorphisms of C n[END_REF]); in particular the group of holomorphic automorphisms of C n is isomorphic to the group of holomorphic automorphisms of C m if and only if n = m. The same argument holds for Aut(C n ) and Aut(C m ).

φ( f ) = τ(ψ f ψ -1 ), ∀ f ∈ Aut(C 2 ).
Remark 6.3.2. -Let us mention the existence of a similar result for the subgroup of tame automorphisms of Aut(C n ): every automorphism of the group of polynomial automorphisms of complex affine n-space inner up to field automorphisms when restricted to the subgroup of tame automorphisms ( [START_REF] Kraft | Automorphisms of the affine Cremona group[END_REF]).

The section is devoted to the proof of Theorem 6.3.1 which uses the well known amalgamated product structure of Aut(C 2 ) (Theorem 2.1.2). Let us recall that a Hénon automorphism is an automorphism of the type ϕg 1 . . .

g p ϕ -1 ϕ ∈ Aut(C 2 ), g i = (y, P i (y) -δ i x), P i ∈ C[y], deg P i ≥ 2, δ i ∈ C * ,
and that

A = (a 1 x + b 1 y + c 1 , a 2 x + b 2 y + c 2 ) a i , b i , c i ∈ C, a 1 b 2 -a 2 b 1 = 0 , E = (αx + P(y), βy + γ) α, β, γ ∈ C, αβ = 0, P ∈ C[y] .
Let us also recall the two following statements.

Proposition 6.3.3 ([97]

). -Let f be an element of Aut(C 2 ).

Either f is conjugate to an element of E, or f is a Hénon automorphism.

H = f ∈ Aut(C 2 ) f is a Hénon automorphism .
We also have the following: for any f in E, φ( f ) is up to conjugacy in E. But Lamy proved that a non-abelian subgroup whose each element is conjugate to an element of E is conjugate either to a subgroup of A, or to a subgroup or E. So we will try to "distinguish" A and E.

We set

E (1) = [E, E] = (x, y) → (x + P(y), y + α) α ∈ C, P ∈ C[y]
and (1) , E (1) ] = (x, y) → (x + P(y), y

E (2) = [E
) P ∈ C[y] .
The group E (2) satisfies the following property.

Lemma 6.3.6 ([70]

). -The group E (2) is a maximal abelian subgroup of E.

Proof. -Let K ⊃ E (2) be an abelian group. Let g = (g 1 , g 2 ) be in K. For any polynomial P and for any t in C let us set f tP = (x + tP(y), y). We have

(⋆) f tP g = g f tP .
If we consider the derivative of (⋆) with respect to t at t = 0 we obtain (⋄) ∂g 1 ∂x P(y) = P(g 2 ), (⋄⋄) ∂g 2 ∂x P(y) = 0.

The equality (⋄⋄) implies that g 2 depends only on y. Thus from (⋆⋆) we get: ∂g 1 ∂x is a function of y, i.e. ∂g 1 ∂x = R(y) and g 1 (x, y) = R(y)x + Q(y). As g is an automorphism, R is a constant α which is non-zero. Then (⋆⋆) can be rewritten αP(y) = P(g 2 ). For P ≡ 1 we obtain that α = 1 and for P(y) = y we have g 2 (y) = y. In other words g = (x + Q(y), y) belongs to E (2) .

Let G be a group; set

G (0) = G, G (1) = [G, G], . . . , G (p) = [G (p-1) , G (p-1) ], . . .
The group G is solvable if there exists an integer k such that G (k) = id; the smallest integer k such that G (k) = id is the length of G. The Lemma 6.3.6 allows us to establish the following statement.

Proposition 6.3.7 ([70]

). -The group E is maximal among the solvable subgroups of Aut(C 2 ) of length 3.

Proof. -Let K be a solvable group of length 3. Assume that K ⊃ E. The group K (2) is abelian and contains E (2) . As E (2) is maximal, K (2) = E (2) . The group K (2) is a normal subgroup of K so for all f = ( f 1 , f 2 ) ∈ K and g = (x + P(y), y) ∈ K (2) = E (2) we have

(⋆) f 1 (x + P(y), y) = f 1 (x, y) + Θ(P)( f 2 (x, y)) (⋆⋆) f 2 (x + P(y), y) = f 2 (x, y)
where Θ : C[y] → C[y] depends on f . The second equality implies that f 2 = f 2 (y). The derivative of (⋆) with respect to x implies ∂ f 1 ∂x (x + P(y), y)

= ∂ f 1 ∂x (x, y) thus ∂ f 1 ∂x = R(y) and f 1 (x, y) = R(y)x + Q(y), Q, R ∈ C[y].
As f is an automorphism we have f 1 (x, y) = αx + Q(y), α = 0. In other words K = E.

This algebraic characterization of E and the fact that a non-abelian subgroup whose each element is conjugate to an element of E is conjugate either to a subgroup of A or to a subgroup or E (see [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]) allow us to establish a rigidity property concerning E.

Proposition 6.3.8 ([70]

). -Let φ be an automorphism of Aut(C 2 ). There exists a polynomial automorphism ψ of C 2 such that φ(E) = ψEψ -1 .

Assume that φ(E) = E; we can show that φ(D) = D and φ(T i ) = T i where

D = (x, y) → (αx, βy) α, β ∈ C * , T 1 = (x, y) → (x + α, y) α ∈ C , T 2 = (x, y) → (x, y + β) β ∈ C .
With an argument similar to the one used in §6.2 we obtain the following statement.

Proposition 6.3.9 ([70]

). -Let φ be an automorphism of Aut(C 2 ). Then up to inner conjugacies and up to the action of an automorphism of the field C the group E is pointwise invariant by φ.

It is thus not difficult to check that if E is pointwise invariant, then φ(x, x + y) = (x, x + y). We conclude using the following fact: E and (x, x + y) generate Aut(C 2 ). Proof. -Let φ be an isomorphism of the semi-group End(C 2 ) in itself; φ induces an automorphism of C 2 . We can assume that, up to the action of an inner automorphism and up to the action of an automorphism of the field C, the restriction of φ to Aut(C 2 ) is trivial (Theorem 6.3.1).

For any α in C 2 , let us denote by f α the constant endomorphism of C 2 , equal to α. For any g in End(C 2 ) we have f α g = f α . This equality implies that φ sends constant endomorphisms onto constant endomorphisms; this defines an invertible map κ from C 2 into itself such that φ( f α ) = f κ(α) . Since g f α = f g(α) for any g in End(C 2 ) and any α in C 2 we get: φ(g) = κgκ -1 .

The restriction φ |Aut(C 2 ) is trivial so κ is trivial. Let us recall the definition of a foliation on a compact complex surface. Let S be a compact complex surface; let (U i ) be a collection of open sets which cover S. A foliation F on S is given by a family (χ i ) i of holomorphic vector fields with isolated zeros defined on the U ′ i s. The vector fields χ i satisfy some conditions on U i ∩ U j we have χ i = g i j χ j ,

g i j ∈ O * (U i ∩ U j ).
Note that a non trivial vector field χ on S defines such a foliation. The keypoint of the proof of Theorem 6.4.1 is the following Lemma.

Lemma 6.4.2 ([71]

). -Let G be an uncountable maximal abelian subgroup of Bir(P 2 ). There exists a rational vector field χ such that

f * χ = χ, ∀ f ∈ G.
In particular G preserves a foliation.

Proof. -The group G is uncountable so there exists an integer n such that

G n = f ∈ G deg f = n is uncountable. Then the Zariski's closure G n of G n in Bir n = f ∈ Bir(P 2 ) deg f ≤ n is an algebraic set and dim G n ≥ 1. Let us consider a curve in G n , i.e. a map η : D → G n , t → η(t).
Remark that the elements of G n are commuting birational maps.

For each p in P 2 (C) \ Ind η(0) -1 set

χ(p) = ∂η(s) ∂s s=0 (η(0) -1 (p)).
This formula defines a rational vector field on P 2 (C) which is non identically zero. By derivating the equality f η(s) f -1 (p) = η(s)(p) we obtain f * χ = χ. Then χ is invariant by G n ; we note that in fact χ is invariant by G.

So take an uncountable maximal abelian subgroup G of Bir(P 2 ) without periodic element and an automorphism φ of Bir(P 2 ). Then φ(G) is an uncountable maximal abelian subgroup of Bir(P 2 ) which preserves a foliation F .

Let F be an holomorphic singular foliation on a compact complex projective surface S.

Such foliations have been classified up to birational equivalence by Brunella, McQuillan and

Mendes ([40, 145, 146]). Let Bir(S, F ) (resp. Aut(S, F )) be the group of birational (resp. biholomorphic) symmetries of F , i.e. mappings g which send leaf to leaf. For a foliation F of general type, Bir(S, F ) = Aut(S, F ) is a finite group. In [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF] the authors classify those triples (S, F , g) for which Bir(S, F ) (or Aut(S, F )) is infinite. The classification leads to five classes of foliations listed below:

• F is left invariant by a holomorphic vector field;

• F is an elliptic fibration;

• S = T /G is the quotient of a complex 2-torus T by a finite group and F is the projection of the stable foliation of some Anosov diffeomorphism of T ;

• F is a rational fibration;

• F is a monomial foliation on P 1 (C) × P 1 (C) (or on the desingularisation of the quotient

P 1 (C) × P 1 (C) by the involution (z, w) → (1/z, 1/w)).
We prove that as φ(G) is uncountable, maximal and abelian without periodic element, F is a rational fibration (1) . In other words φ(G) is up to conjugacy a subgroup of

dJ = PGL 2 (C(y)) ⋊ PGL 2 (C).
The groups dJ a = (x, y) → (x + a(y), y) a ∈ C(y)

and

T = (x, y) → (x + α, y + β) α, β ∈ C
are uncountable, maximal, abelian subgroups of the Cremona group; moreover they have no periodic element. So φ(dJ a ) and φ(T) are contained in dJ. After some computations and algebraic considerations we obtain that, up to conjugacy (by a birational map), φ(dJ a ) = dJ a and φ(T) = T.

As D = (αx, βy) α, β ∈ C * acts by conjugacy on T we establish that φ(D) = D. After conjugating φ by an inner automorphism and an automorphism of the field C the groups T and D are pointwise invariant by φ. Finally we show that φ preserves (y, x) and 1

x , 1 y ; in particular we use the following identity due to Gizatullin ([105])

(hσ) 3 = id, h = x x -1 , x -y x -1 .
Since Bir(P 2 ) is generated by Aut(P 2 ) = PGL 3 (C) and 1

x , 1 y (Theorem 2.1.4) we have after conjugating φ by an inner automorphism and an automorphism of the field C: φ |Bir(P 2 ) = id.

We will give another proof of Theorem 6.4.1 in Chapter 7.

6.4.2.

Corollaries. -We obtain a similar result as Corollary 6.3.11.

Corollary 6.4.3 ([71]

). -An isomorphism of the semi-group of the rational maps from P 2 (C) into itself is inner up to the action of an automorphism of the field C.

We also can prove the following statement.

Corollary 6.4.4 ([71]

). -Let S be a complex projective surface and let ϕ be an isomorphism between Bir(S) and Bir(P 2 ). There exists a birational map ψ : S P 2 (C) and an automorphism of the field C such that

ϕ( f ) = τ(ψ f ψ -1 ) ∀ f ∈ Bir(S).

CHAPTER 7

CREMONA GROUP AND ZIMMER CONJECTURE

Introduction

In the 80's Zimmer suggests to generalise the works of Margulis on the linear representations of the lattices of simple, real Lie groups of real rank strictly greater than 1 (see [START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF][START_REF] Vinberg | Discrete subgroups of Lie groups[END_REF]) to the non-linear ones. He thus establishes a program containing several conjectures ( [START_REF] Zimmer | Kazhdan groups acting on compact manifolds[END_REF][START_REF] Zimmer | On connection-preserving actions of discrete linear groups. Ergodic Theory Dynam[END_REF][START_REF] Zimmer | Actions of semisimple groups and discrete subgroups[END_REF][START_REF] Zimmer | Lattices in semisimple groups and invariant geometric structures on compact manifolds[END_REF]); among them there is the following one.

Conjecture (Zimmer).

Let G be a real, simple, connected Lie group and let Γ be a lattice of G. If there exists a morphism of infinite image from Γ into the diffeomorphisms group of a compact manifold M, the real rank of G is bounded by the dimension of M.

There are a lot of results about this conjecture (see for example [100, 185, 101, 41, 42, 149, 164, 95, 46]). In the case of the Cremona group we have the following statement.

Theorem 7.1.1 ([69]

). -1) The image of an embedding of a subgroup of finite index of SL 3 (Z) into Bir(P 2 ) is, up to conjugation, a subgroup of PGL 3 (C).

More precisely let Γ be a subgroup of finite index of SL 3 (Z) and let ρ be an embedding of Γ into Bir(P 2 ). Then ρ is, up to conjugation, either the canonical embedding or the involution u → t (u -1 ).

2) Let Γ be a subgroup of finite index of SL n (Z) and let ρ be an embedding of Γ into the Cremona group. If ρ has infinite image, then n is less or equal to 3.

In the same context Cantat proves the following statement.

Theorem 7.1.2 ([47]

). -Let Γ be an infinite countable subgroup of Bir(P 2 ). Assume that Γ has Kazhdan's property (1) ; then up to birational conjugacy Γ is a subgroup of PGL 3 (C).

The proof uses the tools presented in Chapter 3 and in particular Theorem 3.4.6. Let us give an idea of the proof: since Γ has Kazhdan property the image of Γ by any ρ : Γ → Bir(P 2 ) is a subgroup of Bir(P 2 ) whose all elements are elliptic. According to Theorem 3.4.6 we have the following alternative: either ρ(Γ) is conjugate to a subgroup of PGL 3 (C), or ρ(Γ) preserves a rational fibration that implies that ρ has finite image (Lemma 7.4.3).

Let τ be an automorphism of the field C ; we can associate to a birational map f the birational map τ( f ) obtained by the action of τ on the coefficients of f given in a fixed system of homogeneous coordinates. Theorem 7.1.1 allows us to give another proof of the following result.

Theorem 7.1.3 ([71]

). -Let φ be an automorphism of the Cremona group. There exist a birational map ψ and an automorphism τ of the field C such that

φ( f ) = τ(ψ f ψ -1 ), ∀ f ∈ Bir(P 2 ).
The Cremona group has a lot of common points with linear groups nevertheless we have the following statement. The images by ς of the three generators satisfy Lemma 7.1.5 so p ≤ n ; as it is possible for every prime p we obtain a contradiction.

This Chapter is devoted to the proof of Theorem 7.1.1. Let us describe the steps of the proof. First of all let us assume to simplify that Γ = SL 3 (Z). Let ρ denote an embedding of Γ into Bir(P 2 ). The group SL 3 (Z) contains many Heisenberg groups, i.e. groups having the following presentation

H = f , g, h | [ f , g] = h, [ f , h] = [g, h] = id .
The key Lemma (Lemma 7.4.2) says if ς is an embedding of H into Bir(P 2 ) then λ(ς(h)) = 1.

Then either ς(h) is an elliptic birational map, or ς(h) is a de Jonquières or Halphen twist (Theorem 3.2.1). Using the well-known presentation of SL 3 (Z) (Proposition 7.2.4) we know that the image of any generator e i j of SL 3 (Z) satisfies this alternative; moreover the relations satisfied by the e i j 's imply the following alternative

• one of the ρ(e i j ) is a de Jonquières or Halphen twist;

• any ρ(e i j ) is an elliptic birational map.

In the first situation ρ(SL 3 (Z)) thus preserves a rational or elliptic fibration that never happen because of the group properties of SL 3 (Z) (Proposition 7.4.4). In the second situation the first step is to prove that the Heisenberg group ρ(e 12 ), ρ(e 13 ), ρ(e 23 ) is, up to finite index and up to conjugacy, a subgroup of Aut(S) where S is either P 2 (C), or a Hirzebuch surface ( §7.3).

In both cases we will prove that ρ(Γ) is up to conjugacy a subgroup of Aut(P 2 ) = PGL 3 (C) (Lemmas 7.4.5, 7.4.6). Idea of the proof (for details see [START_REF] Cantat | Groupes d'automorphismes polynomiaux du plan[END_REF]). The proof of this result uses the amalgamated product structure of Aut(C 2 ) (Theorem 2.1.2). Let us recall that the group of affine automorphisms is given by There exists a tree on which Aut(C 2 ) acts by translation (Bass-Serre theory, see §2.1) ; the stabilizers of the vertex of the tree are conjugate either to A or to E. So if a group G can be embedded into Aut(C 2 ), then :

First

A = (x, y) → (a 1 x + b 1 y + c 1 , a 2 x + b 2 y + c 2 ) a i , b i , c i ∈ C,
• either G acts on a tree without fixing a vertex;

• or G embeds into either A or E. Using this fact, Cantat and Lamy study the embeddings of Kazhdan groups (see [START_REF] De La Harpe | La propriété (T ) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger)[END_REF], chapter I or [START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF], chapter III) having (FA) property and thus the embeddings of lattices of Lie groups with real rank greater or equal to 2.

The groups SL n (Z)

. -Let us recall some properties of the groups SL n (Z) (see [START_REF] Steinberg | Some consequences of the elementary relations in SL n[END_REF] for more details).

For any integer q let us denote by Θ q : SL n (Z) → SL n (Z/qZ) the morphism which sends M onto M modulo q. Let Γ n (q) be the kernel of Θ q and let Γ n (q) be the reciprocical image of the diagonal group of SL n (Z/qZ) by Θ q ; the Γ n (q) are normal subgroups of SL n (Z), called congruence groups.

Theorem 7.2.3 ([11]

). -Let n ≥ 3 be an integer and let Γ be a subgroup of SL n (Z).

If Γ is of finite index, there exists an integer q such that Γ contains a subgroup Γ n (q) and is contained in

Γ n (q).
If Γ is of infinite index, then Γ is central and, in particular, finite.

Let δ i j be the Kronecker matrix 3 × 3 and let us set e i j = id + δ i j .

Proposition 7.2.4. -The group SL 3 (Z) admits the following presentation :

e i j,i = j | [e i j , e kℓ ] =    id if i = ℓ & j = k e iℓ if i = ℓ & j = k e -1 k j if i = ℓ & j = k
, (e 12 e -1 21 e 12 ) 4 = id

The e q i j generate Γ 3 (q) and satisfy equalities similar to those verified by the e i j except (e 12 e -1 21 e 12 ) 4 = id ; we will call them standard generators of Γ 3 (q). The system of roots of sl 3 (C) is of type A 2 (see [START_REF] Fulton | Representation theory[END_REF]) :

r 3 r 2 r 1 r 6 r 5 r 4
Each standard generator of a Γ 3 (q) is an element of the group of one parameter associated to a root r i of the system ; the system of roots thus allows us to find most of the relations which appear in the presentation of SL 3 (Z). For example r 1 + r 3 = r 2 corresponds to [e 12 , e 23 ] = e 13 , the relation r 2 + r 4 = r 3 to [e 13 , e 21 ] = e -1 23 and the fact that r 1 + r 2 is not a root to [e 12 , e 13 ] = id.

Heisenberg groups. -

Definition. -Let k be an integer. We call k-Heisenberg group a group with the presentation :

H k = f, g, h | [f, h] = [g, h] = id, [f, g] = h k .
By convention H = H 1 ; it is a Heisenberg group.

Let us remark that the Heisenberg group generated by f, g and h k is a subgroup of index k of H k . We call f, g and h the standard generators of H k .

Remark 7.2.5. -Each e q 2 i j can be written as the commutator of two e q kℓ with whom it commutes. The group SL 3 (Z) thus contains a lot of k-Heisenberg groups ; for example e q 12 , e q 13 , e q 23 is one (for k = q).

Representations of Heisenberg groups

As we said the groups SL n (Z) contain Heisenberg groups, we thus naturally study the representations of those ones in the automorphisms groups of Hirzebruch surfaces and of P 2 (C). Let us begin with some definitions and properties.

Definition. -Let S be a compact complex surface. The birational map f : S S is an elliptic birational map if there exist a birational map η : S S and an integer n > 0 such that η f n η -1 is an automorphism of S isotopic to the identity (i.e. η f n η -1 ∈ Aut 0 (S)).

Two birational maps f and g on S are simultaneously elliptic if the pair (η, S) is common to f and g. Remark 7.3.1. -Let C 1 and C 2 be two irreducible homologous curves of negative autointersection then C 1 and C 2 coincide. Thus an automorphism f of S isotopic to the identity fixes each curve of negative self-intersection; for any sequence of blow-downs ψ from S to a minimal model S of S, the element ψ f ψ -1 is an automorphism of S isotopic to the identity.

Lemma 7.3.2 ([69]

). -Let f and g be two birational elliptic maps on a surface S. Assume that f and g commute; then f and g are simultaneously elliptic.

Proof. -By hypothesis there exist a surface S, a birational map ζ : S S and an integer n such that ζ -1 f n ζ is an automorphism of S isotopic to the identity. Let us work on S ; to simplify we will still denote by f (resp. g) the automorphism ζ -1 f n ζ (resp. ζ -1 gζ).

First let us prove that there exists a birational map η : Y S such that η -1 f ℓ η is an automorphism of Y isotopic to the identity for some integer ℓ and that η -1 gη is algebraically stable. Let us denote by N(g) the minimal number of blow-ups needed to make g algebraically stable.

If N(g) is zero, then we can take η = id. Assume that the result is true for the maps f and g satisfying N(g) ≤ j; let us consider the pair ( f , g) and assume that it satisfies the assumption of the statement and that N( g) = j + 1. As g is not algebraically stable, there exists a curve V in Exc g and an integer q such that g q (V ) is a point of indeterminacy p of g. As f and g commute, f k fixes the irreducible components of Ind g for some integer k. Let us consider κ the blow-up of p; this point being fixed by f k , on the one hand κ -1 f k κ is an automorphism and on the other hand N(κ -1 gκ) = j. Then, by induction, there exists η : Y S and ℓ such that η -1 f ℓ η is an automorphism isotopic to the identity and that η -1 gη is algebraically stable.

Let us set f = η -1 f ℓ η and g = η -1 gη. Using [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], Lemma 4.1, the maps f and g are simultaneously elliptic. Indeed the first step to get an automorphism from g is to consider the blow-down ε 1 of a curve of Exc g -1 ; as the curves contracted by g -1 are of negative selfintersection and as f is isotopic to the identity, these curves are fixed by f so by ε 1 f ε -1

1 . The i-th step is to repeat the first one with ε i-1 . . .

ε 1 f ε -1 1 . . . ε -1 i-1 and ε i-1 . . . ε 1 gε -1 1 . . . ε -1 i-1
, we then obtain the result. According to [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] the process ends and a power of ε -1 gε is isotopic to the identity.

We have a similar result for the standard generators of a k-Heisenberg group.

Proposition 7.3.3 ([69]

). -Let ς be a representation of H k into the Cremona group. Assume that each standard generator of ς(H k ) is elliptic. Then ς(f), ς(g) and ς(h) are simultaneously elliptic.

Proof. -According to Lemma 7.3.2 the maps ς(f) and ς(h) are simultaneously elliptic. Since g and h commute, Exc ς(g) and Ind ς(g) are invariant by ς(h). The relation [f, g] = h k implies that Exc ς(g) and Ind ς(g) are invariant by ς(f). Using the idea of the proof of Lemma 7.3.2 and ([77], Lemma 4.1), we obtain the result.

In the sequel we are interested in the representations of H k in the automorphisms groups of minimal surfaces which are P 1 (C) × P 1 (C), P 2 (C) and the Hirzebruch surfaces F m . In an affine chart (x, y) of such a surface S, if f is an element of Bir(S), we will denote f by its two components ( f 1 (x, y), f 2 (x, y)). Let us recall that in some good affine charts we have

Aut(P 1 (C) × P 1 (C)) = (PGL 2 (C) × PGL 2 (C)) ⋊ (y, x) and (7.3.1) Aut(F m ) = ζx + P(y) (cy + d) m , ay + b cy + d a b c d ∈ PGL 2 (C), ζ ∈ C * , P ∈ C[y], deg P ≤ m .

Lemma 7.3.4 ([69]

). -Let ς be a morphism from H k into Aut(P 1 (C) × P 1 (C)). The morphism ς is not an embedding.

Proof. -We can assume that f, g and h fixe the two standard fibrations (if it is not the case we can consider H 2k ⊂ H k ), i.e. im ς is contained in PGL 2 (C) × PGL 2 (C). For j = 1, 2 let us denote by π j the j-th projection. The image of ς(H 2k ) by π j is a solvable subgroup of PGL 2 (C); as π j (ς(h k )) is a commutator, this homography is conjugate to the translation z + β j . Assume that β j is nonzero ; then π j (ς(f)) and π j (ς(g)) are also some translations (they commute with π j (ς(h k ))). The relation [π j (ς(f)), π j (ς(g))] = π j (ς(h k )) thus implies that β j is zero : contradiction. So β j is zero and the image of h 2k by ς is trivial : ς is not an embedding.

Concerning the morphisms from H k to Aut(F m ), m ≥ 1, we obtain a different statement.

Let us note that we can see Aut(C 2 ) as a subgroup of Bir(P 2 ); indeed any automorphism ( f 1 (x, y), f 2 (x, y)) of C 2 can be extended to a birational map:

(z n f 1 (x/z, y/z) : z n f 2 (x/
h =      0 ζ β 0 0 γ 0 0 0   ζ, β, γ ∈ C    .
Let us denote by π the canonical projection from SL 3 (C) into PGL 3 (C). The Lie algebra of π -1 (ς(H k )) is, up to conjugation, equal to h. The exponential map sends h in the group H of the upper triangular matrices which is a connected algebraic group. Therefore the identity component of π -1 (ς(H k )) coincides with H. Any element g of π -1 (ς(H k )) acts by conjugation on H so belongs to the group generated by H and j.id where j 3 = id. Since π(j.id) is trivial, the restriction of π to H is surjective on ς(H k ) ; but it is injective so it is an isomorphism.

Therefore ς can be lifted in a representation ς from H k into H :

H k ς G G ς 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ H π |H ς(H k )
As ς(h k ) can be written as a commutator, it is unipotent. The relations satisfied by the generators imply that we have up to conjugation in SL 3 (C) ς(h k ) = (x + k, y), ς(f) = (x + ζy, y + β) and ς(g) = (x + γy, y + δ) with ζδβγ = k. -Let G be a finitely generated group, let a 1 , . . . , a n be a part which generates G and let f be an element of G.

• The length of f , denoted by | f |, is the smallest integer k such that there exists a sequence (s 1 , . . . , s k ), s i ∈ a 1 , . . . , a n , a -1 1 , . . . , a -1 n , with f = s 1 . . . s k .

• The quantity lim

k→+∞ | f k | k
is the stable length of f (see [START_REF] De La Harpe | Topics in geometric group theory[END_REF]).

• An element f of G is distorted if it is of infinite order and if its stable length is zero. This notion is invariant by conjugation.

Lemma 7.4.1 ([69]

).

-Let H k = f, g, h be a k-Heisenberg group. The element h k is distorted. In particular the standard generators of SL n (Z) are distorded.

Proof. -As [f, h] = [g, h] = id, we have h knm = [f n , g m ] for any pair (n, m) of integers. For n = m we obtain h kn 2 = [f n , g n ] ; therefore |h kn 2 | ≤ 4n.
Each standard generator e i j of SL n (Z) can be written as follows e i j = [e ik , e k j ], moreover we have [e i j , e ik ] = [e i j , e k j ] = id (Remark 7.2.5).

Lemma 7.4.2 ([69]

). -Let G be a finitely generated group and let a 1 , . . . , a n be a set which generates G. Let f be an element of G and let ς be an embedding of G into Bir(P 2 ). There exists a constant m ≥ 0 such that

1 ≤ λ(ς( f )) ≤ exp m | f n | n .
In particular, if f is distorted, the stable length of f is zero and the first dynamical degree of ς( f ) is 1.

Proof. -The inequalities λ(ς( f )) n ≤ deg ς( f ) n ≤ max i (deg ς(a i )) | f n | imply 0 ≤ log λ(ς( f )) ≤ | f n | n log(max i (deg ς(a i ))). If f is distorted, the quantity lim k→∞ | f k | k
is zero and the first dynamical degree of ς( f ) is 1.

7.4.2.

Notations. -In the sequel, ρ will denote an embedding of SL 3 (Z) into Bir(P 2 ). Lemmas 7.4.1 and 7.4.2 imply that λ(ρ(e i j )) = 1. Thanks to Proposition 7.2.4 and Theorem 3.2.1, we have :

• either one of the ρ(e i j ) preserves a unique fibration, rational or elliptic;

• or each standard generator of Γ 3 (q) is an elliptic birational map.

We will study these two possibilities. A finitely generated Q-group is isomorphic to a subfield of C so Q(a i (y), b i (y), c i (y), d i (y)) is isomorphic to a subfield of C and we can assume that im ρ ⊂ PGL 2 (C) = Isom(H 3 ). As Γ has property (T), each continuous action of Γ by isometries on a real or complex hyperbolic space has a fixed point ; the image of ρ is thus, up to conjugacy, a subgroup of SO 3 (R). A result of Zimmer implies that the image of ρ is finite (see [START_REF] De La Harpe | La propriété (T ) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger)[END_REF]).

Proposition 7.4.4 ([69]

). -Let ρ be a morphism from a congruence subgroup Γ 3 (q) of SL 3 (Z) into Bir(P 2 ). If one of the ρ(e q i j ) preserves a unique fibration, then the image of ρ is finite.

Proof. -Let us denote by e q i j the image of e q i j by ρ ; Remark 7.2.5 implies that the different generators play a similar role; we can thus assume, without loss of generality, that e q 12 preserves a unique fibration F .

The relations imply that F is invariant by all the e q 2 i j 's. Indeed as e q 12 commutes with e q 13 and e q 32 , the elements e q 13 and e q 32 preserve F (it's the unicity) ; then the relation [ e q

12 , e q 23 ] = e q 2 13 , which can also be written e q 23 e q 12 e -q 23 = e q 2 13 e 12 , implies that e q 23 preserves F . Thanks to [ e q 12 , e q 31 ] = e -q 2 32 we obtain that F is invariant by e q 31 . Finally as [ e q 23 , e q 31 ] = e q 2 21 , the element e q 2 21 preserves F . Then, for each e q 2 i j , there exists h i j in PGL 2 (C) and

F : P 2 (C) → Aut(P 1 (C))
defining F such that F • e q 2 i j = h i j • F. Let us consider the morphism ς given by Γ 3 (q 2 ) → PGL 2 (C), e q 2 i j → h i j .

As Γ 3 (q 2 ) has Kazhdan's property (T) the group Γ = ker ς is of finite index (Lemma 7.4.3) so it also has Kazhdan's property (T). If F is rational, we can assume that F = (y = cte)

where y is a coordinate in an affine chart of P 2 (C) ; as the group of birational maps which preserve the fibration y = cte can be identified with PGL 2 (C(y)) ⋊ PGL 2 (C), the image of Γ by ρ is contained in PGL 2 (C(y)). In this case ρ(Γ) is thus finite (Lemma 7.4.3) which implies that ρ(Γ 3 (q 2 )) and ρ(Γ 3 (q)) are also finite. The fibration F cannot be elliptic ; indeed the group of birational maps which preserve pointwise an elliptic fibration is metabelian and a subgroup of Γ 3 (q 2 ) cannot be metabelian. The following statement deals with the case of Hirzebruch surfaces.

Factorisation in an automorphism

Lemma 7.4.6 ([69]

). -Let ρ be a morphism from SL 3 (Z) to Bir(P 2 ). Assume that ρ(e n 12 ), ρ(e n 13 ) and ρ(e n 23 ) are, for some integer n, simultaneously conjugate some elements of Aut(F m ) with m ≥ 1 ; then the image of ρ is either finite, or contained, up to conjugation, in PGL 3 (C). The restriction of ρ to Γ 3 (n 2 ) can be extended to an endomorphism of Lie group of PGL 3 (C) (see [START_REF] Steinberg | Some consequences of the elementary relations in SL n[END_REF]); as PGL 3 (C) is simple, this extension is injective and thus surjective. According to [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF], chapter IV, the automorphisms of PGL 3 (C) are obtained from inner automorphisms, automorphisms of the field C and the involution u → t (u -1 ) ; since automorphisms of the field C don't act on Γ 3 (n 2 ), we can assume, up to linear conjugation, that the restriction of ρ to Γ 3 (n 2 ) coincides, up to conjugation, with the identity or the involution u → t (u -1 ).

Let f be an element of ρ(SL 3 (Z)) \ ρ(Γ 3 (n 2 )) which contracts at least one curve C = Exc f . The group Γ 3 (n 2 ) is normal in Γ ; therefore the curve C is invariant by ρ(Γ 3 (n 2 )) and so by Proof. -Let Γ be a subgroup of finite index of SL 4 (Z) and let ρ be a morphism from Γ into Bir(P 2 ). To simplify we will assume that Γ = SL 4 (Z). Let us denote by E i j the images of the standard generators of SL 4 (Z) by ρ. The morphism ρ induces a faithful representation ρ from SL 3 (Z) into Bir(P 2 ) :

ρ(Γ 3 (n 2 )) = PGL 3 (C) (
SL 4 (Z) ⊃ SL 3 (Z) 0 0 1 → Bir(P 2 ).
According to the first assertion of Theorem 7.1.1, the map ρ is, up to conjugation, either the identity or the involution u → t (u -1 ).

Let us begin with the first case. The element E 34 commutes with E 31 and E 32 so ρ(E 14 ) commutes with (x, y, ax + by + z) where a and b are two complex numbers and Exc ρ(E 34 ) is invariant by (x, y, ax + by + z). Moreover E 34 commutes with E 12 and E 21 , in other words with the following SL 2 (Z):

SL 4 (Z) ⊃   SL 2 (Z) 0 0 0 1 0 0 0 1   → Bir(P 2 ).
But the action of SL 2 (Z) on C 2 has no invariant curve; the curves contracted by ρ(E 34 ) are contained in the line at infinity. The image of this one by (x, y, ax + by + z) intersects C 2 ; so Exc ρ(E 34 ) is empty and ρ(E 34 ) belongs to PGL 3 (C). With a similar argument we show that ρ(E 43 ) belongs to PGL 3 (C). The relations thus imply that ρ(Γ 4 (q)) is in PGL 3 (C) ; so the image of ρ is finite. We can use a similar idea when ρ is the involution u → t (u -1 ).

Conclusion of the proof of Theorem 7.1.1.

-Let n be an integer greater or equal to 4 and let Γ be a subgroup of finite index of SL n (Z). Let ρ be a morphism from Γ to Bir(P 2 ) ; let us denote by Γ n (q) the congruence subgroup contained in Γ (Theorem 7.2.3). The morphism ρ induces a representation from Γ 4 (q) to Bir(P 2 ); according to Theorem 7.4.7 its kernel is finite, so ker ρ is finite.

Automorphisms and endomorphisms of the Cremona group

We will prove Theorem 7.1.3. To do it we will use that (Theorem 2.1.4)

Bir(P 2 ) = Aut(P 2 ) = PGL 3 (C), 1
H =      1 a b 0 1 c 0 0 1   a, b, c ∈ C    .
The groups H and SL 3 (Z) generate PGL 3 (C) so PGL Let T denote by the group of translations in C 2 ; each element of T can be written

(x + a, y)(x, y + b).
As f b , resp. h c is of the type (x + η(b), y), resp. (x + η(c), y + η(c)), the image of T by φ is a subgroup of T. The group of translations is a maximal abelian subgroup of Bir(P 2 ), so does φ(T) and the inclusion φ(T) ⊂ T is an equality. The map η is thus surjective and φ(H) = H. So φ induces an automorphism of PGL 3 (C) trivial on SL 3 (Z). But the automorphisms of PGL 3 (C) are generated by inner automorphisms, automorphisms of the field C and the involution u → t (u -1 ) (see [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF]). Then up to conjugation and up to the action of an automorphism of the field C, φ |PGL 3 (C) is trivial (the involution u → t (u -1 ) on SL 3 (Z) is not the restriction of an inner automorphism). and G are rational. The equality σ(βx, µy) = (β -1 x, µ -1 y)σ implies (F, G)(βx, µy) = (F, G) ; as this equality is true for any pair (β, µ) of nonzero complex numbers, the functions F and G are constant.

The involution u → t (u -1 ) preserves the diagonal group; so φ |PGL 3 (C) coincides with u → t (u -1 ).

Proof of Theorem 7.1.3. -Theorem 7.1.1, Corollary 7.5.2 and Lemma 7.5.1 allow us to assume that up to conjugation and up to the action of an automorphism of the field C, φ |PGL 3 (C) is trivial or is the involution u → t (u -1 ). Assume we are in the last case and let us set h = (x, xy, xz) ; the map (hσ) 3 is trivial (see [START_REF] Gizatullin | Rational G-surfaces[END_REF]). But φ(h) = (x + y + z, -y, -z) and φ(σ) = a x , b y , 1 z (Lemma 7.5.3) so φ(hσ) 3 = id: contradiction. We thus can assume that φ |PGL 3 (C) is trivial ; the equality (hσ) 3 = id implies φ(σ) = σ and Theorem 2.1.4 allows us to conclude.

Using the same type of arguments we can describe the endomorphisms of the Cremona group.

Theorem 7.5.4 ([72]

). -Let φ be a non-trivial endomorphism of Bir(P 2 ). There exists an embedding τ of the field C into itself and a birational map

ψ of P 2 (C) such that φ( f ) = τ(ψ f ψ -1 ), ∀ f ∈ Bir(P 2 ).
This allows us to state the following corollary.

Corollary 7.5.5 ([72]

). -The Cremona group is hopfian: any surjective endomorphism of Bir(P 2) is an automorphism.

CHAPTER 8 CENTRALIZERS IN THE CREMONA GROUP

Introduction

The description of the centralizers of the discrete dynamical systems is an important problem in real and complex dynamic. Julia ([127, 126]) and then Ritt ([165]) show that the set Cent( f , Rat P 1 ) = ψ : P 1 → P 1 f ψ = ψ f of rational functions commuting with a fixed rational function f is in general f N 0 = f n 0 n ∈ N for some f 0 in Cent( f , Rat P 1 ) except in some special cases (up to conjugacy z → z k , Tchebychev polynomials, Lattès examples...) In the 60's Smale asks if the centralizer of a generic diffeomorphism f : M → M of a compact manifold is trivial, i.e. if Let us precise some of these works. In [START_REF] Kopell | Commuting diffeomorphisms[END_REF] Kopell proves the existence of a dense open subset Ω of Diff ∞ (S 1 ) having the following property: the centralizer of any element of Ω is trivial.

Cent( f , Diff ∞ (M)) = g ∈ Diff ∞ (M) f ψ = ψ f coincides with f Z = f n n ∈ Z . A
Let f be a C r -diffeomorphism of a compact manifold M without boundary. A point p of M is non-wandering if for any neighborhood U of p and for any integer n 0 > 0 there exists an integer n > n 0 such that f n U ∩ U = / 0. The set of such points is denoted by

Ω( f ), it is a closed invariant set; Ω( f ) is hyperbolic if
• the tangent bundle of M restricted to Ω( f ) can be written as a continuous direct sum of two subbundles T Ω( f ) M = E s ⊕ E u which are invariant by the differential D f of f ; • there exists a riemannian metric on M and a constant 0 < µ < 1 such that for any p ∈ Ω( f

), v ∈ E s p , w ∈ E u p ||D f p v|| ≤ µ||v||, ||D f -1 p w|| ≤ µ||w||.
In this case the sets

W s (p) = z ∈ M d( f n (p), f n (z)) → 0 as n → ∞ and W u (p) = z ∈ M d( f -n (p), f -n (z)) → 0 as n → ∞
are some immersed submanifolds of M called stable and unstable manifolds of p ∈ Ω( f ). We say that f satisfies axiom A if Ω( f ) is hyperbolic and if Ω( f ) coincides with the closure of periodic points of f (see [START_REF] Smale | Differentiable dynamical systems[END_REF]). Finally we impose a "strong" transversality condition: for any p ∈ Ω( f ) the stable W s (p) and unstable W u (p) manifolds are transverse. In [START_REF] Palis | Rigidity of the centralizers of diffeomorphisms and structural stability of suspended foliations[END_REF] Palis proves that the set of diffeomorphisms of M satisfying axiom A and the strong transversality condition contains a dense open subset Λ such that: the centralizer of any f in Λ is trivial.

Anderson shows a similar result for the Morse-Smale diffeomorphisms ( [START_REF] Anderson | Diffeomorphisms with discrete centralizer[END_REF]).

In the study of the elements of the group Diff(C, 0) of the germs of holomorphic diffeomorphism at the origin of C, the description of the centralizers is very important. Ecalle proves that if f ∈ Diff(C, 0) is tangent to the identity, then, except for some exceptional cases, its centralizer is a f Z 0 (see [START_REF] Écalle | Les fonctions résurgentes. Tome II[END_REF][START_REF] Écalle | L'équation du pont et la classification analytique des objects locaux[END_REF]); it allows for example to describe the solvable non abelian subgroups of Diff(C, 0) (see [START_REF] Cerveau | Groupes d'automorphismes de (C, 0) et équations différentielles ydy + • • • = 0[END_REF]). Conversely Perez-Marco gets the existence of uncountable, non linearizable abelian subgroups of Diff(C, 0) related to some difficult questions of small divisors ( [START_REF] Marco | Nonlinearizable holomorphic dynamics having an uncountable number of symmetries[END_REF]).

In the context of polynomial automorphisms of the plane, Lamy obtains that the centralizer of a Hénon automorphism is almost trivial. More precisely we have the following statement: let f be a polynomial automorphism of C 2 ; then

• either f is conjugate to an element of the type

(αx + P(y), βy + γ), P ∈ C[y], α, β, γ ∈ C, αβ = 0
and its centralizer is uncountable, • or f is a Hénon automorphism ψg 1 . . . g n ψ -1 where

ψ ∈ Aut(C 2 ), g i = (y, P i (y) -δ i x), P i ∈ C[y], deg P i ≥ 2, δ i ∈ C *
and its centralizer is isomorphic to Z ⋊ Z/pZ (see [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]Proposition 4.8]). We will not give the proof of Lamy but will give a "related" result due to Cantat (Corollary 8.2.4)

Let us also mention the recent work [START_REF] Dinh | Groupes commutatifs d'automorphismes d'une variété kählérienne compacte[END_REF] of Dinh and Sibony.

Dynamics and centralizer of hyperbolic diffeomorphisms

Let S be a complex surface and let f : S → S be a holomorphic map. Let q be a periodic point of period k for f , i.e. f k (q) = q and f ℓ (q) = q for all 1 ≤ ℓ ≤ k -1. Let λ u (q) and λ s (q) be the eigenvalues of D f (q) . We say that f is hyperbolic if

|λ s (q)| < 1 < |λ u (q)|.
Let us denote by P k ( f ) the set hyperbolic periodic points of period k of f . Let us consider q ∈ P k ( f ); locally around q the map f is well defined. We can linearize f k . The local stable manifold W s loc (q) and local unstable manifold W u loc (q) of f k in q are the image by the linearizing map of the eigenvectors of D f k q . To simplify we can assume that up to conjugation D f k q is given by α 0 0 β with |α| < 1 < |β|; there exists a holomorphic diffeomorphism κ : (U, q) → (C 2 , 0) where U is a neighborhood of q such that κ f k κ -1 = α 0 0 β . Then W s loc (q) = κ -1 (y = 0) and W u loc (q) = κ -1 (x = 0):

W s loc (q) W u loc (q)
In the sequel, to simplify, we will denote f instead of f k .

Lemma 8.2.1.

-There exist entire curves ξ s q , ξ u q : C → S such that • ξ u q (0) = ξ s q (0) = q; • the global stable and global unstable manifolds of f in q are defined by

W s (q) = n>0 f n (W s loc (q)), W u (q) = n>0 f n (W u loc (q)).
• f (ξ u q (z)) = ξ u q (α u (z)), f (ξ s q (z)) = ξ s q (α s (z)) for all z ∈ C; • if η u q : C → S (resp. η s q : C → S) satisfies the first three properties, then η u q (z) = ξ u q (µz) (resp. η s q (z) = ξ s q (µ ′ z)) for some µ ∈ C * (resp. µ ′ ∈ C * ).

Proof. -As we just see there exists a holomorphic diffeomorphism κ : (U, q) → D where U is a neighborhood of q and D a small disk centered at the origin such that κ f k κ -1 = α 0 0 β . Moreover W u loc (q) = κ -1 (x = 0) and W s loc (q) = κ -1 (y = 0). Let us extend κ. Let z be a point which does not belong to D; there exist an integer m such that z/α m belongs to D. We then set

ξ u q (z) = f m κ -1 z α m . Let us note that if z α m and z α k both belong to D we have f m κ -1 z α m = f k κ -1 z α k
and ξ s q (z) is well-defined. By construction we get

• ξ u q (0) = ξ s q (0) = q; • W s (q) = n>0 f n (W s loc (q)), W u (q) = n>0 f n (W u loc (q)). • f (ξ u q (z)) = ξ u q (α u (z)), f (ξ s q (z)) = ξ s q (α s (z)) for all z ∈ C.
The map ξ s q is the analytic extension of κ -1 |y=0 . Let ∆ be a subset of y = 0 containing 0. Set q = ξ s q (1). Let η s q : ∆ → W s loc (q) be a non-constant map such that • η s q (0) = q, • η s q (αz) = f (η s q (z)) for any z in ∆ such that αz belongs to ∆.

Working with η s q •(z → µz) for some good choice of µ instead of η s q we can assume that η s q (1) = q. Since η s q (0) = ξ s q (0), η s q (1) = ξ s q (1),

η s q 1 α n = ξ s q 1 α n ∀ n ∈ Z we have η s q = ξ s q .
Let ψ be an automorphism of S which commutes with f . The map ψ permutes the elements of P k ( f ). If P k ( f ) is finite, of cardinal N k > 0, the map ψ N k ! fixes any element of P k ( f ). The stable and unstable manifolds of the points q of P k ( f ) are also invariant under the action of ψ. When the union of W u (q) and W s (q) is Zariski dense in S, then the restrictions of ψ to W u loc (q) and W s loc (q) completely determine the map ψ : S → S. Let us denote by A k the subgroup of Cent( f , Aut(S)) which contains the automorphisms of S fixing any of the N k points of P k ( f ). Then ψ preserves W u (q) and W s (q). We thus can define the morphism

α : A k → C * × C * , ψ → α(ψ) = (α s (ψ), α u (ψ)) such that ∀ z ∈ C, ξ s q (α s (ψ)z) = ψ(ξ s q (z)) and ξ u q (α u (ψ)z) = ψ(ξ u q (z)).
When the union of W s (q) and W u (q) is Zariski dense, this morphism is injective. In particular A k is abelian and Cent( f , Aut(S)) contains an abelian subgroup of finite index with index ≤ N k !.

Lemma 8.2.2 ([47]

). -The subset Λ of C × C defined by

Λ = (x, y) ∈ C × C ξ u q (x) = ξ s q (y) is a discrete subset of C × C.
The set Λ intersects {0} × C (resp. C × {0}) only at (0, 0).

Proof. -Let (x, y) be an element of Λ and let m be the point of S defined by m = ξ s q (x) = ξ u q (y). In a sufficiently small neighborhood of m, the connected components of W s (q) and W u (q) which contain m are two distinct complex submanifolds and so intersect in a finite number of points. Therefore there exist a neighborhood U of x and a neighborhood V of y such

that ξ s q (U) ∩ ξ u q (V ) = {m}. The point (x, y) is thus the unique point of Λ in U × V so Λ is discrete.
Since ξ u q and ξ s q are injective, we have the second assertion.

Proposition 8.2.3 ([47]

). -Let f be a holomorphic diffeomorphism of a connected complex surface S. Assume that there exists an integer k such that • the set P k ( f ) is finite and non empty;

• for at least one point q in P k ( f ) we have # (W s (q) ∩ W u (q)) ≥ 2. Then the cyclic group generated by f is of finite index in the group of holomorphic diffeomorphisms of S which commute to f . on a point p by f and if W s (q) intersects E infinitely many times, then W s (q) passes through p infinitely many times.

Lemma 8.3.1 ([47]

). -Let Λ be the set of pairs (x, y) such that ξ u q (x) = ξ s q (y). The set Λ is a discrete subset of C × C which intersects the coordinate axis only at the origin.

Proof. -Let (x, y) be a point of Λ and set m = ξ u q (x) = ξ s q (y). The unstable and stable manifolds can a priori pass through m infinitely many times. But since each of these manifolds is the union of the f ±n (W u/s loc (q)), there exist two open subsets U ∋ x and V ∋ y of C and an open subset W of S containing m such that ξ u q (U) ∩ W and ξ s q (V ) ∩ W are two distinct analytic curves of W . We can assume that # ξ u q (U) ∩ ξ s q (V ) = 1 (if it is not the case we can consi-

der U ′ ⊂ U and V ′ ⊂ V such that # ξ u q (U ′ ) ∩ ξ s q (V ′ ) = 1); therefore (x, y) is the only point of Λ contained in U × V .
The set Λ is thus discrete. Since q is periodic there is no curve contracted onto q by an iterate of f , the map ξ u q (resp. ξ s q ) doesn't pass again through q. So Λ intersects the axis-coordinates only at (0, 0).

Let us recall that if a map f is algebraically stable then the positive orbits f n (p), n ≥ 0, of the elements p of Ind f -1 do not intersect Ind f . We say that f satisfies the Bedford-Diller condition if the sum

∑ n≥0 1 λ( f ) n log(dist( f n (p), Ind f ))
is finite for any p in Ind f -1 ; in other words the positive orbit f n (p), n ≥ 0, of the elements p of Ind f -1 does not go too fast to Ind f . Note that this condition is verified by automorphisms of P 2 (C) or also by birational maps whose points of indeterminacy have finite orbit. Let us mention the following statement.

Theorem 8.3.2 ([20, 87]

). -Let f be a hyperbolic birational map of complex projective surface. Assume that f satisfies the Bedford-Diller condition. Then there is a infinite number of hyperbolic periodic points whose stable and unstable manifolds intersect.

Birational maps satisfying Bedford-Diller condition. -Proposition 8.3.3 ([47]). -Let f be a hyperbolic birational map of a complex projective surface S. If f satisfies the Bedford-Diller condition, then the cyclic subgroup generated by f is of finite index in the group of birational maps of S which commute with f .

Proof. -The set of hyperbolic periodic points of f of period k is a finite set. According to Theorem 8.3.2 there exists an integer k such that

• q is a hyperbolic periodic point of period k;

• W s (q) and W u (q) are Zariski dense in S;

• # (W s (q) ∩ W u (q)) is not finite. Let ψ be a birational map of S which commutes with f . The map ψ permutes the unstable and stable manifolds of hyperbolic periodic points of f even if these manifolds pass through a point of indeterminacy of ψ. Indeed, if q is a periodic point of f and W u (q) is Zariskidense, then ψ is holomorphic in any generic point of W u (q) so we can extend ψ analytically along W u (q). Since f has ν k hyperbolic periodic points of period k, there exists a subgroup B k of Cent( f , Bir(S)) of index less than ν k !; any element of B k fixes W s (q) and W u (q). More precisely there exists a morphism α :

B k → C * × C * , ψ → (α u (ψ), α s (ψ))
such that ψ(ξ

u/s q (z)) = ξ u/s
q (α u/s (ψ)z) for any ψ of B k and for any z of C such that ψ is holomorphic on a neighborhood of ξ u/s q (z). As W s (q) and W u (q) are Zariski dense, α is injective. Then we can apply the arguments of Proposition 8.2.3.

Birational maps that don't satisfy

Bedford-Diller condition. -Let f be a birational map of a complex surface S; assume that f is algebraically stable. Let p be a point of indeterminacy of f . If C is a curve contracted on p by an iterate f -n , n > 0, of f , then we say that C comes from p. If q is a point of S for which there exists an integer m such that

∀ 0 ≤ ℓ < m, f ℓ (q) ∈ Ind f , f m (q) = p
we say that q is a point of indeterminacy of f passing through p at the time m. Since f is algebraically stable, the iterates f -m of f , m ≥ 0, are all holomorphic in a neighborhood of p so the unique point passing through p at the time m is f -m (p). We say that p has an infinite negative orbit if the set f -m (p) | m ≥ 0 is infinite.

Lemma 8.3.4 ([47]

). -Let f be a birational map of S. Assume that f is algebraically stable. Let p be a point of indeterminacy of f having an infinite negative orbit. One of the following holds: i. there exist an infinite number of irreducible curves contracted on p by the iterates f -n of f , n ∈ N; ii. there exists a birational morphism π : S → S ′ such that π f π -1 is an algebraically stable birational map of S ′ whose all iterates are holomorphic in a neighborhood of π(p).

We will say that a point of indeterminacy p is persistent if there exists no birational morphism π : S → S ′ satisfying property ii.

Proof. -Assume that the union of the curves contracted by f -n , n ≥ 0, onto p is a finite union C of curves.

Let us consider a curve C in C such that -Let S be a compact complex surface and let f be a birational map of S. If f preserves an infinite number of curves, then f preserves a fibration.

• f m is holomorphic on C; • f m (C)

Proposition 8.3.6 ([47]

). -Let f be an algebraically stable birational map of a compact complex surface S. Let p be a persistent point of indeterminacy of f whose negative orbit is infinite. If ψ is a birational map of S which commutes with f then • either ψ preserves a pencil of rational curves;

• or an iterate ψ m of ψ, m = 0, coincides with an iterate f n of f .

Proof. -Let us set ν := # Ind f , and consider ψ ν! instead of ψ. Since the negative orbit of p is infinite, there exists an integer k 0 such that ψ is holomorphic around the points f -k (p) for any k ≥ k 0 . For any n ≥ 0 let us denote by C n the union of curves coming from p. The periodic point p is persistent, so according to Lemma 8.3.4 there is an infinite number of curves coming from p. Hence there exists an integer n 0 such that for any n ≥ n 0 the map ψ does not contract C n .

Since f and ψ commute,

ψ( f -k (p)) is a point of indeterminacy of f m for at least an integer 0 ≤ m ≤ n 0 + k + 1 (∀ k ≥ k 0 ).
This point of indeterminacy passes through p. Let us consider ψ f ℓ for some good choice of ℓ; we can thus assume that ψ( f -k (p)) is a point of indeterminacy of f passing through p at the time k and so ψ( f -k (p)) = f -k (p) for any k ≥ k 0 . Moreover for n sufficiently large we have

ψ(C n ) = C n .
We conclude with Lemma 8.3.5.

Corollary 8.3.7 ([47]

). -Let f be a birational map of a compact complex surface S which is algebraically stable. Assume that • the map f is hyperbolic;

• f has a persistent point of indeterminacy whose negative orbit is infinite. If ψ is a birational map of S which commutes with f , there exists m ∈ Z \ {0} and n ∈ Z such that ψ m = f n .

Proof. -Let ψ be in Cent( f , Bir(P 2 )). Assume that ψ preserves a pencil of curves P . As f is hyperbolic, f doesn't preserve a pencil of curves so ψ preserves two distinct pencils P and f (P ). According to [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] an iterate of ψ is conjugate to an automorphism isotopic to the identity on a minimal rational surface S ′ ; let us still denote by f and by ψ the maps of S ′ obtained from f and ψ by conjugation. Assume that ψ has infinite order; let us denote by G the Zariski closure of the cyclic group generated by ψ in Aut(S ′ ). It is an abelian Lie group which commutes with f . Any subgroup of one parameter of G determines a flow which commutes with f : f φ t = φ t f . If the orbits of φ t are algebraic curves, f preserves a pencil of curves: contradiction with λ( f ) > 1. Otherwise φ t fixes a finite number of algebraic curves and among these we find all the curves contracted by f or by some f n ; hence there is a finite number of such curves: contradiction with the second assumption.

Since then Blanc and Cantat got a more precise statement.

Theorem 8.3.8 ([33]

). -Let f be a hyperbolic birational map. Then

Cent( f , Bir(P 2 )) ≃ Z ⋊ F
where F denotes a finite group.

Centralizer of elliptic birational maps of infinite order

Let us recall ([34, Proposition 1.3]) that an elliptic birational map f of P 2 (C) of infinite order is conjugate to an automorphism of P 2 (C) which restricts to one of the following automorphisms on some open subset isomorphic to C 2 :

• (αx, βy), where α, β ∈ C * , and where the kernel of the group homomorphism Z 2 → C * given by (i, j) → α i β j is generated by (k, 0) for some k ∈ Z. • (αx, y + 1), where α ∈ C * . We can describe the centralizers of such maps.

Lemma 8.4.1 ([34]

). -Let us consider f = (αx, βy) where α, β are in C * , and where the kernel of the group homomorphism Z 2 → C * given by (i, j) → α i β j is generated by (k, 0) for some k ∈ Z. Then the centralizer of f in Bir(P 2 ) is

Cent( f , Bir(P 2 )) = (η(x), yR(x k )) R ∈ C(x), η ∈ PGL 2 (C), η(αx) = αη(x) .

Lemma 8.4.2 ([34]

). -Let us consider f = (αx, y+β) where α, β ∈ C * . Then Cent( f , Bir(P 2 )) is equal to

(η(x), y + R(x)) η ∈ PGL 2 (C), η(αx) = αη(x), R ∈ C(x), R(αx) = R(x) .

Centralizer of de Jonquières twists

Let us denote by π 2 the morphism from dJ (see Chapter 2, §2.3) into PGL 2 (C), i.e. π 2 ( f ) is the second component of f ∈ dJ. The elements of dJ which preserve the fibration with a trivial action on the basis of the fibration form a normal subgroup dJ 0 of dJ (kernel of the morphism π 2 ); of course dJ 0 ≃ PGL 2 (C(y)). Let f be an element of dJ 0 ; it is, up to conjugacy, of one of the following form (see for example [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF])

a (x + a(y), y), b (b(y)x, y), c c(y)x + F(y) x + c(y) , y , with a in C(y), b in C(y) * and c, F in C[y], F being not a square (if F is a square, then f is conjugate to an element of type b).
The non finite maximal abelian subgroups of dJ 0 are

dJ a = (x + a(y), y) a ∈ C(y) , dJ m = (b(y)x, y) b ∈ C(y) * , dJ F = (x, y), c(y)x + F(y) x + c(y) , y c ∈ C(y)
where F denotes an element of C[y] which is not a square ( [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF]). We can assume that F is a polynomial with roots of multiplicity one (up to conjugation by a map (a(y)x, y)). Therefore if f belongs to dJ 0 and if Ab( f ) is the non finite maximal abelian subgroup of dJ 0 that contains f then, up to conjugacy, Ab( f ) is either dJ a , or dJ m , or dJ F . More precisely if f is of type a (resp. b, resp. c), then Ab( f ) = dJ a (resp. Ab( f ) = dJ m , resp. Ab( f ) = dJ F ).

In [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF] we first establish the following property.

Proposition 8.5.1 ([55]

). -Let f be an element of dJ 0 . Then • either Cent( f , Bir(P 2 )) is contained in dJ;

• or f is periodic.

Proof. -Let f = (ψ(x, y), y) be an element of dJ 0 , i.e. ψ ∈ PGL 2 (C(y)).

Let ϕ = (P(x, y), Q(x, y)) be a rational map that commutes with f . If ϕ does not belong to dJ, then Q = cte is a fibration invariant by f which is not y = cte. Hence f preserves two distinct fibrations and the action on the basis is trivial in both cases so f is periodic. This allows us to prove the following statement.

Theorem 8.5.2 ([55]

). -Let f be a birational map which preserves a rational fibration, the action on the basis being trivial. If f is a Jonquières twist, then Cent( f , Bir(P 2 )) is a finite extension of Ab( f ).

This result allows us to describe, up to finite index, the centralisers of the elements of dJ\dJ 0 , question related to classical problems of difference equations. A generic element of dJ \ dJ 0 has a trivial centralizer.

In this section we will give an idea of the proof of Theorem 8.5.2.

Maps of dJ a . -Proposition 8.5.3 ([55]

). -The centralizer of f = (x + 1, y) is

(x + b(y), ν(y)) b ∈ C(y), ν ∈ PGL 2 (C) ≃ dJ a ⋊ PGL 2 (C).
Proof. -The map f is not periodic and so, according to Proposition 8.5.1, any map ψ which commutes with f can be written as y) and ∂ψ 1 ∂x depends only on y, i.e. ψ 1 (x, y) = A(y)x + B(y).

(ψ 1 (x, y), ν(y)) with ν in PGL 2 (C). The equality f ψ = ψ f implies ψ 1 (x + 1, y) = ψ 1 (x, y) + 1. Thus ∂ψ 1 ∂x (x + 1, y) = ∂ψ 1 ∂x (x,
Writing again ψ 1 (x + 1, y) = ψ 1 (x, y) + 1 we get A = 1. Hence Proof. -Let f = (x + a(y), y) be a non trivial element of dJ a , i.e. a = 0; up to conjugation by (a(y)x, y) we can assume that f = (x + 1, y). Let us also introduce the subgroup

ψ = (x + B(y), ν(y)), B ∈ C(y) ν ∈ PGL 2 (C).
Stab(a) = ν ∈ PGL 2 (C) a(ν(y)) = a(y) ±1 .
We remark that stab(a) is a normal subgroup of Stab(a). If f is a Jonquières twist, then Cent( f , Bir(P 2 )) = dJ m ⋊ Stab(a).

Remarks 8.5.7. -• For generic a the group Stab(a) is trivial; so for generic f ∈ dJ m , the group Cent( f , Bir(P 2 )) coincides with dJ m = Ab( f ).

• If f = (a(y)x, y) with a non constant, then Cent( f , Bir(P 2 )) is a finite extension of dJ m = Ab( f ). • If f = (ax, y), a ∈ C * , we have Cent( f , Bir(P 2 )) = dJ m ⋊ Stab(a) (here we can define Stab(a) = PGL 2 (C)).

Maps of dJ

F . -Let us now consider the elements of dJ F ; as we said we can assume that F only has roots with multiplicity one. We can thus write f as follows:

f = c(y)x + F(y) x + c(y) , y c ∈ C(y);
the curve of fixed points C of f is given by x 2 = F(y). Since the eigenvalues of c(y) F(y) 1 c(y) are c(y) ± F(y) we note that f is periodic if and only if c is zero; in that case f is periodic of period 2. Assume now that f is not periodic. As F has simple roots the genus of C is ≥ 2 for deg F ≥ 5, is equal to 1 for deg F ∈ {3, 4}; finally C is rational when deg F ∈ {1, 2}. 8.5.3.1. Assume that the genus of C is positive. -Since f is a Jonquières twist, f is not periodic. The map f has two fixed points on a generic fiber which correspond to the two points on the curve x 2 = F(y). The curves x 2 = F(y) and the fibers y = constant are invariant by f and there is no other invariant curve. Indeed an invariant curve which is not a fiber y = constant intersects a generic fiber in a finite number of points necessary invariant by f ; since f is of infinite order it is impossible (a Moebius transformation which preserves a set of more than three elements is periodic).

Proposition 8.5.8 ([55]

). -Let f = c(y)x+F (y) x+c(y) , y be a non periodic map (i.e. c = 0), where F is a polynomial of degree ≥ 3 with simple roots (i.e. the genus of

C is ≥ 1). Then if F is generic, Cent( f , Bir(P 2 )) coincides with dJ F ; if it is not, Cent( f , Bir(P 2 )) is a finite extension of dJ F = Ab( f ). 8.5.3.2. Suppose that C is rational. -Let f be an element of dJ F ; assume that f is a Jon- quières twist.
The curve of fixed points C of f is given by x 2 = F(y). Let ψ be an element of Cent( f , Bir(P 2 )); either ψ contracts C , or ψ preserves C . According to Proposition 8.

S(c; α) = stab 4c(y) 2 c(y) 2 -y ∩ D ∞ (α).
The description of Cent( f , Bir(P 2 )) with f in dJ F and C = Fix f rational is given by:

Centralizer of Halphen twists

For the definition of Halphen twists, see Chapter 3, §3.2.

Proposition 8.6.1 ([47, 104]

). -Let f be an Halphen twist. The centralizer of f in Bir(P 2 ) contains a subgroup of finite index which is abelian, free and of rank ≤ 8.

Proof. -Up to a birational change of coordinates, we can assume that f is an element of a rational surface with an elliptic fibration π : S → P 1 and that this fibration is f -invariant. Moreover we can assume that this fibration is minimal (there is no smooth curve of self intersection -1 in the fibers) and so f is an automorphism. The elliptic fibration is the unique fibration invariant by f (see [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]) so it is invariant by Cent( f , Bir(P 2 )); thus Cent( f , Bir(P 2 )) is contained in Aut(S).

As the fibration is minimal, the surface S is obtained by blowing up P 2 (C) in the nine base-points of an Halphen pencil (1) and the rank of its Neron-Severi group is equal to 10 (Proposition 1.1.8). The automorphism group of S can be embedded in the endomorphisms of H 2 (S, Z) for the intersection form and preserves the class [K S ] of the canonical divisor, i.e. the class of the elliptic fibration. The dimension of the orthogonal hyperplane to [K S ] is 9 and the restriction of the intersection form on its hyperplane is semi-negative: its kernel coincides with Z[K S ]. Hence Aut(S) contains an abelian group of finite index whose rank is ≤ 8.

1. An Halphen pencil is a pencil of plane algebraic curves of degree 3n with nine n-tuple base-points.

CHAPTER 9 AUTOMORPHISMS WITH POSITIVE ENTROPY, FIRST DEFINITIONS AND PROPERTIES

Let V be a complex projective manifold. Let φ be a rational or holomorphic map on V. When we iterate this map we obtain a "dynamical system": a point p of V moves to p 1 = φ(p), then to p 2 = φ(p 1 ), to p 3 = φ(p 2 ) . . . So φ "induces a movement on V". The set p, p 1 , p 2 , p 3 , . . . is the orbit of p.

Let A be a projective manifold; A is an Abelian variety of dimension k if A(C) is isomorphic to a compact quotient of C k by an additive subgroup.

Multiplication by an integer m > 1 on an Abelian variety, endomorphisms of degree d > 1 on projective spaces are studied since XIXth century in particular by Julia and Fatou ([4]). These two families of maps "have an interesting dynamic". Consider the first case; let f m denote the multiplication by m. Periodic points of f m are repulsive and dense in A(C) : a point is periodic if and only if it is a torsion point of A; the differential of f n m at a periodic point of period n is an homothety of ratio m n > 1.

Around 1964 Adler, Konheim and McAndrew introduce a new way to measure the complexity of a dynamical system: the topological entropy ( [START_REF] Adler | Topological entropy[END_REF]). Let X be a compact metric space. Let φ be a continuous map from X into itself. Let ε be a strictly positif real number. For all integer n let N(n, ε) be the minimal cardinal of a part X n of X such that for all y in X there exists x in X satisfying dist(

f j (x), f j (y)) ≤ ε, ∀ 0 ≤ j ≤ n.
We introduce h top ( f , ε) defined by

h top ( f , ε) = lim sup n→+∞ 1 n log N(n, ε).
The topological entropy of f is given by

h top ( f ) = lim ε→0 h top ( f , ε).
For an isometry of X the topological entropy is zero. For the multiplication by m on a complex Abelian variety of dimension k we have: h top ( f ) = 2k log m. For an endomorphism of P k (C) defined by homogeneous polynomials of degree d we have: h top ( f ) = k log d (see [START_REF] Gromov | On the entropy of holomorphic maps[END_REF]). Let V be a complex projective manifold. On which conditions do rational maps with chaotic behavior exist ? The existence of such rational maps implies a lot of constraints on V : Theorem 9.0.2 ([16]). -A smooth complex projective hypersurface of dimension greater than 1 and degree greater than 2 admits no endomorphism of degree greater than 1.

Let us consider the case of compact homogeneous manifolds V : the group of holomorphic diffeomorphisms acts faithfully on V and there are a lot of holomorphic maps on it. Meanwhile in this context all endomorphisms with topological degree strictly greater than 1 come from endomorphisms on projective manifolds and nilvarieties.

So the "idea" is that complex projective manifolds with rich polynomial dynamic are rare; moreover it is not easy to describe the set of rational or holomorphic maps on such manifolds. 9.1. Some dynamics 9.1.1. Smale horseshoe. -The Smale horsehoe is the hallmark of chaos. Let us now describe it (see for example [START_REF] Shub | What is . . . a horseshoe? Notices[END_REF]). Consider the embedding f of the disc ∆ into itself. Assume that

• f contracts the semi-discs f (A) and f (E) in A;

• f sends the rectangles B and D linearly to the rectangles f (B) and f (D) stretching them vertically and shrinking them horizontally, in the case of D it also rotates by 180 degrees. We don't care what the image

f (C) of C is, as long as f (C) ∩ (B ∪C ∪ D) = / 0.
In other words we have the following situation

E D C B A f (C) f (D) f (B) f (A) f (E)
There are three fixed points:

p ∈ f (B), q ∈ A, s ∈ f (D).
The points q is a sink in the sense that for all z ∈ A ∪C ∪ E we have lim n→+∞ f n (z) = q. The points p and s are saddle points: if m lies on the horizontal through p then f n squeezes it to p as n → +∞, while if m lies on the vertical through p then f -n squeezes it to p as n → +∞. In some coordinates centered in p we have

∀(x, y) ∈ B, f (x, y) = (kx, my)
for some 0 < k < 1 < m; similarly f (x, y) = (-kx, -my) on D for some coordinates centered at s. Let us recall that the sets

W s (p) = z f n (z) → p as n → +∞ , W u (p) = z f n (z) → p as n → -∞
are called stable and unstable manifolds of p. They intersect at r, which is what Poincaré called a homoclinic point. Homoclinic points are dense in m ∈ ∆ f n (m) ∈ ∆, n ∈ Z . The keypart of the dynamic of f happens on the horseshoe

Λ = z f n (z) ∈ B ∪ D ∀ n ∈ Z .
Let us introduce the shift map on the space of two symbols. Take two symbols 0 and 1, and look at the set Σ = 0, 1 Z of all bi-infinite sequences a = (a n ) n∈Z where, for each n, a n is 0

or 1. The map σ : Σ → Σ that sends a = (a n ) to σ(a) = (a n+1
) is a homeomorphism called the shift map. Let us consider the itinerary map i : Λ → Σ defined as follows:

i(p) = (s n ) n∈Z where s n = 1 if f n (p) is in B and s n = 0 if f n (p) belongs to D. The diagram Σ i σ G G Σ i Λ f G G Λ
commutes so every dynamical property of the shift map is possessed equally by f |Λ . Due to conjugacy the chaos of σ is reproduced exactly in the horseshoe: the map σ has positive entropy: log 2; it has 2 n periodic orbits of period n, and so must be the set of periodic orbits of f |Λ .

To summarize: every dynamical system having a transverse homoclinic point also has a horseshoe and thus has a shift chaos, even in higher dimensions. The mere existence of a transverse intersection between the stable and unstable manifolds of a periodic orbit implies a horseshoe; since transversality persists under perturbation, it follows that so does the horseshoe and so does the chaos.

The concepts of horseshoe and hyperbolicity are related. In the description of the horseshoe the derivative of f stretches tangent vectors that are parallel to the vertical and contracts vectors parallel to the horizontal, not only at the saddle points, but uniformly throughout Λ. In general, hyperbolicity of a compact invariant set such as Λ is expressed in terms of expansion and contraction of the derivative on subbundles of the tangent bundle.

Two examples. -Let us consider

P c (z) = z 2 +c. A periodic point p of P c with period n is repelling if |(P n c (p)) ′ | > 1 and
the Julia set of P c is the closure of the set of repelling periodic points. P c is a complex horseshoe if it is hyperbolic (i.e. uniformly expanding on the Julia set) and conjugate to the shift on two symbols. The Mandelbrot set M is defined as the set of all points c such that the sequence (P n c (0)) n does not escape to infinity

M = c ∈ C ∃ s ∈ R, ∀ n ∈ N, P n c (0) ≤ s .
The complex horseshoe locus is the complement of the Mandelbrot set.

Let us consider the Hénon family of quadratic maps

φ a,b : R 2 → R 2 , φ a,b (x, y) = (x 2 + a -by, x).
For fixed parameters a and b, φ a,b defines a dynamical system, and we are interested in the way that the dynamic varies with the parameters. The parameter b is equal to det jac φ a,b ; when b = 0, the map has a one-dimensional image and is equivalent to P c . As soon as b is non zero, these maps are diffeomorphisms, and maps similar to Smale's horseshoe example occur when a << 0 (see [START_REF] Devaney | Shift automorphisms in the Hénon mapping[END_REF]).

In the 60's it was hoped that uniformly hyperbolic dynamical systems might be in some sense typical. While they form a large open sets on all manifolds, they are not dense. The search for typical dynamical systems continues to be a great problem, in order to find new phenomena we try the framework of compact complex surfaces. 9.2. Some algebraic geometry 9.2.1. Compact complex surfaces. -Let us recall some notions introduced in Chapters 1 and 3 and some others.

To any surface S we associate its Dolbeault cohomology groups H p,q (S) and the cohomo-

logical groups H k (S, Z), H k (S, R) and H k (S, C). Set H 1,1 R (S) = H 1,1 (S) ∩ H 2 (S, R).
Let f : X S be a dominating meromorphic map between compact complex surfaces, let Γ be a desingularization of its graph and let π 1 , π 2 be the natural projections. A smooth form α in C ∞ p,q (S) can be pulled back as a smooth form π * 2 α ∈ C ∞ p,q (Γ) and then pushed forward as a current. We define f * by

f * α = π 1 * π *
2 α which gives a L 1 loc form on X that is smooth outside Ind f . The action of f * satisfies: f * (dα) = d( f * α) so descends to a linear action on Dolbeault cohomology.

Let {α} ∈ H p,q (S) be the Dolbeault class of some smooth form α. We set

f * {α} = {π 1 * π * 2 α} ∈ H p,q (X).
This defines a linear map f * from H p,q (S) into H p,q (X). Similarly we can define the pushforward f * = π 2 * π * 1 from H p,q (X) into H p,q (S). When f is bimeromorphic, we have f * = ( f -1 ) * . The operation (α, β) → α ∧ β on smooth 2-forms induced a quadratic intersection form, called product intersection, denoted by (•, •) on H 2 (S, C). Its structure is given by the following fundamental statement.

Theorem 9.2.1 ([9]

). -Let S be a compact Kähler surface and let h 1,1 denote the dimension of H 1,1 (S, R) ⊂ H 2 (S, R). Then the signature of the restriction of the intersection product to H 1,1 (S, R) is (1, h 1,1 -1). In particular, there is no 2-dimensional linear subspace L in H 1,1 (S, R) with the property that (v, v) = 0 forall v in L.

The Picard group Pic(P 2 ) is isomorphic to Z (see Chapter 1, Example 1.1.2); similarly H 2 (P 2 (C), Z) is isomorphic to Z. We may identify Pic(P 2 ) and H 2 (P 2 (C), Z). 9.2.2. Exceptional configurations and characteristic matrices. -Let f ∈ Bir(P 2 ) be a birational map of degree ν. By Theorem 1.3.1 there exist a smooth projective surface S ′ and π, η two sequences of blow-ups such that

S π } } ③ ③ ③ ③ ③ ③ ③ ③ η 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ P 2 (C) f G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
We can rewrite π as follows

π : S = S k π k → S k-1 π k-1 → . . . π 2 → S 1 π 1 → S 0 = P 2 (C)
where π i is the blow-up of the point p i-1 in S i-1 . Let us set

E i = π -1 i (p i ), E i = (π i+1 • . . . • π k ) * E i .
The divisors E i are called the exceptional configurations of π and the p i base-points of f .

For any effective divisor D = 0 on P 2 (C) let mult p i D be defined inductively in the following way. We set mult p 1 D to be the usual multiplicity of D at p 1 : it is defined as the largest integer m such that the local equation of D at p 1 belongs to the m-th power of the maximal ideal m P 2 ,p 1 . Suppose that mult p 1 D is defined. We take the proper inverse transform π -1 i D of D in S i and define mult p i+1 D = mult p i+1 π -1 i D. It follows from the definition that

π -1 D = π * (D) - k ∑ i=1 m i E i
where m i = mult p i D.

There are two relationships between ν and the m i 's (Chapter 1, §1.2):

1 = ν 2 - k ∑ i=1 m 2 i , 3 = 3ν - k ∑ i=1 m i .
An ordered resolution of f is a decomposition f = ηπ -1 where η and π are ordered sequences of blow-ups. An ordered resolution of f induces two basis of Pic(S)

• B = e 0 = π * H, e 1 = [E 1 ], . . . , e k = [E k ] , • B ′ = e ′ 0 = η * H, e ′ 1 = [E ′ 1 ], . . . , e ′ k = [E ′ k ]
, where H is a generic line. We can write e ′ i as follows

e ′ 0 = νe 0 - k ∑ i=1 m i e i , e ′ j = ν j e 0 - k ∑ i=1 m i j e i , j ≥ 1.
The matrix of change of basis

M =      ν ν 1 . . . ν k -m 1 -m 11 . . . -m 1k . . . . . . . . . -m k -m k1 . . . -m kk     

and

(r 3 , s 3 ) → (r 3 s 3 , s 3 ) (x,y) → (1 : r 3 :

r 3 s 3 ) = 1 r 3 s 3 , 1 s 3 (x,y) → 1 r 3 , 1 s 3 (r 3 ,s 3 ) .
Thus G → ∆ ′ 3 and ∆ ′ 3 → G. There are no more points of indeterminacy, no more exceptional curves; in other words σ is conjugate to an automorphism of Bl P,Q 1 ,R 2 P 2 .

Let H be a generic line. Note that E 1 = E, E 2 = F, E 3 = H. Consider the basis {H, E, F, G}.

After the first blow-up ∆ and E are swapped; the point blown up is the intersection of ∆ ′ and ∆ ′′ so ∆ → ∆ + F + G. Then σ * E = H -F -G. Similarly we have:

σ * F = H -E -G and σ * G = H -E -F.
It remains to determine σ * H. The image of a generic line by σ is a conic hence

σ * H = 2H - m 1 E -m 2 F -m 3 G.
Let L be a generic line described by a 0 x + a 1 y + a 2 z. A computation shows that

(u 1 , v 1 ) → (u 1 , u 1 v 1 ) (y,z) → (u 2 1 v 1 : u 1 v 1 : u 1 ) → u 1 (a 0 v 2 + a 1 u 2 v 2 + a 2 )
vanishes to order 1 on E = {u 1 = 0} thus m 1 = 1. Note also that

(u 2 , v 2 ) → (u 2 , u 2 v 2 ) (x,z) → (u 2 v 2 : u 2 2 v 2 : u 2 ) → u 2 (a 0 v 2 + a 1 u 2 v 2 + a 2 ), respectively (u 3 , v 3 ) → (u 3 , u 3 v 3 ) (x,y) → (u 3 v 3 : u 3 : u 2 3 v 3 ) → u 3 (a 0 v 3 + a 1 + a 2 u 3 v 3 )
vanishes to order 1 on F = {u 2 = 0}, resp. G = {u 3 = 0} so m 2 = 1, resp. m 3 = 1. Therefore σ * H = 2H -E -F -G and the characteristic matrix of σ in the basis H, E, F, G is We can verify that M τ = M σ .

M σ =     2 1 1 1 -1 0 -1 -1 -1 -1 0 -1 -1 -1 -1 0     .

9.3.

Where can we find automorphisms with positive entropy ? 9.3.1. Some properties about the entropy. -Let f be a map of class C ∞ on a compact manifold V; the topological entropy is greater than the logarithm of the spectral radius of the linear map induced by f on H * (V, R), direct sum of the cohomological groups of V:

h top ( f ) ≥ log r( f * ).
Remark that the inequality h top ( f ) ≥ log r( f * ) is still true in the meromorphic case ( [START_REF] Dinh | Regularization of currents and entropy[END_REF]). Before stating a more precise result when V is Kähler we introduce some notation: for all integer p such that 0 ≤ p ≤ dim C V we denote by λ p ( f ) the spectral radius of the map f * acting on the Dolbeault cohomological group H p,p (V, R).

( f ) = λ 1 ( f ) > 1.
In other words in order to know if the entropy of f is positive we just have to study the growth of ( f n ) * {α} where {α} is a Kähler form.

Examples 9.3.3. -• Let V be a compact Kähler manifold and Aut 0 (V) be the connected component of Aut(V) which contains the identity element. The topological entropy of each element of Aut 0 (V) is zero.

• The topological entropy of an holomorphic endomorphism f of the projective sapce is equal to the logarithm of the topological degree of f . • Whereas the topological entropy of an elementary automorphism is zero, the topological entropy of an Hénon automorphism is positive.

A theorem of Cantat.

-Before describing the pairs (S, f ) of compact complex surfaces S carrying an automorphism f with positive entropy, let us recall that a surface S is rational if it is birational to P 2 (C). A rational surface is always projective ( [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). A K3 surface is a complex, compact, simply connected surface S with a trivial canonical bundle. Equivalently there exists a holomorphic 2-form ω on S which is never zero; ω is unique modulo multiplication by a scalar. Let S be a K3 surface with a holomorphic involution ι. If ι has no fixed point the quotient is an Enriques surface, otherwise it is a rational surface. As Enriques surfaces are quotients of K3 surfaces by a group of order 2 acting without fixed points, their theory is similar to that of algebraic K3 surfaces.

Theorem 9.3.4 ([44]

). -Let S be a compact complex surface. Assume that S has an automorphism f with positive entropy. Then • either f is conjugate to an automorphism on the unique minimal model of S which is either a torus, or a K3 surface, or an Enriques surface;

• or S is rational, obtained from P 2 (C) by blowing up P 2 (C) in at least 10 points and f is birationally conjugate to a birational map of P 2 (C). In particular S is kählerian.

Examples 9.3.5. -

• Set Λ = Z[i] and E = C/Λ. The group SL 2 (Λ) acts linearly on C 2 and preserves the lattice Λ × Λ; then each element A of SL 2 (Λ) induces an automorphism f A on E × E which commutes with ι(x, y) = (ix, iy). Each automorphism f A can be lifted to an automorphism f A on the desingularization of (E × E)/ι which is a K3 surface. The entropy of f A is positive as soon as the modulus of one eigenvalue of A is strictly greater than 1.

• We have the following statement due to Torelli. Theorem 9.3.6. -Let S be a K3 surface. The morphism

Aut(S) → GL(H 2 (S, Z)), f → f * is injective.
Conversely assume that ψ is an element of GL(H 2 (S, Z)) which preserves the intersection form on H 2 (S, Z), the Hodge decomposition of H 2 (S, Z) and the Kähler cone of H 2 (S, Z). Then there exists an automorphism f on S such that f * = ψ.

The case of K3 surfaces has been studied by Cantat, McMullen, Silverman, Wang and others (see for example [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF][START_REF] Mcmullen | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF][START_REF] Silverman | Rational points on K3 surfaces: a new canonical height[END_REF][START_REF] Wang | Rational points and canonical heights on K3-surfaces in P 1 × P 1 × P 1[END_REF]). The context of rational surfaces produces much more examples (see for example [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF][START_REF] Bedford | Periodicities in linear fractional recurrences: degree growth of birational surface maps[END_REF][START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF][START_REF] Bedford | Continuous families of rational surface automorphisms with positive entropy[END_REF][START_REF] Déserti | Special automorphisms of rational surfaces with positive topological entropy[END_REF]). -Let S be a rational surface and let f be an automorphism on S such that f * is of infinite order; then there exists a sequence of holomorphic maps π j+1 : S j+1 → S j such that S 1 = P 2 (C), S N+1 = S and π j+1 is the blow-up of p j ∈ S j .

Remark that a surface obtained from P 2 (C) via generic blow-ups has no nontrivial automorphism ( [START_REF] Hirschowitz | Symétries des surfaces rationnelles génériques[END_REF][START_REF] Koitabashi | Automorphism groups of generic rational surfaces[END_REF]). Moreover we have the following statement which can be found for example in [ Moreover, if n ≤ 8 then there exists an integer k such that f k is birationally conjugate to an automorphism of the complex projective plane.

Proof. -Assume that f has positive entropy log λ( f ) > 0. According to [START_REF] Cantat | Dynamique des automorphismes des surfaces projectives complexes[END_REF] there exists a non-trivial cohomology class θ in H 2 (S, R) such that f * θ = λ( f )θ and θ 2 = 0. Moreover

f * K S = f * K S = K S . Since (θ, K S ) = ( f * θ, f * K S ) = (λ( f )θ, K S )
we have (θ, K S ) = 0. The intersection form on S has signature (1, n -1) and K 2 S ≥ 0 for n ≤ 9 so θ = cK S for some c < 0. But then f * θ = θ = λ( f )θ: contradiction. The map f thus has zero entropy.

If n ≤ 8, then K 2 S > 0. The intersection form is thus strictly negative on the orthogonal complement H ⊂ H 2 (S, R) of K S . But dim H is finite, H is invariant under f * and f * preserves H 2 (S, Z) so f * has finite order on H. Therefore f k * is trivial for some integer k. In particular f k preserves each of the exceptional divisors in X that correspond to the n ≤ 8 points blown up in P 2 (C). So f k descends to a well-defined automorphism of P 2 (C).

Let f be an automorphism with positive entropy on a Kähler surface. The following statement gives properties on the eigenvalues of f * . Theorem 9.3.9 ([19], Theorem 2.8, Corollary 2.9). -Let f be an automorphism with positive entropy log λ( f ) on a Kähler surface. The first dynamical degree λ( f ) is an eigenvalue of f * with multiplicity 1 and this is the unique eigenvalue with modulus strictly greater than 1.

If η is an eigenvalue of f * , then either η belongs to {λ( f ), λ( f ) -1 }, or |η| is equal to 1.

Proof. -Let v 1 , . . ., v k denote the eigenvectors of f * for which the associated eigenvalue µ ℓ has modulus > 1. We have Let us see that λ( f ) has multiplicity one. Assume that it has not; then there exists θ such that f * θ = λ( f )θ + cv. In this case

(v j , v k ) = ( f * v j , f * v k ) = µ j µ k (v j , v k ), ∀ 1 ≤ j ≤ k so (v j , v k ) = 0. Let L be the linear span of v 1 , . . ., v k . Each element v = ∑ i α i v i in L satisfies (v, v) =
(θ, v) = ( f * θ, f * v) = (λ( f )θ + cv, λv) = λ 2 (θ, v)
so (θ, v) = 0. Similarly we have (θ, θ) = 0 so by Theorem 9.2.1 again, the space spanned by θ and v must have dimension 1; in other words λ( f ) is a simple eigenvalue.

We know that λ( f ) is the only eigenvalue of modulus > 1. Since ( f * ) -1 = ( f -1 ) * , if η is an eigenvalue of f * , then 1 η is an eigenvalue of ( f -1 ) * . Applying the first statement to f -1 we obtain that λ is the only eigenvalue of ( f -1 ) * with modulus strictly larger than 1.

Let χ f denote the characteristic polynomial of f * . This is a monic polynomial whose constant term is ±1 (constant term is equal to the determinant of f * ). Let Ψ f be the minimal polynomial of λ( f ). Except for λ( f ) and λ( f ) -1 all zeroes of χ f (and thus of Ψ f ) lie on the unit circle. Such polynomial is a Salem polynomial and such a λ( f ) is a Salem number. So Theorem 9.3.9 says that if f is conjugate to an automorphism then λ( f ) is a Salem number; in fact the converse is true ( [33]). There exists another birational invariant which allows us to characterize birational maps that are conjugate to automorphisms (see [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]33]).

i b 2 = a 2 α 2 -α , . . . , b n = a n + Q n α n -α with Q n ∈ Z[a i , i ≤ n -1, b i , i ≤ n]
we say that f is formally linearizable. If ψ converges, we say that the germ f is analytically linearizable.

Theorem 9.4.1 (Cremer). -If lim inf |α q -α| 1/q = 0, there exists an analytic germ f of the type (9.4.1) which is not analytically linearizable. More precisely if lim inf |α q -α| 1 ν q = 0, then no polynomial germ f (z) = αz + a 2 z 2 + . . . + z ν of degree ν is linearizable. For more details see [START_REF] Arnol ′ D | Chapitres supplémentaires de la théorie des équations différentielles ordinaires[END_REF][START_REF] Blanchard | Complex analytic dynamics on the Riemann sphere[END_REF][START_REF] Herman | Recent results and some open questions on Siegel's linearization theorem of germs of complex analytic diffeomorphisms of C n near a fixed point[END_REF][START_REF] Siegel | Iteration of analytic functions[END_REF]. 9.4.2. Fatou sets. -• If (x, y) belongs to J + there exists no neighborhood U of (x, y) on which the family

f n |U n ≥ 0 is normal.
We have the following statement. Definitions. -Let Ω be a Fatou component; Ω is recurrent if there exist a compact subset C of Ω and a point m in C such that f n j (m) belongs to C for an infinite number of n j → +∞. A recurrent Fatou component is periodic.

A fixed point m of f is a sink if m belongs to the interior of the stable manifold

W s (m) = p lim n→+∞ dist( f n (m), f n (p)) = 0 .
We say that W s (m) is the basin of m. If m is a sink, the eigenvalues of D f m have all modulus less than 1.

A Siegel disk (resp. Herman ring) is the image of a disk (resp. of an annulus) ∆ by an injective holomorphic map ϕ having the following property: for any z in ∆ we have

f ϕ(z) = ϕ(αz), α = e 2iπθ , θ ∈ R \ Q.
We can describe the recurrent Fatou components of a Hénon map. -If S is a complex torus, an automorphism of positive entropy is essentially an element of GL 2 (Z); since the entropy is positive, the eigenvalues satisfy:

|λ 1 | < 1 < |λ 2 |
and the Fatou set is empty. Assume that S is a K3 surface or a quotient of a K3 surface. Since there exists a volume form, the only possible Fatou components are rotation domains. McMullen proved there exist non algebraic K3 surfaces with rotation domains of rank 2 (see [START_REF] Mcmullen | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF]); we can also look at [START_REF] Oguiso | The third smallest Salem number in automorphisms of K3 surfaces. In Algebraic geometry in East Asia-Seoul[END_REF].

The other compact surfaces carrying automorphisms with positive entropy are rational ones; in this case there are rotation domains of rank 1, 2 (see [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF][START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]). Other phenomena like attractive, repulsive basins can happen ( [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF][START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]).

CHAPTER 10 WEYL GROUPS AND AUTOMORPHISMS OF POSITIVE ENTROPY

In [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF] McMullen, thanks to Nagata's works and Harbourne's works, establishes a result similar to Torelli's theorem for K3 surfaces: he constructs automorphisms on some rational surfaces prescribing the action of the automorphisms on cohomological groups of the surface. These rational surfaces own, up to multiplication by a constant, a unique meromorphic nowhere vanishing 2-form Ω. If f is an automorphism on S obtained via this construction, f * Ω is proportional to Ω and f preserves the poles of Ω. When we project S on the complex projective plane, f induces a birational map preserving a cubic.

The relationship of the Weyl group to the birational geometry of the plane, used by Mc-Mullen, is discussed since 1895 in [START_REF] Kantor | Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene[END_REF]

Weyl groups

Let S be a surface obtained by blowing up the complex projective plane in a finite number of points. Let e 0 , . . . , e n be a basis of H 2 (S, Z); if For all j in {0, . . . , n-1} we have α j •α j = -2. When j is nonzero the reflection R α j induces a permutation on {e j , e j+1 }. The subgroup generated by the R α j 's, with 1 ≤ j ≤ n -1, is the set of permutations on the elements {e 1 , . . . , e n }. Let W n ⊂ O(Z 1,n ) denote the group

e 0 • e 0 = 1, e j • e j = -1, ∀ 1 ≤ j ≤ k, e i • e j = 0, ∀ 0 ≤ i = j ≤
R α j | 0 ≤ j ≤ n -1 which is called Weyl group.
The Weyl groups are, for 3 ≤ n ≤ 8, isomorphic to the following finite groups

A 1 × A 2 , A 4 , D 5 , E 6 , E 7 , E 8
and are associated to del Pezzo surfaces. For n ≥ 9 Weyl groups are infinite and for n ≥ 10 Weyl groups contain elements with a spectral radius strictly greater than 1.

If Y and S are two projective surfaces, let us recall that Y dominates S if there exists a surjective algebraic birational morphism from Y to S.

Theorem 10.1.1 ([82]

). -Let S be a rational surface which dominates P 2 (C).

• The Weyl group W k ⊂ GL(Pic(S)) does not depend on the chosen exceptional configuration.

• If E and E ′ are two distinct exceptional configurations, there exists w in W k such that w(E) = E ′ . • If S is obtained by blowing up k generic points and if E is an exceptional configuration, then for any w in the Weyl group w(E) is an exceptional configuration.

If f is an automorphism of S, by a theorem of Nagata there exists a unique element w in W n such that

Z 1,n ϕ w G G Z 1,n ϕ H 2 (S, Z) f * G G H 2 (S, Z)
commutes; we said that the automorphism f realizes ω.

A product of generators R α j is a Coxeter element of W n . Note that all Coxeter elements are conjugate so the spectral radius of a Coxeter element is well defined.

The map σ is represented by the reflection κ i jk = R α i jk where α i jk = e 0e ie je k and i, j, k ≥ 1 are distinct elements; it acts as follows e 0 → 2e 0e ie je k , e i → e 0e je k , e j → e 0e ie k e k → e 0e ie j , e ℓ → e ℓ if ℓ ∈ {0, i, j, k}

When n = 3, we say that κ 123 is the standard element of W 3 . Consider the cyclic permutation

(123 . . . n) = κ 123 R α 1 . . . R α n-1 ∈ Σ n ⊂ W n ;

Statements

In [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF] McMullen constructs examples of automorphisms with positive entropy "thanks to" elements of Weyl groups.

Theorem 10.2.1 ([144]

). -For n ≥ 10, the standard element of W n can be realizable by an automorphism f n with positive entropy log(λ n ) of a rational surface S n .

More precisely the automorphism f n : S n → S n can be chosen to have the following additional properties:

• S n is the complex projective plane blown up in n distinct points p 1 , . . . , p n lying on a cuspidal cubic curve C ,

• there exists a nowhere vanishing meromorphic 2-form η on S n with a simple pole along the proper transform of C ,

• f * n (η) = λ n • η, • ( f n , S n ) is
minimal in the sense of Manin (1) . The first three properties determine f n uniquely. The points p i admit a simple description which leads to concrete formulas for f n .

The smallest known Salem number is a root λ Lehmer ∼ 1.17628081 of Lehmer's polynom

L(t) = t 10 + t 9 -t 7 -t 6 -t 5 -t 4 -t 3 + t + 1.

Theorem 10.2.2 ([144]

). -If f is an automorphism of a compact complex surface with positive entropy, then h top ( f ) ≥ log λ Lehmer .

Corollary 10.2.3 ([144]

). -The map f 10 : S 10 → S 10 is an automorphism of S 10 with the smallest possible positive entropy.

Theorem 10.2.4 ([144]

). -There is an infinite number of n for which the standard element of W n can be realized as an automorphism of P 2 (C) blown up in a finite number of points having a Siegel disk.

Let us also mention a more recent work in this direction ( [START_REF] Uehara | Rational surface automorphisms with positive entropy[END_REF]). Diller also find examples using plane cubics ( [START_REF] Diller | Cremona transformations, surface automorphisms, and plane cubics[END_REF]). of the Picard group of such a curve (see [START_REF] Harris | Moduli of curves[END_REF] for more details). We have the following exact sequence 0

-→ Pic 0 (C ) -→ Pic(C ) -→ H 2 (C , Z) -→ 0
where Pic 0 (C ) is isomorphic to 1. Let Z be a surface and G be a subgroup of Aut(S). A birational map f : S S is G-equivariant if G = f G f -1 ⊂ Aut( S). The pair (G, S) is minimal if every G-equivariant birational morphism is an isomorphism.

• either a torus C/Λ (when C is smooth); • or to the multiplicative group C * (it corresponds to the following case: C is either a nodal cubic or the union of a cubic curve and a transverse line, or the union of three line in general position);

• or to the additive group C (when C is either a cuspidal cubic, or the union of a conic and a tangent line, or the union of three lines through a single point).

A cubic marked curve is a pair (C , η) of an abstract curve C equipped with a homomorphism Let (C , η) and (C ′ , η ′ ) be two cubic marked curves; an isomorphism between (C , η) and

(C ′ , η ′ ) is a biholomorphic application f : C → C ′ such that η ′ = f * • η.
Let (C , η) be a cubic marked curve; let us set

W (C , η) = w ∈ W n (C , ηw) is a cubic marked curve , Aut(C , η) = w ∈ W (C , η) (C , η) & (C ′ , η ′ ) are isomorphic .
We can decompose the marking η of C in two pieces

η 0 : ker(deg •η) → Pic 0 (C ), deg •η : Z 1,n → H 2 (C , Z).
We have the following property.

Theorem 10.3.2 ([144]

). -Let (C , η) be a marked cubic curve. The applications η 0 and deg •η determine (C , η) up to isomorphism.

A consequence of this statement is the following. The marking determines the morphism π : S → P 2 (C) up to the action of an automorphism of P 2 (C).

Let (S, Φ) and (S ′ , Φ) be two marked blow-ups; an isomorphism between (S, Φ) and (S ′ , Φ ′ ) is a biholomorphic application F : S → S ′ such that the following diagram

Z 1,n Φ z z ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ Φ ′ 6 6 ■ ■ ■ ■ ■ ■ ■ ■ ■ H 2 (S, Z) F * G G H 2 (S ′ , Z)
commutes. If (S, Φ) and (S ′ , Φ ′ ) are isomorphic, there exists an automorphism ϕ of P 2 (C) such that p ′ i = ϕ(p i ). Assume that there exist two birational morphisms π, π ′ : S → P 2 (C) such that S is the surface obtained by blowing up P 2 (C) in p 1 , . . . , p n (resp. p ′ 1 , . . . , p ′ n ) via π (resp. π ′ ).There exists a birational map f : P

2 (C) P 2 (C) such that the diagram S π } } ③ ③ ③ ③ ③ ③ ③ ③ π ′ 3 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ P 2 (C) f G G ❴ ❴ ❴ ❴ ❴ ❴ ❴ P 2 (C)
commutes; moreover there exists a unique element w in Z 1,n such that Φ ′ = Φw.

The Weyl group satisfies the following property due to Nagata: let (S, Φ) be a marked blowup and let w be an element of Z 1,n . If (S, Φw) is still a marked blow-up, then w belongs to the Weyl group W n . Let (S, Φ) be a marked blow-up; let us denote by W (S, Φ) the set of elements w of W n such that (S, Φw) is a marked blow-up:

W (S, Φ) = w ∈ W n (S, Φw) is a marked blow-up .
The right action of the symmetric group reorders the base-points of a blow-up so the group of permutations is contained in W (S, Φ). The following statement gives other examples of elements of W (S, Φ).

Theorem 10.3.4 ([144]

). -Let (S, Φ) be a marked blow-up and let σ be the involution (x : y : z) (yz : xz : xy). Let us denote by p 1 , . . . , p n the base-points of (S, Φ). If, for any 4 ≤ k ≤ n, the point p k does not belong to the line through p i and p j , where 1 ≤ i, j ≤ 3, i = j, then (S, Φκ 123 ) is a marked blow-up.

Proof. -Let π : S → P 2 (C) be the birational morphism associated to the marked blow-up (S, Φ). Let us denote by q 1 , q 2 and q 3 the points of indeterminacy of σ. Let us choose some coordinates for which p i = q i for i = 1, 2, 3; then π ′ = σπ : S → P 2 (C) is a birational morphism which allows us to see (S, Φκ 123 ) as a marked blow-up with base-points p 1 , p 2 , p 3 and σ(p i ) for i ≥ 4. These points are distinct since, by hypothesis, p 4 , . . . , p n do not belong to the lines contracted by σ.

A root α of Θ n is a nodal root for (S, Φ) if Φ(α) is represented by an effective divisor D. In this case D projects to a curve of degree d > 0 on P 2 (C); thus α = de 0 -∑ i≥1 m i e i is a positive root. A nodal root is geometric if we can write D as a sum of smooth rational curves. Proof. -After reordering the base-points p 1 , . . . , p n , we can assume that p 1 , p 2 and p 3 are colinear; let us denote by L the line through these three points. We can suppose that the basepoints which belong to L are p 1 , . . . , p k . The strict transform L of L induces a smooth rational curve on S with

[ L] = [H -∑ k i=1 E i ] so Φ(α 123 ) = [ L + k ∑ i=1 E i ]. Theorem
[Y ] = [3H -∑ i E i ] = -K S .
A marked pair (S, Φ,Y ) is the data of a marked blow-up (S, Φ) and an anticanonical curve Y. An isomorphism between marked pairs (S, Φ,Y ) and (S ′ , Φ ′ ,Y ′ ) is a biholomorphism f from S into S ′ , compatible with markings and which sends Y to Y ′ . If n ≥ 10, then S contains at most one irreducible anticanonical curve; indeed if such a curve Y exists, then Y 2 = 9n < 0.

η : Z 1,n Φ -→ H 2 (S, Z) = Pic(S) r -→ Pic(Y ) π * -→ Pic(C )
where r is the restriction r : Pic(S) → Pic(Y ). Therefore a marked pair (S,Y, Φ) determines canonically a marked cubic curve (C , η). This construction inverts the previous one, in other words we have the following statement.

Proposition 10.3.8 ([144]

). -A marked pair determines canonically a marked cubic curve and conversely.

Idea of the proof

The automorphisms constructed to prove the previous results are obtained from a birational map by blowing up base-points on a cubic curve C ; the cubic curves play a very special role because its transforms Y are anticanonical curves. Assume that w ∈ W n is realized by an automorphism F of a rational surface S which preserve an anticanonical curve Y . A marked cubic curve (C , η) is canonically associated to a marked pair (S, Φ,Y ) (Theorem 10.3.8). Then there exists a birational map f : P 2 (C) P 2 (C) such that:

• the lift of f to S coincides with F,

• f preserves C ,

• and f induces an automorphism f * of Pic 0 (C ) which satisfies η 0 w = f * η 0 . In other words [η 0 ] is a fixed point for the natural action of w on the moduli space of markings.

Conversely to realize a given element w of the group W n we search a fixed point η 0 in the moduli space of markings. We can associate to η 0 a marked cubic (C , η) up to isomorphism (Corollary 10.3.3). Let us denote by (S, Φ,Y ) the marked pair canonically determined by (C , η). Assume that, for any α in Θ n , η 0 (α) is non zero (which is a generic condition); the base-points p i do not satisfy some nodal relation (they all are distinct, no three are on a line, no six are on a conic, etc). According to a theorem of Nagata there exists a second projection π ′ : S → P 2 (C) which corresponds to the marking Φw. Let us denote by C ′ the cubic π ′ (Y ).

Since [η 0 ] is a fixed point of w, the marked cubics (C ′ , ηw) and (C , η) are isomorphic. But such an isomorphism is an automorphism F of S satisfying F * Φ = Φw.

Let us remark that in [START_REF] Hirschowitz | Symétries des surfaces rationnelles génériques[END_REF][START_REF] Harbourne | Blowings-up of P 2 and their blowings-down[END_REF][START_REF] Penrose | A quadratic mapping with invariant cubic curve[END_REF][START_REF] Diller | Cremona transformations, surface automorphisms, and plane cubics[END_REF] there are also constructions with automorphisms of surfaces and cubic curves.

Examples

Let us consider the family of birational maps f : P

2 (C) P 2 (C) given in the affine chart z = 1 by f (x, y) = a + y, b + y x , a, b ∈ C.
Let us remark that the case b = -a has been studied in [START_REF] Penrose | A quadratic mapping with invariant cubic curve[END_REF] and [START_REF] Bastien | Global behavior of the solutions of Lyness' difference equation u n+2 u n = u n+1 + a[END_REF]. The points of indeterminacy of f are p 1 = (0 : 0 : 1), p 2 = (0 : 1 : 0) and p 3 = (1 : 0 : 0). Let us set p 4 = (a : b : 1) and let us denote by ∆ (resp. ∆ ′ ) the triangle whose vertex are p 1 , p 2 , p 3 (resp. p 2 , p 3 , p 4 ). The map f sends ∆ onto ∆ ′ : the point p 1 (resp. p 2 , resp. p 3 ) is blown up on the line (p 1 p 4 ) (resp. (p 2 p 3 ), resp. (p 3 p 4 )) and the lines (p 1 p 2 ) (resp. (p 1 p 3 ), resp. (p 2 p 3 )) are contracted on p 2 (resp. p 4 , resp. p 3 ).

If a and b are chosen such that p 1 = p 4 , then ∆ is invariant by f and if we blow up P 2 (C) at p 1 , p 2 , p 3 we obtain a realization of the standard Coxeter element of W 3 . Indeed, f sends a generic line onto a conic through the p i ; so w(e 0 ) = 2e 0e 1e 2e 3 . The point p 1 (resp. p 2 , resp. p 3 ) is blown up on the line through p 2 and p 3 (resp. p 1 and p 3 , resp. p 1 and p 2 ). Therefore Proof. -Assume that there exists an integer i such that f i (p 4 ) = p i+4 . Let (S, π) be the marked blow-up with base-points p i . The map f lifts to a morphism F 0 : S → P 2 (C). Since any p i is now the image F 0 (ℓ i ) of a line in S, the morphism F 0 lifts to an automorphism F of S such that f lifts to F. Let us find the element w realized by F. Let us remark that f sends a generic line onto a conic through p 2 , p

CHAPTER 11 AUTOMORPHISMS OF POSITIVE ENTROPY: SOME EXAMPLES

A possibility to produce an automorphism f on a rational surface S is the following: starting with a birational map f of P 2 (C), we find a sequence of blow-ups π : S → P 2 (C) such that the induced map f S = π f π -1 is an automorphism of S. The difficulty is to find such a sequence π... If f is not an automorphism of the complex projective plane, then f contracts a curve C 1 onto a point p 1 ; the first thing to do to obtain an automorphism from f is to blow up the point p 1 via π 1 : S 1 → P 2 (C). In the best case f S 1 = π 1 f π -1 1 sends the strict transform of C 1 onto the exceptional divisor E 1 . But if p 1 is not a point of indeterminacy, f S 1 contracts E 1 onto p 2 = f (p 1 ). This process thus finishes only if f is not algebraically stable.

In [START_REF] Bedford | Continuous families of rational surface automorphisms with positive entropy[END_REF] Bedford and Kim exhibit a continuous family of birational maps ( f a ) a∈C k-2 . We will see that this family is conjugate to automorphisms with positive entropy on some rational surface S a (Theorem 11.6.1). Let us hold the parameter c fixed; the family f a induces a family of dynamical systems of dimension k/2 -1: there exists a neighborhood U of 0 in C k/2-1 such that if a = (a 0 , a 2 , . . . , a k-2 ), b = (b 0 , b 2 , . . . , b k-2 ) are in U then f a and f b are not smoothly conjugate (Theorem 11.6.3). Moreover they show, for k ≥ 4, the existence of a neighborhood U of 0 in C k/2-1 such that if a, b are two distinct points of U, then S a is not biholomorphically equivalent to S b (Theorem 11.6.4).

The results evoked in the last section are also due to Bedford and Kim ( [START_REF] Bedford | Dynamics of rational surface automorphisms: rotations domains[END_REF]); they concern the Fatou sets of automorphisms with positive entropy on rational non-minimal surfaces obtained from birational maps of the complex projective plane. Bedford and Kim prove that such automorphisms can have large rotation domains (Theorem 11.7.1).

Description of the sequence of blow-ups ([21])

Let f a,b be the birational map of the complex projective plane given by

Σ 0 = {x = 0}, Σ β = {bx + y = 0}, Σ γ = {ax + z = 0}. Σ γ Σ B Σ C Σ 0 p * p 2 q Σ β p 1 Set Y = Bl p 1 ,p 2 P 2 , π : Y → P 2 (C) and f a,b,Y = π -1 f a,b π.
Let us prove that after these two blow-ups Σ 0 does not belong to Exc f a,b,Y .

To begin let us blow up p 2 . Let us set x = r 2 and y = r 2 s 2 ; then (r 2 , s 2 ) is a system of local coordinates in which Σ β = {s 2 + b = 0} and E 2 = {r 2 = 0}. We remark that

(r 2 , s 2 ) → (r 2 , r 2 s 2 ) (x,y) → (r 2 (b + s 2 ) : b + s 2 : ar 2 + 1) = r 2 (b + s 2 ) ar 2 + 1 , b + s 2 ar 2 + 1 (x,y) → r 2 (b + s 2 ) ar 2 + 1 , 1 r 2 (r 2 ,s 2 ) .
Thus Σ β is sent onto E 2 and E 2 sur Σ 0 .

Let us now blow up p 1 . Set x = u 2 v 2 and y = v 2 ; the exceptional divisor E 2 is given by v 2 = 0 and Σ 0 by u 2 = 0. We have

(u 2 , v 2 ) → (u 2 v 2 , v 2 ) (x,y) → (u 2 v 2 (bu 2 + 1) : bu 2 + 1 : u 2 (au 2 v 2 + 1)) = v 2 (bu 2 + 1) au 2 v 2 + 1 , bu 2 + 1 u 2 (au 2 v 2 + 1) (x,y) → u 2 v 2 , bu 2 + 1 u 2 (au 2 v 2 + 1) (u 2 ,v 2 ) ; therefore E 2 is sent onto Σ 0 . Let us set x = r 1 , z = r 1 s 1 ; in the coordinates (r 1 , s 1 ) we have E 1 = {r 1 = 0}. Moreover (r 1 , s 1 ) → (r 1 , r 1 s 1 ) (x,z) → (br 1 + 1 : b + s 1 (br 1 + 1) : r 1 (a + s 1 )). Hence E 1 is sent onto Σ B . Set x = u 1 v 1 and z = v 1 ; in these coordinates Σ 0 = {u 1 = 0}, E 1 = {v 1 = 0} and (u 1 , v 1 ) → (u 1 v 1 , v 1 ) (x,z) → (u 1 (bu 1 v 1 + 1) : bu 1 v 1 + 1 : u 1 v 1 (au 1 + 1)) = u 1 , u 1 v 1 (au 1 + 1) bu 1 v 1 + 1 (x,z) → u 1 , v 1 (au 1 + 1) bu 1 v 1 + 1 (r 1 ,s 1 ) . So Σ 0 → E 1 and Σ β → E 2 → Σ 0 → E 1 → Σ B . In particular Ind f a,b,Y = {p * } & Exc f a,b,Y = {Σ γ }.
We remark that H, E 1 , E 2 is a basis of Pic(Y ). The exceptional divisor E 1 is sent on Σ B ; since p 1 belongs to Σ B we have E 1 → Σ B → Σ B + E 1 . On the other hand E 2 is sent onto Σ 0 ; as p 1 and p 2 belong to Σ 0 we have

E 2 → Σ 0 → Σ 0 + E 1 + E 2 .
Let H be a generic line of P 2 (C); it is given by ℓ = 0 with ℓ

= a 0 x + a 1 y + a 2 z. Its image by f a,b,Y is a conic thus f * a,b,Y H = 2H - 2 ∑ i=1 m i E i . Let us find the m i 's. As (r 2 , s 2 ) → (r 2 , r 2 s 2 ) (x,y) → (r 2 (b + s 2 ) : b + s 2 : ar 2 + 1) → r 2 a 0 r 2 (b + s 2 ) + a 1 (b + s 2 ) + a 2 (ar 2 + 1) and E 2 = {r 2 = 0} the integer m 2 is equal to 1. Since (r 1 , s 1 ) → (r 1 , r 1 s 1 ) (x,z) → (br 1 + 1 : b + s 1 (br 1 + 1) : r 1 (a + s 1 )) → s 1 r 1 a 0 (bs 1 r 1 + 1) + a 1 s 1 (bs 1 r 1 + 1) + s 1 r 1 (a + s 1 ) and E 1 = {s 1 = 0} we get m 1 = 1. That's why M f a,b,Y =   2 1 1 -1 -1 -1 -1 0 -1   .
(L), then f 2 a,b,Y (L) = M 2 f a,b   1 0 0   = 2H -E 2 ; in other words f 2 a,b,Y (L) is a conic which intersects E 2 but not E 1 . If p * does not belong to L ∪ f a,b,Y (L) ∪ f 2 a,b,Y (L), then f 3 a,b,Y (L) = M 3 f a,b   1 0 0   = 3H -E 1 -E 2 , i.e. f

Construction of surfaces and automorphisms ([21])

Let us consider the subset V n of C 2 given by Conversely assume that there exists an integer n such that (a, b) belongs to V n . Let S be the surface obtained from Y by blowing up the points q, f a,b,Y (q), . . . , f n a,b,Y (q) = p * of the orbit of q. We can check that the induced map f a,b,S is an automorphism of S.

V n = (a, b) ∈ C 2 f j a,b,Y (q) = p * ∀ 0 ≤ j ≤ n -1, f n a,b,Y (q) = p * . Theorem
χ n (t) = t n+1 (t 3 -t -1) + t 3 + t 2 -1.
Moreover, when n tends to infinity, λ( f a,b ) tends to the largest eigenvalue of t 3t -1.

The action f a,b,S * on the cohomology is given by

E 2 → Σ 0 = H -E 1 -E 2 → E 1 → Σ B = H -E 1 -Q
where Q denotes the divisor obtained by blowing up the point q which is on Σ

B . As p * is blown-up by f a,b on Σ C , we have Q → f a,b (Q) → . . . → f n a,b (Q) → Σ C = H -E 2 -Q.
Finally a generic line L intersects Σ 0 , Σ β and Σ γ with multiplicity 1; the image of L is thus a conic through q, p 1 and p 2 so H

→ 2H -E 1 -E 2 -Q. In the basis H, E 1 , E 2 , Q, f a,b (Q), . . . , f n a,b (Q) we have M f a,b =                   2 1 1 0 0 . . . . . . 0 1 -1 -1 -1 0 0 . . . . . . 0 0 -1 0 -1 0 0 . . . . . . 0 -1 -1 -1 0 0 0 . . . . . . 0 -1 0 0 0 1
                 
.

Invariant curves ([22])

In the spirit of [START_REF] Diller | Invariant curves for birational surface maps[END_REF] (see Chapter 5, §5.4) Bedford and Kim study the curves invariant by f a,b . There exists rational maps ϕ j : C → C 2 such that if (a, b) = ϕ j (t) for some complex number t, then f a,b has an invariant curve C with j irreducible components. Let us set There is a relationship between the parameters (a, b) for which there exists a complex number t such that ϕ j (t) = (a, b) and the roots of the characteristic polynomial χ n . Theorem 11.3.2. -Let n be an integer, let 1 ≤ j ≤ 3 be an integer and let t be a complex number. Assume that (a, b) := ϕ j (t) does not belong to any V k for k < n. Then (a, b) belongs to V n if and only if j divides n and t is a root of χ n .

ϕ 1 (t) = t -t 3 -t 4 1 + 2t + t 2 , 1 -t 5 t 2 + t 3 , ϕ 2 (t) = t + t 2 + t 3 1 + 2t + t 2 , t 3 -1 t + t 2 , ϕ 3 (t) = 1 + t,
We can write χ n as C n ψ n where C n is the product of cyclotomic factors and ψ n is the minimal polynomial of λ( f a,b ).

Theorem 11.3.3. -Assume that n ≥ 7. Let t be a root of χ n not equal to 1. Then either t is a root of ψ n , or t is a root of χ j for some 0 ≤ j ≤ 5.

Bedford and Kim prove that # (Γ j ∩ V n ) is, for n ≥ 7, determined by the number of Galois conjugates of the unique root of ψ n strictly greater than 1 : if n ≥ 7 and 1 ≤ j ≤ 3 divides n, then

Γ j ∩ V n = ϕ j (t) t root of ψ n ; in particular Γ j ∩ V n is not empty.
Let X be a rational surface and let g be an automorphism of X . The pair (X , g) is said minimal if any birational morphism π : X → X ′ which sends (X , g) on (X ′ , g ′ ), where g ′ is an automorphism of X ′ , is an isomorphism. Let us recall a question of [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]. Let X be a rational surface and let g be an automorphism of X . Assume that (X , g) is minimal. Does there exist a negative power of the class of the canonical divisor K X which admits an holomorphic section ? We know since [START_REF] Hartshorne | Algebraic geometry[END_REF] that the answer is no if we remove the assumption "(X , g) minimal". If g is an automorphism of a rational surface X such that a negative power of K X admits an holomorphic section, g preserves a curve; so Theorem 11.3.4 gives an answer to McMullen's question.

Rotation domains ([22])

Assume that n ≥ 7 (so f is not periodic); if there is a rotation domain, then its rank is 1 or 2 (Theorem 9.4.4). We will see that both happen; let us begin with rotation domains of rank 1.

Theorem 11.5.2. -Let X be a rational surface and let F be an automorphism on X which represents the standard element of the Weyl group W n . There exist

• a surface Y obtained by blowing up Y in a finite number of distinct points π : Y → Y,

• an automorphism g on Y ,

• (a, b) in V n-3 such that (F, X ) is conjugate to (g, Y ) and πg = f a,b,Y π.

Continuous families of automorphisms with positive entropy ([23])

In [START_REF] Bedford | Continuous families of rational surface automorphisms with positive entropy[END_REF] Bedford and Kim introduce the following family:

(11.6.1) f a (y, z) = z, -y + cz + k-2 ∑ j=1 j pair a j y j + 1 y k , a = (a 1 , . . . , a k-2 ) ∈ C k-2 , c ∈ R, k ≥ 2.
Theorem 11.6.1. -Let us consider the family ( f a ) of birational maps given by (11.6.1). Let j, n be two integers relatively prime and such that 1 ≤ j ≤ n. There exists a non-empty subset C n of R such that, for any even k ≥ 2 and for any (c, a j ) in C n × C, the map f a is conjugate to an automorphism of a rational surface S a with entropy log λ n,k where log λ n,k is the largest root of the polynomial

χ n,k = 1 -k n-1 ∑ j=1 x j + x n .
Let us explain briefly the construction of C n . The line ∆ = {x = 0} is invariant by f a . An element of ∆ \ {(0 : 0 : 1)} can be written as (0 : 1 : w) and f (0 : 1 : w) = 0 : 1 : c -1 w . The restriction of f a to ∆ coincides with g(w) = c -1 w . The set of values of c for which g is periodic of period n is 2 cos( jπ/n) 0 < j < n, ( j, n) = 1 .

Let us set w s = g s-1 (c) for 1 ≤ s ≤ n -1, in other words the w i 's encode the orbit of (0 : 1 : 0) under the action of f . The w j satisfy the following properties:

• w j w n-1-j = 1;

• if n is even, then w 1 . . . Proof. -Let us denote by P 1 (resp. P 2 ) the point infinitely near obtained by blowing up Q, Q 1 , U, W and Y (resp. R, S, T, V and X ). By following the sequence of blow-ups we get that f induces an isomorphism between Bl P 1 P 2 and Bl P 2 P 2 , the components being switched as follows

E → F, ∆ ′′ → Ω, K → L, M → H, Λ → ∆ ′′ , G → N.
A conjugate of f has positive entropy on P 2 (C) blown up in ℓ points if ℓ ≥ 10; we thus search an automorphism A of P 2 (C) such that (A f ) 2 A sends P 2 onto P 1 . We remark that f (R) = (0 : 1 : 1) and f 2 (R) = Q then that f 2 ( P 2 ) = P 1 so A = id is such that (A f ) 2 A sends P 2 onto P 1 .

The components are switched as follows

∆ ′′ → f Ω, E → f F, G → f N, K → f L, M → f H, Λ → f ∆ ′′ , f F → f 2 F, f N → f 2 N, f L → f 2 L, f H → f 2 H, f Ω → f 2 Ω, f 2 F → E, f 2 N → G, f 2 L → K, f 2 H → M, f 2 Ω → Λ.
Therefore the matrix of f * is given in the basis

{∆ ′′ , E, G, K, M, Λ, f F, f N, f L, f H, f Ω, f 2 F, f 2 N, f 2 L, f 2 H, f 2 Ω} by                               
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

                              
; the largest root of the characteristic polynomial (X 2 -3X + 1)(X 2 -X + 1)(X + 1) 2 (X 2 + X + 1) 3 (X -1) 4 is 3+ √ 5 2 , i.e. the first dynamical degree of f is 3+ √ 5

2 . Let us remark that the polynomial χ 3,2 introduced in Theorem 11.6.1 is 1 -2X -2X 2 + X 3 whose the largest root is 3+ √ 5

2 .

The considered family of birational maps is not trivial, i.e. parameters are effective. The rank of the map T is equal to k 2 -1 at a = 0. In fact the fixed points of f a can be written (ξ s , ξ s ) where ξ s is a root of (11.6. If S is a compact complex surface carrying an automorphism with positive entropy f , a theorem of Cantat (Theorem 9.3.4) says that

• either f is conjugate to an automorphism of the unique minimal model of S which has to be a torus, a K3 surface or an Enriques surface; • or f is birationally conjugate to a birational map of the complex projective plane ( [START_REF] Cantat | Dynamique des automorphismes des surfaces projectives complexes[END_REF]). We also see that if S is a complex torus, the Fatou set of f is empty. If S is a K3 surface or a quotient of a K3 surface, the existence of a volume form implies that the only possible Fatou components are the rotation domains. McMullen proved the existence of non-algebraic K3 surfaces with rotation domains of rank 2 (see [START_REF] Mcmullen | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF]). What happen if S is a rational nonminimal surface ? The automorphisms with positive entropy on rational non-minimal surfaces can have large rotation domains. In particular, h has no periodic point on U \ {z = 0}.

Let us consider for n, m ≥ 1 the polynomial P n,m (t) = t(t nm -1)(t n -2t n-1 + 1) (t n -1)(t -1) + 1.

If n ≥ 4, m ≥ 1 or if n = 3, m ≥ 2 this polynomial is a Salem polynomial. There exists a rational surface S obtained by blowing up P 2 (C) in a finite number of points π : S → P 2 (C) such that π -1 f π is an automorphism on S.

Moreover, the entropy of f is the largest root of the polynomial P n,m .

Bedford and Kim use the pair ( f k , S) to prove the statements 11.7.1 and 11.7.2.

CHAPTER 12 A "SYSTEMATIC" WAY TO CONSTRUCT AUTOMORPHISMS OF POSITIVE ENTROPY

This section is devoted to a "systematic" construction of examples of rational surfaces with biholomorphisms of positive entropy. The strategy is the following: start with a birational map f of P 2 (C). By the standard factorization theorem for birational maps on surfaces as a composition of blow-ups and blow-downs, there exist two sets of (possibly infinitely near) points P 1 and P 2 in P 2 (C) such that f can be lifted to an automorphism between Bl P 1 P 2 and Bl P 2 P 2 . The data of P 1 and P 2 allows to get automorphisms of rational surfaces in the left PGL 3 (C)-orbit of f : assume that k ∈ N is fixed and let ϕ be an element of PGL 3 (C) such that P 1 , ϕ P 2 , (ϕ f )ϕ P 2 , . . . , (ϕ f ) k-1 ϕ P 2 have all distinct supports in P 2 (C) and (ϕ f ) k ϕ P 2 = P 1 . Then ϕ f can be lifted to an automorphism of P 2 (C) blown up at P 1 , ϕ P 2 , (ϕ f )ϕ P 2 , . . . , (ϕ f ) k-1 ϕ P 2 . Furthermore, if the conditions above are satisfied for a holomorphic family of ϕ, we get a holomorphic family of rational surfaces (whose dimension is at most eight). Therefore, we see that the problem of lifting an element in the PGL 3 (C)-orbit of f to an automorphism is strongly related to the equation u( P 2 ) = P 1 , where u is a germ of biholomorphism of P 2 (C) mapping the support of P 2 to the support of P 1 . In concrete examples, when P 1 and P 2 are known, this equation can actually be solved and involves polynomial equations in the Taylor expansions of u at the various points of the support of P 2 . It is worth pointing out that in the generic case, P 1 and P 2 consist of the same number d of distinct points in the projective plane, and the equation u( P 2 ) = P 1 gives 2d independent conditions on u (which is the maximum possible number if P 1 and P 2 have length d). Conversely, infinitely near points can considerably decrease the number of conditions on u as shown in our examples. This explains why holomorphic families of automorphisms of rational surfaces occur when blow-ups on infinitely near point are made. We illustrate the method on two examples.

We end the chapter with a summary about the current knowledge on automorphisms of rational surfaces with positive entropy.

Then u can be lifted to a germ of biholomorphism between Bl P 2 P 2 and Bl P 1 P 2 if and only if m 0,0 = n 0,0 = n 1,0 = m 3 1,0 + n 2 0,1 = 0, n 2,0 = 3m 0,1 n 0,1 2m 1,0 . The different components are swapped as follows ( §12.1.1)

∆ → ϕ α M, E → ϕ α E, F → ϕ α K, G → ϕ α G, H → ϕ α F, L → ϕ α ∆, ϕ α E → ϕ α Φ 3 ϕ α E, ϕ α F → ϕ α Φ 3 ϕ α F, ϕ α G → ϕ α Φ 3 ϕ α G, ϕ α K → ϕ α Φ 3 ϕ α K, ϕ α M → ϕ α Φ 3 ϕ α M, ϕ α Φ 3 ϕ α E → E, ϕ α Φ 3 ϕ α F → F, ϕ α Φ 3 ϕ α G → G, ϕ α Φ 3 ϕ α K → H, ϕ α Φ 3 ϕ α M → L.
So, in the basis ∆, E, F, G, H, L, ϕ α E, ϕ α F, ϕ α G, ϕ α K, ϕ α M ϕ α Φ 3 ϕ α E, ϕ α Φ 3 ϕ α F, ϕ α Φ 3 ϕ α G, ϕ α Φ 3 ϕ α K, ϕ α Φ 3 ϕ α M , the matrix of (ϕ α Φ 3 ) * is

                              
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

                              
and its characteristic polynomial is (X 2 -3X + 1)(X 2 -X + 1)(X + 1) 2 (X 2 + X + 1) 3 (X -1) 4 .

Thus

λ(ϕ α Φ 3 ) = 3 + √ 5 2 > 1.
Fix a point α 0 in C \ {0, 1}. We can find locally around α 0 a matrix M α depending holomorphically on α such that for all α near α 0 we have

ϕ α Φ 3 = M -1 α ϕ α 0 Φ 3 M α :
if µ is a local holomorphic solution of the equation α = µ n α 0 such that µ 0 = 1 we can take

M α =   1 0 α 0 -α 0 1 0 0 0 1   .

A birational cubic map blowing down one conic and one line

Let ψ denote the following birational map ψ = y

-α 7 (2i √ 3 + 3) 0 0       , α ∈ C * .
The map ϕ α ψ is conjugate to an automorphism of P 2 blown up in 15 points. The first dynamical degree of ϕ α ψ is λ(ϕ α ψ) = 3+ √ 5

2 . The family ϕ α ψ is locally holomorphically trivial.

Proof. -In the basis ∆ ′ , E, F, H, L, N, ϕ α E, ϕ α G, ϕ α K, ϕ α M, ϕ α Ω, ϕ α ψϕ α E, ϕ α ψϕ α G, ϕ α ψϕ α K, ϕ α ψϕ α M, ϕ α ψϕ α Ω

the matrix M of (ϕ α ψ) * is                               
0 0 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -2 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 -3 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 -4 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

                              
.

Its characteristic polynomial is (X -1) 4 (X + 1) 2 (X 2 -X + 1)(X 2 + X + 1) 3 (X 2 -3X + 1).

Hence λ(ϕ α ψ) = 3+ √ 5

2 . Fix a point α 0 in C * . We can find locally around α 0 a matrix M α depending holomorphically on α such that for all α near α 0 , we have ϕ α ψ = M -1 α ϕ α 0 ψM α : take

M α =    1 0 0 0 α α 0 0 0 0 α 2 α 2 0    .

Scholium

There are now two different points of view to construct automorphisms with positive entropy on rational non-minimal surfaces obtained from birational maps of the complex projective plane.

The first one is to start with birational maps of P 2 (C) and to adjust their coefficients such that after a finite number of blow-ups the maps become automorphisms on some rational surfaces S. Then we compute the action of these maps on the Picard group of S and in particular obtain the entropy. There is a systematic way to do explained in [START_REF] Déserti | Special automorphisms of rational surfaces with positive topological entropy[END_REF] and applied to produce examples. Using examples coming from physicists Bedford and Kim

• exhibit continuous families of birational maps conjugate to automorphisms with positive entropy on some rational surfaces; • show that automorphisms with positive entropy on rational non-minimal surfaces obtained from birational maps of P 2 (C) can have large rotation domains and that rotation domains of rank 1 and 2 coexist. Let us also mention the idea of [START_REF] Diller | Cremona transformations, surface automorphisms, and plane cubics[END_REF]: the author begins with a quadratic birational map that fixes some cubic curve and then use the "group law" on the cubic to understand when the indeterminacy and exceptional behavior of the transformation can be eliminated by repeated blowing up.

The second point of view is to construct automorphisms on some rational surfaces prescribing the action of the automorphisms on cohomological groups; this is exactly what does McMullen in [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]: for n ≥ 10, the standard element of the Weyl group W n can be realized by an automorphism f n with positive entropy log(λ n ) of a rational surface S n . This result has been improved in [START_REF] Uehara | Rational surface automorphisms with positive entropy[END_REF]: λ( f ) | f is an automorphism on some rational surface = spectral radius of w ≥ 1 | w ∈ W n , n ≥ 3 .

In [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF] the authors classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible; it can be rephrased in terms of periodic orbits of birational actions of infinite Coxeter groups.

CHAPTER 13

INDEX

Proposition 1 . 1 . 3 (

 113 [START_REF] Hartshorne | Algebraic geometry[END_REF]). -Let S be a smooth projective surface. There exists a unique bilinear symmetric form Div(S) × Div(S) → Z, (C, D) → C • D having the following properties: • if C and D are smooth curves meeting transversally then C • D = #(C ∩ D); • if C and C ′ are linearly equivalent then C • D = C ′ • D. In particular this yields an intersection form Pic(S) × Pic(S) → Z, (C, D) → C • D.

2 .Proposition 1 . 1 . 6 (

 2116 Let us recall that if X is a quasi-projective variety and if x is a point of X, then O p,X is the set of equivalence classes of pairs (U, f ) where U ⊂ X is an open subset x ∈ U and f ∈ C[U]. [115], Chapter V, Proposition 1.4). -If C and D are distinct curves without any common irreducible component on a smooth surface, we have C • D = ∑ p∈C∩D (C • D) p ; in particular C • D ≥ 0.

Lemma 1 . 1 . 7 .

 117 -Let π : Bl p S → S be the blow-up of a point p ∈ S. We have in Pic(Bl p S)π * (C) = C + m p (C)Ewhere C is the strict transform of C and E = π -1 (p).

  p S is induced by the intersection form on S via the following formulas• π * C • π * D = C • D for any C, D ∈ Pic(S); • π * C • E = 0 for any C ∈ Pic(S); • E 2 = E • E = -1; • C 2 = C 2 -1 for any smooth curve C passing through p and where C is the strict transform of C.

Examples 1 . 2 . 2 .

 122 -• For any f in PGL 3 (C) = Aut(P 2 ) we have Ind f = Exc f = / 0.

2 .

 2 -The involution σ : P 2 (C) P 2 (C), (x : y : z) (yz : xz : xy)

  where L is a line, L is called line at infinity; • f induces an isomorphism between S \ D and P 2 (C) \ L. If f : S P 2 (C) is induced by a polynomial automorphism of C 2 it satisfies some properties: Lemma 1.3.5.

Theorem 2 . 1 . 2 (

 212 [129], Jung's Theorem). -The group Aut(C 2 ) is the amalgamated product of A and E along S : Aut(C 2 ) = A * S E.

P 2

 2 (C) P 2 (C), (x : y : z) → ((αx + y)z : βy(x + z) : z(x + z)), α, β ∈ C * , |α| = |β| = 1 so in the affine chart z = 1 f α,β (x, y) = αx + y x + 1 , βy . Theorem 2.4.1 ([70]).

Ω 1

 1 (0.4 + 10 -4 i, 0.4 + 10 -4 i) Ω 2 (0.4 + 10 -4 i, 0.4 + 10 -4 i)

Theorem 3 . 1 . 5 (

 315 [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF], theorem 0.1).

Proposition 3 . 3 . 2 (Example 3 . 3 . 3 .

 332333 [140], p.197). -The morphism NS(S)×Ec(S) → Z(S) induces an isometry between (NS(S), (•, •)) ⊕ (Ec(S), (•, •) E ) and (Z(S), (•, •)). -Let us consider a point p of P 2 (C), Bl p P 2 the blow-up of p and let us denote by E p the exceptional divisor.

Example 3 . 3 . 4 .

 334 -For S = P 2 (C) the Néron-Severi group NS(S) is isomorphic to Z[H] where H is a line. Thus the elements of Z(S) are given by

  k , Bir k and Bir k are invariant under these two actions. Let us denote by O dyn (Q) (resp. O l.r. (Q)) the orbit of Q ∈ Rat k under the action of PGL 3 (C) by dynamic conjugation (resp. under the l.r. action).Examples 4.1.1.

  , αx : βy : z αβ , S 6 × S 6 α, β ∈ C * where S 6 = id, (x : z : y), (z : y : x), (y : x : z), (y : z : x), (z : x : y) . This implies that dim Isot σ = 2. Proposition 4.3.2 ([56]). -The dimension of Σ 2 = O l.r. (ρ) is 13. Proof. -We will compute Isot ρ, i.e. let A and C be two elements of SL 3 (C) such that Aρ = ηρC where η is in C * . Let us recall that Ind ρ = (0 : 1 : 0), (1 : 0 : 0) ; the equality Aρ = ηρC implies that C preserves Ind ρ. But the points of indetermincay of ρ "are not the same", they don't have the same multiplicity so C fixes (0 : 1 : 0) and (1 : 0 : 0); thus C = (ax + bz : cy + dz : ez), where ace = 0. A computation shows that A = (ηγδx + ηβδz : ηα 2 y : ηαδz), C = (γx + βz : δy : αz) with η 3 α 2 δ = αγδ = 1. The dimension of the isotropy group is then 3.

4. 5 .

 5 A geometric description of quadratic birational maps 4.5.1. First definitions and first properties. -In a plane P let us consider a net of conics,

Proposition 4 . 5 . 1 .

 451 -To give a quadratic birational map between two planes is, up to an automorphism, the same as giving a homaloidal net of conics in one of them.Remark 4.5.2.-To a base-point of one of the two nets is associated a line in the other plane which is an exceptional line.4.5.2. Classification of the quadratic birational maps between planes. -We can deducethe classification of the quadratic birational maps between planes from the description of the homaloidal nets Λ of conics in P .

5. 1 .

 1 Birational involutions 5.1.1. Geiser involutions. -Let p 1 , . . . , p 7 be seven points of P 2 (C) in general position.

  has nine base-points (indeed by Bezout's theorem the intersection of two cubics is 3 × 3 = 9 points); so we define by I G (p) the ninth base-point of L p . The involution I G = I G (p 1 , . . . , p 7 ) which sends p to I G (p) is a Geiser involution.

5. 1 . 2 .

 12 Bertini involutions. -Let p 1 , . . . , p 8 be eight points of P 2 (C) in general position. Let us consider the set of sextics S = S(p 1 , . . . , p 8 ) with double points in p 1 , . . . , p 8 . Let m be a point of P 2 (C). The pencil given by the elements of S having a double point in m has a tenth base double point m ′ . The involution which swaps m and m ′ is a Bertini involution

5. 1 . 4 .

 14 Classification of birational involutions. -Definition. -We say that an involution is of de Jonquières type it is birationally conjugate to a de Jonquières involution. We can also speak about involution of Geiser type, resp. Bertini type. Theorem 5.1.1 ([27, 14]). -A non-trivial birational involution of P 2 (C) is either of de Jonquières type, or Bertini type, or Geiser type.More precisely Bayle and Beauville obtained the following statement.Theorem 5.1.2 ([14]). -The map which associates to a birational involution of P 2 its normalized fixed curve establishes a one-to-one correspondence between:• conjugacy classes of de Jonquières involutions of degree d and isomorphism classes of hyperelliptic curves of genus d -2 (d ≥ 3);• conjugacy classes of Geiser involutions and isomorphism classes of non-hyperelliptic curves of genus 3; • conjugacy classes of Bertini involutions and isomorphism classes of non-hyperellipticcurves of genus 4 whose canonical model lies on a singular quadric. The de Jonquières involutions of degree 2 form one conjugacy class.5.2. Birational involutionsand foliations 5.2.1. Foliations: first definitions. -A holomorphic foliation F of codimension 1 and degree ν on P 2 (C) is given by a 1-form ω = u(x, y, z)dx + v(x, y, z)dy + w(x, y, z)dz where u, v and w are homogeneous polynomials of degree ν + 1 without common component and satisfying the Euler identity xu + vy + wz = 0. The singular locus Sing F of F is the projectivization of the singular locus of ω Sing

x 2 y

 2 + ax 2 + bxy + cx + ey ∂ ∂x + xy 2 + Ay 2 + Bxy +Cx + Ey ∂ ∂y with 1 + a + b + c + e = 1 + A + B +C + E = 0. Then the construction detailed in 5.1.1 allows us to give an explicit expression for the involution I F . Remark 5.2.3.

  that C contains exactly k ≥ 2 distinct fibers of π, and C has at most dim Pic(S)+ k -1 irreducible components. Several mathematicians have been interested in and are still interested in the algebraic properties of the diffeomorphisms groups of manifolds. Let us for example mention the following result. Let M and N be two smooth manifolds without boundary and let Diff p (M) denote the group of C p -diffeomorphisms of M. In 1982 Filipkiewicz proves that if Diff p (M) and

  be the affine group of the complex line. Theorem 6.2.1. -Let φ be an automorphism of Aff(C). Then there exist τ an automorphism of the field C and ψ an element of Aff(C) such that φ

6. 3 . 6 . 3 . 1 .

 3631 The group of polynomial automorphisms of the plane Description of the automorphisms group of Aut(C 2 ). -Theorem 6.3.1 ([70]). -Let φ be an automorphism of Aut(C 2 ). There exist ψ in Aut(C 2 ) and an automorphism τ of the field C such that

6. 3 . 2 .

 32 Corollaries. -Corollary 6.3.10 ([70]). -An automorphism φ of Aut(C 2 ) is inner if and only if for any f in Aut(C 2 ) we have jac φ( f ) = jac f where jac f is the determinant of the jacobian matrix of f .Proof. -There exists an automorphism τ of the field C and a polynomial automorphism ψ such that for any polynomial automorphism f we have φ( f ) = τ(ψ -1 f ψ). Hence jac φ( f ) = jac τ( f ) = τ(jac f ), so jac φ( f ) = jac f for any f ifand only if τ is trivial. Corollary 6.3.11. -An isomorphism of the semi-group End(C 2 ) in itself is inner up to the action of an automorphism of the field C.

6. 4 .

 4 The Cremona group 6.4.1. Description of the automorphisms group of Bir(P 2 ). -Theorem 6.4.1 ([71]). -Any automorphism of the Cremona group is the composition of an inner automorphism and an automorphism of the field C.

Proposition 7 . 1 . 4 (

 714 [START_REF] Cerveau | Transformations birationnelles de petit degré[END_REF]). -The Cremona group cannot be embedded into GL n (k) where k is a field of characteristic zero.First let us recall a result of linear algebra due to Birkhoff. Lemma 7.1.5 ([28]). -Let k be a field of characteristic zero and let A, B, C be three elements of GL n (k) such that [A, B] = C, [A,C] = [B,C] = id and C p = id with p prime. Then p ≤ n. Proof of Proposition 7.1.4. -Assume that there exists an embedding ς of the Cremona group into GL n (k). For all prime p let us consider in the affine chart z = 1 the group exp -2iπ p x, y , (x, xy), x, exp 2iπ p y .

a 1 b 2 -

 2 a 2 b 1 = 0 and the group of elementary automorphisms by E = (x, y) → (αx + P(y), βy + γ) α, β ∈ C * , γ ∈ C, P ∈ C[y] . Theorem 7.2.2 ([129, 136]). -The group Aut(C 2 ) is the amalgamated product of A and E along A ∩ E.

7. 4 .

 4 Quasi-rigidity of SL 3 (Z) 7.4.1. Dynamic of the image of an Heisenberg group. -Definition.

7. 4 . 3 .Lemma 7 . 4 . 3 (

 43743 Invariant fibration. -[START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF]). -Let Γ be a finitely generated group with the Kazhdan's property (T). Let ρ be a morphism from Γ to PGL 2 (C(y)) (resp. PGL 2 (C)). Then the image of ρ is finite.Proof. -Let us denote by γ i the generators of Γ and let a i (y) b i (y) c i (y) d i (y) be their image by ρ.

7. 4 . 5 .

 45 Proof of Theorem 7.1.1 1). -According to Proposition 7.4.4 any standard generator of SL 3 (Z) is virtually isotopic to the identity. The maps ρ(e n 12 ), ρ(e n 13 ) and ρ(e n 23 ) are, for some integer n, conjugate to automorphisms of a minimal surface S (Proposition 7.3.3); we don't have to consider the case S = P 1 (C) × P 1 (C) (Lemma 7.3.4). We finally obtain that ρ(Γ 3 (n 2 )) is, up to conjugation, a subgroup of PGL 3 (C) (Lemmas 7.4.5 and 7.4.6).

  where the closure is the Zariski closure) which is impossible. So f belongs to PGL 3 (C) and ρ(SL 3 (Z)) is contained in PGL 3 (C).

7. 4 . 6 .

 46 Proof of Theorem 7.1.1 2). -Theorem 7.4.7 ([69]). -Each morphism from a subgroup of finite index of SL 4 (Z) in the Cremona group is of finite image.

  3 (C) is invariant by φ if and only if φ(H) = H. Let us set : f b (x, y) = φ(x + b, y), g a (x, y) = φ(x + ay, y) and h c (x, y) = φ(x, y + c). The birational map f b (resp. h c ) commutes with (x + 1, y) and (x, y + 1) so f b (resp. h c ) can be written as (x + η(b), y + ζ(b)) (resp. (x + γ(c), y + β(c))) where η and ζ (resp. γ and β) are two additive morphisms; as g a commute with (x + y, y) and (x + 1, y) we have: g a = (x + A a (y), y). The equality (x + ay, y)(x, y + c)(x + ay, y) -1 (x, y + c) -1 = (x + ac, y) implies that, for any complex numbers a and c, we have: g a h c = f ac h c g a . Therefore f b = (x + η(b), y), g a = (x + µ(a)y + δ(a), y) and µ(a)β(c) = η(ac). In particular φ(H) is contained in H. Since µ(a)β(c) = η(ac) we have η = µ = β (because η(1) = µ(1) = β(1) = 1); let us note that this equality also implies that η is multiplicative.

Corollary 7 . 5 . 2 (

 752 [START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF]). -Let φ be an automorphism of the Cremona group. If φ |SL 3 (Z) is the involution u → t (u -1 ) then φ |PGL 3 (C) also.Proof. -Let us denote by ψ the composition of φ |SL 3 (Z) with the restriction C of the involution u → t (u -1 ) to SL 3 (Z). The morphism ψ can be extended to a morphism ψ from PGL 3 (C) into Bir(P 2 ) by ψ = φ |PGL 3 (C) • C. The kernel of ψ contains SL 3 (Z) ; as the group PGL 3 (C) is simple, ψ is trivial.Lemma 7.5.3 ([69]). -Let φ be an automorphism of the Cremona group such that φ |PGL 3 (C) is trivial or is the involution u → t (u -1 ). There exist a, b two nonzero complex numbers such that φ(σ) = ax , b y where σ is the involution 1 x , 1 y .Proof. -Assume that φ |PGL 3 (C) is trivial. The map φ(σ) can be written F x , G y where F

Lemma 8 . 3 . 5 (

 835 is a point. We can then contract the divisor C by a birational map π : S → S ′ and the map π f π -1 is still algebraically stable. By induction we can suppose that there is no such curve C in C .If C is empty the second assertion of the statement is satisfied.Assume that C is not empty. If C belongs to C and f m (C) does not belong to C then f m (C)is a point which does not belong to C and f m is holomorphic along C: contradiction. So for any curve C of C , f m (C), m ≥ 0, belongs to C . We can hence assume that C is invariant by any f m with m ≥ 0. The set C is invariant by f n for any n in Z so f -n (p), n > 0, is a sequence of points of C . Let C be an irreducible component of C passing through p. Since C contains curves coming from p there exists an integer k such that f -k is holomorphic along C and contracts C onto p. Therefore the negative orbit of p passes periodically through p and cannot be infinite: contradiction. [62, 78]).

Corollary 8 . 5 . 4 .

 854 -The centralizer of a non trivial element (x + b(y), y) is thus conjugate to dJ a ⋊ PGL 2 (C).

8. 5 . 2 .

 52 Maps of dJ m . -If a ∈ C(y) is non constant, we denote stab(a) the finite subgroup of PGL 2 (C) defined by stab(a) = ν ∈ PGL 2 (C) a(ν(y)) = a(y) .

Example 8 . 5 . 5 .

 855 -If k is an integer and if a(y) = y k , then stab(a) = ω k y ω k = 1 & Stab(a) = 1 y , ω k y ω k = 1 . Let us denote by stab(a) the linear group stab(a) = (x, ν(y)) ν ∈ stab(a) . By definition the group Stab(a) is generated by stab(a) and the elements 1 x , ν(y) , with ν in Stab(a) \ stab(a). Proposition 8.5.6 ([55]). -Let f = (a(y)x, y) be a non periodic element of dJ m . If f is an elliptic birational map, i.e. a is a constant, the centralizer of f is b(y)x, ν(y) b ∈ C(y) * , ν ∈ PGL 2 (C) .

Lemma 8 . 5 . 9 (

 859 5.1 the map ψ preserves the fibration y = cte; the curve C is transverse to the fibration so ψ cannot contract C . Therefore ψ belongs to dJ and preserves C . As soon as deg F ≥ 3 the assumptions of Proposition 8.5.8 are satisfied; so assume that deg F ≤ 2. The case deg F = 2 can be deduced from the case deg F = 1. Indeed let us consider f = c(y)x+y x+c(y) , y . Let us set ϕ = x cy+d , ay+b cy+d . We can check that ϕ -1 f ϕ can be written c(y)x + (ay + b)(cy + d) x + c(y) , y , and this allows to obtain all polynomials of degree 2 with simple roots. If deg F = 1, i.e. F(y) = ay + b, we have, up to conjugation by x, y-b a , F(y) = y. [55]). -Let f be a map of the form c(y)x+y x+c(y) , y with c in C(y) * . If ψ is an element of Cent( f , Bir(P 2 )), then π 2 (ψ) is either α y , α ∈ C * , or ξy, ξ root of unity; moreover, π 2 (ψ) belongs to stab 4c(y) 2 c(y) 2 -y . For α in C * we denote by D ∞ (α) the infinite dihedral group D ∞ (α) = α y , ωy ω root of unity ; let us remark that any D ∞ (α) is conjugate to D ∞ (1). If c is a non constant element of C(y) * , then S(c; α) is the finite subgroup of PGL 2 (C) given by

Proposition 8 . 5 . 10 (

 8510 [55]). -Let us consider f = c(y)x+y x+c(y) , y with c in C(y) * , c non constant. There exists α in C * such that Cent( f , Bir(P 2 )) = dJ y ⋊ S(c; α). Propositions 8.5.3, 8.5.6, 8.5.8 and 8.5.10 imply Theorem 8.5.2.

Example 9 . 2 . 3 .

 923 -Let us consider the involution given by ρ : P 2 (C) P 2 (C), (x : y : z) (xy : z 2 : yz). We can show that M ρ = M σ . Example 9.2.4. -Consider the birational map τ : P 2 (C) P 2 (C), (x : y : z) (x 2 : xy : y 2xz).

9. 3 . 3 .

 33 Case of rational surfaces. -Let us recall the following statement due to Nagata. Proposition 9.3.7 ([147], Theorem 5).

  0. According to Theorem 9.2.1 dim L ≤ 1. But since λ( f ) > 1, L is spanned by a unique nontrivial eigenvector. If v has eigenvalue µ, then v has eigenvalue µ so we must have µ = µ = λ( f ).

9. 4 .

 4 Linearization and Fatou sets 9.4.1. Linearization. -Let us recall some facts about linearization of germs of holomorphic diffeomorphism in dimension 1 when the modulus of the multipliers is 1. Let us consider (9.4.1) f (z) = αz + a 2 z 2 + a 3 z 3 + . . . , α = e 2iπθ , θ ∈ R \ Q We are looking for ψ(z) = z + b 2 z 2 + . . . such that f ψ(z) = ψ(αz). Since we can formally compute the coefficients b

Theorem 9 . 4 . 2 ( 2 . 9 . 4 . 3 .

 9422943 Siegel). -If there exist two constants c and M strictly positive such that |α q -α| ≥ c q M then any germ f (z) = αz + a 2 z 2 + . . . is analytically linearizable. Let us now deal with the case of two variables. Let us consider f (x, y) = (αx, βy) + h.o.t. with α, β of modulus 1 but not root of unity. The pair (α, β) is resonant if there exists a relation of the form α = α a β b or β = α a β b where a, b are some positive integers such that a + b ≥ 2. A resonant monomial is a monomial of the form x a y b . We say that α and β are multiplicatively independent if the unique solution of α a β b = 1 with a, b in Z is (0, 0). The numbers α and β are simultaneously diophantine if there exist two positive constants c and M such that min |α a β b -α|, |α a β b -β| ≥ c |a + b| M ∀a, b ∈ N, a + b ≥ Theorem -If α and β are simultaneously diophantine then f is linearizable. If α and β are algebraic and multiplicatively independent then they are simultaneously diophantine.

Proposition 9 . 4 . 5 .

 945 -The Fatou set of a Hénon map is C 2 \ J + .

Theorem 9 . 4 . 6 (Proposition 9 . 4 . 7 . 9 . 4 . 3 .

 946947943 [START_REF] Bedford | Polynomial diffeomorphisms of C 2 . II. Stable manifolds and recurrence[END_REF]). -Let f be a Hénon map with jacobian < 1 and let Ω be a recurrent Fatou component. Then Ω is• either the basin of a sink;• or the basin of a Siegel disk;• or a Herman ring.Under some assumptions the Fatou component of a Hénon automorphisms are recurrent. -The Fatou component of a Hénon map which preserves the volume are periodic and recurrent. Fatou sets of automorphisms with positive entropy on torus, (quotients of) K3, rational surfaces.

  and has been much developed since then ([86,

n then e 0

 0 , . . . , e n is a geometric basis.Consider α in H 2 (S, Z) such that α•α = -2, then R α (x) = x + (x • α)α sends α on -α and R α fixes each element of α ⊥ ; in other words R α is a reflection in the direction α.Consider the vectors given by α 0 = e 0e 1e 2e 3 , α j = e j+1e j , 1 ≤ j ≤ n -1.

  let us denote it by π n . For n ≥ 4 we define the standard element w of W n by w = π n κ 123 . It satisfies w(e 0 ) = 2e 0e 2e 3e 4 , w(e 1 ) = e 0e 3e 4 , w(e 2 ) = e 0e 2e 4 , w(e 3 ) = e 0e 2e 3 , w(e j ) = e j+1 , 4 ≤ j ≤ n -2, w(e n-1 ) = e 1 .

1 .

 1 Marked cubic curves. -A cubic curve C ⊂ P 2 (C) is a reduced curve of degree 3. It can be singular or reducible; let us denote by C * its smooth part. Let us recall some properties

η

  : Z 1,n → Pic(C ) such that • the sections of the line bundle η(e 0 ) provide an embedding of C into P 2 (C); • there exist distinct base-points p i on C * for which η(e i ) = [p i ] for any i = 2, . . . , n. The base-points p i are uniquely determined by η since C * can be embedded into Pic(C ). Conversely a cubic curve C which embeds into P 2 (C) and a collection of distinct points on C * determine a marking of C . Remark 10.3.1. -Different markings of C can yield different projective embeddings C ֒→ P 2 (C) but all these embeddings are equivalent under the action of Aut(C ).

Corollary 10 . 3 . 3 ( 10 . 3 . 2 .

 10331032 [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]). -An irreducible marked cubic curve (C , η) is determined, up to isomorphism, by η 0 : L n → Pic 0 (C ). Marked blow-ups. -A marked blow-up (S, Φ) is the data of a smooth projective surface S and an isomorphism Φ :Z 1,n → H 2 (S, Z) such that • Φ sends the Minkowski inner product (x • x) = x 2 = x 2 0x 2 1 -. . .x 2n on the intersection pairing on H 2 (S, Z); • there exists a birational morphism π : S → P 2 (C) presenting S as the blow-up of P 2 (C) in n distinct base-points p 1 , . . . , p n ; • Φ(e 0 ) = [H] and Φ(e i ) = [E i ] for any i = 1, . . ., n where H is the pre-image of a generic line in P 2 (C) and E i the divisor obtained by blowing up p i .

Theorem 10 . 3 . 5 (

 1035 [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]). -Let (S, Φ) be a marked blow-up. If three of the base-points are colinear, (S, Φ) has a geometric nodal root.

10. 3 . 3 . 2 .

 332 From surfaces to cubic curves. -Let us consider a marked pair (S, Φ,Y ). Let π be the projection of S to P 2 (C) compatible with Φ. The equality (10.3.1) implies that C = π(Y ) is a cubic curve through any base-point p i with multiplicity 1. Moreover, E i •Y = 1 implies that π : Y → C is an isomorphism. The identification of H 2 (S, Z) and Pic(S) allows us to obtain the natural marking

10. 3 . 3 . 3 .

 333 From cubic curves to surfaces. -Conversely let us consider a marked cubic curve (C , η). Then we have base-points p i ∈ C determined by (η(e i )) 1≤i≤n and an embedding C ⊂ P 2 (C) determined by η(e 0 ). Let (S, Φ) be the marked blow-up with base-points p i and Y ⊂ S the strict transform of C . Hence we obtain a marked pair (S, Φ,Y ) called blow-up of (C , η) and denoted Bl(C , η).

w(e 1 ) 1 -e 2 .

 112 = e 0e 2e 3 , w(e 2 ) = e 0e 1e 3 , w(e 3 ) = e 0e More generally we have the following statement. Theorem 10.5.1 ([144]). -Let us denote by p i+4 the i-th iterate f i (p 4 ) of p 4 . The realization of the standard Coxeter element of W n corresponds to the pairs (a, b) of C 2 such that p i ∈ (p 1 p 2 ) ∪ (p 2 p 3 ) ∪ (p 3 p 1 ), p n+1 = p 1 .

2 -e 3 -e 4 . 1 .

 2341 3 and p 4 thus w(e 0 ) = 2e 0e The point p 1 is blown up to the line through p 3 and p 4 so w(e 1 ) = e 0e 3e 4 ; similarly we obtain w(e 2 ) = e 0e 2e 4 , w(e 3 ) = e 0e 2e 3 , w(e i ) = e i+1 for 4 ≤ i < n , w(e n ) = e Conversely if an automorphism F : S → S realizes the standard Coxeter element w = π n κ 123 , we can normalize the base-points such that p 1 , p 2 , p 3 = (0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0) ; the birational map f : P 2 (C) P 2 (C) covered by F is a composition of the standard Cremona involution and an automorphism sending (p 1 , p 2 ) onto (p 2 , p 3 ). Such a map f has the form in the affine chart z = 1 f (x, y) = (a ′ , b ′ ) + (Ay, By/x) so up to conjugacy by (Bx, By/A), we have f (x, y) = (a, b) + (y, y/x).

  f a,b (x, y, z) = x(bx + y) : z(bx + y) : x(ax + z) , or in the affine chart x = 1 f a,b (y, z) = z, a + z b + y . We note that Ind f a,b = {p 1 , p 2 , p * } and Exc f a,b = Σ 0 ∪ Σ β ∪ Σ γ with p 1 = (0 : 1 : 0), p 2 = (0 : 0 : 1), p * = (1 : -b : -a),

  The characteristic polynomial of M f a,b,Y is 1+t -t 3 . Let us explain all the information contained in M f a,b,Y . Let L be a line and L its class in Pic(Y ).If L does not intersect neither E 1 , nor E 2 , then L = H. As f * a,b,Y H = 2H -E 1 -E 2 theimage of L by f a,b,Y is a conic which intersects E 1 and E 2 with multiplicity 1. If L contains p * , then f a,b,Y (L) is the union of Σ C and a second line. Assume that p * does not belong to L ∪ f a,b,Y

Theorem 11 . 1 . 1 .

 1111 3 a,b,Y (L) is a cubic which intersects E 1 and E 2 with multiplicity 1. If p * does not belong toL ∪ f a,b,Y (L) ∪ . . . ∪ f n-1a,b,Y (L), the iterates of f a,b,Y are holomorphic on the neighborhood of L and( f * a,b,Y ) n (H) = f n a,b,Y L.The parameters a and b are said generic if p * does not belong to -Assume that a and b are generic; f a,b,Y is algebraically stable and λ( f a,b ) ∼ 1.324 is the largest eigenvalue of the characteristic polynomial t 3t -1.

  More precisely we have the following description.• If (a, b) = ϕ 1 (t), then Γ 1 = (P t,a,b = 0) is a irreducible cuspidal cubic.The map f a,b has two fixed points, one of them is the singular point of C . • If (a, b) = ϕ 2 (t), then Γ 2 = (P t,a,b = 0) is the union of a conic and a tangent line to it. The map f a,b has two fixed points. • If (a, b) = ϕ 3 (t), then Γ 3 = (P t,a,b = 0) is the union of three concurrent lines; f a,b has two fixed points, one of them is the intersection of the three components of C .

Theorem 11 . 3 . 4 .

 1134 -There exists a surface S and an automorphism with positive entropy f a,b on S such that (S, f a,b ) is minimal and such that f a,b has no invariant curve.

Theorem 11 . 6 . 2 .

 1162 w n-2 = 1; • if n is odd, let us set w * (c) = w (n-1)/2 then w 1 . . . w n-2 = w * .Let us give details about the case n = 3, k = 2, then C 3 = {-1, 1}. Assume that c = 1; in other wordsf a = f = xz 2 : z 3 : x 3 + z 3yz 2 .The map f contracts only one line ∆ ′′ = {z = 0} onto the point R = (0 : 0 : 1) and blows up exactly one point, Q = (0 : 1 : 0). Let us describe the sequence of blow-ups that allows us to "solve indeterminacy":• first blow-up. First of all let us blow up Q in the domain and R in the range. Let us denote by E (resp. F) the exceptional divisor obtained by blowing upQ (resp. R). One can check that E is sent onto F, ∆ ′′ 1 is contracted onto S = (0, 0) (a 1 ,b 1 ) and Q 1 = (0, 0) (u 1 ,v 1) is a point of indeterminacy;• second blow-up. Let us then blow up Q 1 in the domain and S in the range; let G, resp.H be the exceptional divisors. One can verify that the exceptional divisor G is contracted onto T = (0, 0) (c 2 ,d 2 ) , ∆ ′′ 2 onto T and U = (0, 0) (r 2 ,s 2 ) is a point of indeterminacy; • third blow-up. Let us continue by blowing up U in the domain and T in the range, where K and L denote the associated exceptional divisors. One can check that W = (1, 0) (r 3 ,s 3 ) is a point of indeterminacy, K is sent onto L and G 1 is contracted on V = (1, 0) (c 3 ,d 3 ) and ∆ ′′3 on V ; • fourth blow-up. Let us blow up W in the domain and V in the range, let M and N be the associated exceptional divisors. Then ∆ ′′ 4 is contracted on X = (0, 0) (c 4 ,d 4 ) , Y = (0, 0) (r 4 ,s 4 ) is a point of indeterminacy, G 1 is sent onto N and M onto H; • fifth blow-up. Finally let us blow up Y in the domain and X in the range, where Λ, Ω are the associated exceptional divisors. So ∆ ′′ 5 is sent onto Ω and Λ onto ∆ ′′ 5 . -The map f = xz 2 : z 3 : x 3 + z 3yz 2 is conjugate to an automorphism of P 2 (C) blown up in 15 points.The first dynamical degree of f is 3+

Theorem 11 . 6 . 3 .

 1163 -Let us hold the parameter c ∈ C n fixed. The family of maps ( f a ) defined by (11.6.1) induces a family of dynamical systems of dimension k/2 -1. In other words thereis a neighborhood U of 0 in C k/2-1 such that if a = (a 0 , a 2 , . . . , a k-2 ), b = (b 0 , b 2 , . . . , b k-2 ) are in U then f a and f b are not smoothly conjugate.Idea of the proof. -Such a map f a has k+1 fixed points p 1 , . . . , p k+1 . Let us set a = (a 1 , . . . , a k-2 ). Bedford and Kim show that the eigenvalues of D f a at p j (a) depend on a; it follows that the family varies non trivially with a. More precisely they prove that the trace of D f a varies in a non-trivial way. Let τ j (a) denote the trace of D f a at p j (a) and let us consider the map T defined by a → T (a) = (τ 1 (a), . . . , τ k+1 (a)).

∂ξ ∂a ℓ a=0 = 1 ( 2 If we let ξ j range over k 2 -1 distinct choices of roots 1 ( 2 2 - 1 ) × ( k 2 - 1 ) 1 . 2 - 1 2 - 1 → 11 . 7 .

 12212212112121117 2) ξ = (c -1)ξ + When a is zero, we have for any fixed point ξ k+1 = 1 2-c . By differentiating (11.6.2) with respect to a ℓ we get for a = 0 the equality 2 -c)(k + 1)ξ ℓ . The trace of D f a (y,z) is given byτ = c -For y = ξ a we have ∂τ(ξ a ) ∂a ℓ a=0 = -ℓ y ℓ+1 + k(k + 1) -c) k+1, the matrix essentially is a ( k Vandermondian and so of rank k 2 -There exists a neighborhood U of 0 in C k such that, for any a, b in U with a = b, the map f a is not diffeomorphic to f b . In fact the map C k C k+1 , a → T (a) is locally injective in a neighborhood of 0. Moreover, for a = 0, the fixed points p 1 , . . . , p k+1 , and so the valuesτ 1 (0), . . . , τ k+1 (0), are distinct. Thus C k 2 -1 ∋ a → {τ 1 (a), . . . , τ k+1 (a)} is locally injective in 0.So if U is a sufficiently small neighborhood of 0 and if a and b are two distinct elements of U, the sets of multipliers at the fixed points are not the same; it follows that f a and f b are not smoothly conjugate.Let f a be a map which satisfies Theorem 11.6.1. Bedford and Kim show that in all the cases under their consideration the representation Aut(S a ) → GL(Pic(S a )), φ → φ * is at most ((k 2 -1) : 1); moreover if a k-2 is non zero, it is faithful. When n = 2, the image of Aut(S a ) → GL(Pic(S a )), φ → φ * coincides with elements of GL(Pic(S a )) that are isometries with respect to the intersection product, and which preserve the canonical class of S a as well as the semigroup of effective divisors; this subgroup is the infinite dihedral group with generators f a * and ι * where ι denotes the reflection (x, y) → (y, x). They deduce from it that, always for n = 2, the surfaces S a are, in general, not biholomorphically equivalent. Theorem 11.6.4. -Assume that n = 2 and that k ≥ 4 is even. Let a be in C k/2-1 and c be in C 2 . There exists a neighborhood U of 0 in C k/2-1 such that if a, b are two distinct points of U and if a k-1 is nonzero, then S a is not biholomorphically equivalent to S b . Dynamics of automorphisms with positive entropy: rotation domains ([24])

Theorem 11 . 7 . 1 .

 1171 -There exists a rational surface S carrying an automorphism with positive entropy h and a rotation domain U. Moreover, U is a union of invariant Siegel disks, h acting as an irrational rotation on any of these disks.The linearization is a very good tool to prove the existence of rotation domains but it is a local technique. In order to understand the global nature of the Fatou component U, Bedford and Kim introduce a global model and get the following statement. Theorem 11.7.2. -There exist a surface L obtained by blowing up P 2 (C) in a finite number of points, an automorphism L on L, a domain Ω of L and a biholomorphic conjugacy Φ : U → Ω which sends (h, U) onto (L, L).

Theorem 11 . 7 . 3 . 2 √

 11732 -Let us consider the birational map f given in the affine chart z = 1 by f (x, y) = y, -δx + cy + 1 ywhere δ is a root of P n,m which is not a root of unity and c = δ cos( jπ/n) with 1 ≤ j ≤ n -1, ( j, n) = 1.

12. 1 . 3 .Theorem 12 . 1 . 3 . 2 > 1 .

 13121321 Examples. -In this section, we will use the two above steps to produce explicit examples of automorphisms of rational surfaces obtained from birational maps in the PGL 3 (C)orbit of Φ 3 . As we have to blow up P 2 (C) at least ten times to have non zero-entropy, we want to find an automorphism ϕ of P 2 (C) such that (12.1.1)(ϕΦ 3 ) k ϕ( P 2 ) = P 1 with (k + 1)(2n -1) ≥ 10(ϕΦ 3 ) i ϕ(P) = P for 0 ≤ i ≤ k -1First of all let us introduce the following definition.Definition. -Let U be an open subset of C n and let ϕ : U → PGL 3 (C) be a holomorphic map. If f is a birational map of the projective plane, we say that the family of birational maps(ϕ α 1 ,...,α n f ) (α 1 ,...,α n )∈U is holomorphically trivial if for every α 0 = (α 0 1 , . . . , α 0 n ) in U there exists a holomorphic map from a neighborhood U α 0 of α 0 to PGL 3 (C) such that • M α 0 1 ,...,α 0 n = Id, • ∀ (α 1 , . . . , α n ) ∈ U α 0 , ϕ α 1 ,...,α n f = M α 1 ,...,α n (ϕ α 0 1 ,...,α 0 n f )M -1 α 1 ,...,α n .-Let ϕ α be the automorphism of the complex projective plane given byϕ α =   α 2(1 -α) (2 + αα 2 ) ∈ C \ {0, 1}.The map ϕ α Φ 3 is conjugate to an automorphism of P 2 (C) blown up in 15 points. The first dynamical degree of ϕ α Φ 3 is 3+ √ 5 The family ϕ α Φ 3 is holomorphically trivial.Proof. -The first assertion is given by Proposition 12.1.2.

Theorem 12 . 2 . 3 .

 1223 2 z : x(xz + y 2 ) : y(xz + y 2 ) ; it blows up two points and blows down two curves, more precisely Ind ψ = R = (1 : 0 : 0), P = (0 : 0 : 1) , Exc ψ = C = xz + y 2 = 0 ∪ ∆ ′ = y = 0 . -Assume that ψ = y 2 z : x(xz + y 2 ) : y(xz + y 2 ) and that

  2

: Upward induction. - Lemma 2.2.1. -

  Let n ≥ 1 and let h : F n P 2 (C) be a birational map induced by a polynomial automorphism of C 2 . Suppose that h has only one point of indeterminacy p such that

2.2.4. Third step: Downward induction. - Lemma 2.2.2. -

  Let n ≥ 2 and let h : F n P 2 (C) be a birational map induced by a polynomial automorphism of C 2 . Assume that h has only one point of indeterminacy p, and that p belongs to f n but not to s n . Let ϕ : F n F n-1 be the birational map which is the blow-up of p composed with the contraction of the strict transform of f n . Let us consider the birational map h

Remark 3.1.2. -Let

  1. The positive orbit of p 1 under the action of f is the set { f n (p 1 ) | n ≥ 0}. f be a Cremona transformation. The map f is not algebraically stable if and only if there exists an integer k such thatdeg f k < (deg f ) k . Ind σ -1 ; moreover deg σ 2 = 1 and (deg σ) 2 = 4.

	So if f is algebraically stable, then λ( f ) = deg f .
	Examples 3.1.3. -• The involution σ : P 2 (C) • An automorphism of P 2 (C) is algebraically stable. P 2 (C), (x : y : z) → (yz : xz : xy) is not algebraically stable: Ind σ -1 = Examples 3.1.4. -Let A be an automorphism of the complex projective plane and let σ be
	the birational map given by
	σ : P 2 (C)	P 2 (C),	(x : y : z)	(yz : xz : xy).
	Assume that the coefficients of A are positive real numbers. The map Aσ is algebraically stable
	([56]).
	Let A be an automorphism of the complex projective plane and let ρ be the birational map
	given by
	ρ : P 2 (C)	P 2 (C),	(x : y : z)	(xy : z 2 : yz).
	Assume that the coefficients of A are positive real numbers. We can verify that Aρ is alge-
	braically stable. The same holds with
	τ : P 2 (C)	P 2 (C),	(x : y : z)	(x 2 : xy : y 2 -xz).

Theorem 3.2.1 ([104, 77, 34]). -

			Let f be an element of Bir(P 2 ); up to birational conjuga-
	tion, exactly one of the following holds.
	• The sequence |( f n ) * | is bounded, the map f is conjugate either to (αx : βy : z) or to (αx : y + z : z);
	• the sequence |( f n ) * | grows linearly, and f preserves a rational fibration. In this case f cannot be conjugate to an automorphism of a projective surface;
	• the sequence |( f n ) * | grows quadratically, and f is conjugate to an automorphism pre-serving an elliptic fibration.
	• the sequence |( f n ) * | grows exponentially; the spectrum of f * outside the unit disk consists of the single simple eigenvalue λ( f ), the eigenspace associated to λ( f ) is generated by
	a nef class θ + ∈ H 1,1 (P 2 (C)). Moreover f is conjugate to an automorphism if and only if (θ + , θ + ) = 0.
	In the second and third cases, the invariant fibration is unique.
	Definition. -Let f be an element of Bir(P 2 ).
	• If deg f k • if deg f k an Halphen twist); k∈N is bounded, f is elliptic; k∈N grows linearly (resp. quadratically), then f is a de Jonquières twist (resp.
	• if deg f k	k∈N grows exponentially, f is hyperbolic.
	Remark 3.2.2. -If deg f k	k∈N grows linearly (resp.

. Applications 3.4.1. Tits alternative. -Linear groups satisfy Tits alternative. Theorem 3.4.1 ([179]). -

  and only if f is a parabolic map: there exists a non zero ℓ in Z Let k be a field of characteristic zero. Let Γ be a finitely generated subgroup of GL n (k). Then• either Γ contains a non abelian, free group;

	belongs to Z + \ Rℓ -then lim n→+∞	1 λ	n	f n * (p) = v ∈ Rℓ + \ {0};
	there is a similar property for ℓ -and f -1 .
	3.4

* (S) such that f (ℓ) = ℓ. Moreover (ℓ, ℓ) = 0 and Rℓ is the unique invariant line by f * which intersects Z + (S). If p belongs to Z * (S), then lim n→+∞ ( f * ) n (Rp) = Rℓ.

• f * is hyperbolic if and only if f is a hyperbolic map: there exists a real number λ > 1 and two elements ℓ + and ℓ -in Z(S) such that f * (ℓ + ) = λℓ + and f * (ℓ -) = (1/λ)ℓ -. If p

Lemma 3.4.3. -

  Let G be a group acting on a set X. Let us consider Γ 1 and Γ 2 two subgroups of G, and set Γ = Γ 1 , Γ 2 .

		• G contains two Hénon automorphisms with distinct geodesics, G thus contains a free subgroup on two generators.
		One of the common ingredients of the proofs of Theorems 3.4.1, 3.4.2 3.4.6 is the following
		statement, a criterion used by Klein to construct free products.
		Assume that
	bizarre cette hypothèse	• Γ 1 (resp. Γ 2 ) has only 3 (resp. 2) elements,
		3. Let G be a group; let us set G

  Γ 1 and Γ 2 .

	Example 3.4.4. -The matrices	1 2 0 1	and	1 0 2 1	generate a free subgroup of rank 2
	in SL 2 (Z). Indeed let us set

  Is the Theorem 4.2.2 avalaible in degree strictly larger than 2 ? No, as soon as the degree is 3 we can exhibit elements Q contracting det jac Q but which are not birational: -In [56, Chapter 1, §6] we can find another criterion which allows us to determine if a quadratic rational map is rational or not.Proof of Theorem 4.2.2. -Let us see that det jac Q is the union of three lines.

2 (C) into itself belongs to O l.r. (σ) if and only if it has three points of indeterminacy. Remark 4.2.4. -A birational map Q of P 2 (C) into itself contracts det jac Q and doesn't contract any other curve. Q : (x : y : z) (x 2 y : xz 2 : y 2 z). Remark 4.2.5. -We don't know if there is an analogue to Theorem 4.2.2 in any dimension; [160] can maybe help to find an answer in dimension 3. Remark 4.2.6. Assume that det jac Q is irreducible. Let us set Q : (x : y

38, 93, 94, 156, 157, 158]).

  lot of mathematicians have worked on this problem, for example Bonatti, Crovisier, Fisher, Palis, Wilkinson, Yoccoz ([133,

  -The spectral radius of f * is strictly greater than 1 if and only if one of the λ p ( f )'s is and, in fact, if and only if λ

	Theorem 9.3.1 ([110, 108, 186]). -Let f be a holomorphic map on a compact complex Käh-
	ler manifold V; we have
	h top ( f ) = max 0≤p≤dim C V	log λ p ( f ).
	Remark 9.3.2.

  76, Proposition 2.2.]. Let S be a surface obtained from P 2 (C) by blowing up n ≤ 9 points. Let f be an automorphism on S. The topological entropy of f is zero.

	Proposition 9.3.8. -

Corollary 10.3.7 ([144])

	10.3.3. Marked pairs. -
	10.3.3.1. First definitions. -Let (S, Φ) be a marked blow-up. Let us recall that an anti-
	canonical curve is a reduced curve Y ⊂ S such that its class in H 2 (S, Z) satisfies
	(10.3.1)

10.3.6 ([144]

). -Let (S, Φ) be a marked blow-up. If (S, Φ) has no geometric nodal root, then W (S, Φ) = W n .

Proof. -If (S, Φ) has no geometric nodal root and if w belongs to W (S, Φ), then (S, Φw) has no geometric nodal root. It is so sufficient to prove that the generators of W n belong to W (S, Φ). Since the group of permutations is contained in W (S, Φ), it is clear for the transpositions; for κ 123 it is a consequence of Theorems 10.3.4 and 10.3.5. . -A marked surface has a nodal root if and only if it has a geometric nodal root.

The Néron-Severi group of a variety is the group of divisors modulo algebraic equivalence.

Here a rational fibration is a rational application from P 2 (C) into P 1 (C) whose fibers are rational curves.

Proof. -Let us take the notations introduced previously and let us set A := α(A k ). Since # (W s (q) ∩ W u (q)) ≥ 2, the manifolds W s (q) and W u (q) intersect in an infinite number of points and there exists a neighborhood U of q such that any holomorphic function on U which vanishes on U ∩ W u (q) vanishes everywhere. The morphism α is thus injective and Λ is a discrete and infinite subset of C × C invariant under the diagonal action of A.

Let us show that A is discrete. Let A be the closure of A in C * × C * . Since Λ is discrete, Λ is A-invariant. Let us assume that A is not discrete; then A contains a 1-parameter non-trivial subgroup of the type t → (e tu , e tv ). Since Λ is discrete, one of the following property holds:

• Λ = {(0, 0)},

• u = 0 and Λ ⊂ C × {0},

• v = 0 and Λ ⊂ {0} × C. But according to Lemma 8.2.2 none of this possibilities hold. So A doesn't contain a 1parameter non-trivial subgroup and A is discrete. In particular there is a finite index abelian free subgroup A ′ of A such that the rank of A ′ is less or equal to 2. Since f is an element of infinite order of Cent( f , Aut(S)), the group f k is a free subgroup of rank 1 of A k so the lower bound of the rank of A ′ is 1 and if this lower bound is reached then f is of finite index in Cent( f , Aut(S)). Let us consider exp : 4 . Its rank is 3 or 4; indeed the kernel of exp contains 2iπZ × 2iπZ and also (α u ( f ), α s ( f )).

If A ′ is of rank 2, then A ′ is a discrete and co-compact subgroup of C * × C * and there exists an element ψ in Cent( f , Aut(S)) such that

Let (x, y) be a point of Λ \ {(0, 0)}; the sequence

is thus an infinite sequence of elements of Λ and ψ n (x, y) → (0, 0) as n → +∞: contradiction. This implies that A ′ is of rank 1.

Corollary 8.2.4 ([47]

). -Let f be a Hénon automorphism. The cyclic group generated by f is of finite index in the group of biholomorphisms of C 2 which commute with f .

Proof. -According to [START_REF] Bedford | Distribution of periodic points of polynomial diffeomorphisms of C 2[END_REF] if k is large enough, then the automorphism f has n > 0 hyperbolic periodic points of period k whose unstable and stable manifolds intersect each other. Proposition 8.2.3 allows us to conclude.

Centralizer of hyperbolic birational maps

In this context we can also define global stable and unstable manifolds but this time we take the union of strict transforms of W s loc (q) and W u loc (q) by f n . They are parametrized by holomorphic applications ξ u q , ξ s q which are not necessarily injective: if a curve C is contracted is called characteristic matrix of f . The first column of M, which is the characteristic vector of f , is the vector (ν, -m 1 , . . . , -m k ). The other columns (ν i , -m 1i , . . . , -m ki ) describe the "behavior of E ′ i ": if ν j > 0, then π(E ′ j ) is a curve of degree ν j in P 2 (C) through the points p ℓ of f with multiplicity m ℓ j . The points of indeterminacy of σ are P = (1 : 0 : 0), Q = (0 : 1 : 0) and R = (0 : 0 : 1); the exceptional set is the union of the three lines ∆ = {x = 0}, ∆ ′ = {y = 0} and ∆ ′′ = {z = 0}.

First we blow up P; let us denote by E the exceptional divisor and D 1 the strict transform of D. Set

on the other hand (r 1 , s 1 ) → (r 1 s 1 , s 1 ) (y,z) → (r 1 s 1 : 1 :

.

Hence E is sent on ∆ 1 ; as σ is an involution ∆ 1 is sent on E. Now blow up Q 1 ; this time let us denote by F the exceptional divisor and D 2 the strict transform of D 1 :

We have

and

Therefore F → ∆ ′ 2 and ∆ ′ 2 → F. Finally we blow up R 2 ; let us denote by G the exceptional divisor and set

9.4.2.1. Definitions and properties. -Let f be an automorphism on a compact complex manifold M. Let us recall that the Fatou set F ( f ) of f is the set of points which own a neighborhood

linear rotation; in dimension 1 this is equivalent to have a Siegel disk. We have the following properties ( [START_REF] Bedford | Dynamics of rational surface automorphisms: rotations domains[END_REF]).

• If f preserves a smooth volume form, then any Fatou component is a rotation domain.

• A Fatou component U is a rotation domain if and only there exists a subsequence such that (n j ) → +∞ and such that ( f n j ) converges uniformly to the identity on compact subsets of

We say that d is the rank of the rotation domain. The rank is equal to the dimension of the closure of a generic orbit of a point in U.

We have some geometric information on the rotation domains: if U is a rotation domain then it is pseudo-convex ( [START_REF] Bedford | Dynamics of rational surface automorphisms: rotations domains[END_REF]).

Let us give some details when M is a kählerian surface carrying an automorphism with positive entropy.

Theorem 9.4.4 ([24]

). -Let S be a compact, kählerian surface and let f be an automorphism of S with positive entropy. Let U be a rotation domain of rank d. Then d ≤ 2.

, there exists a holomorphic vector field which induces a foliation by Riemann surfaces on S whose any leaf is invariant by G 0 .

We can use an argument of local linearization to show that some fixed points belong to the Fatou set. Conversely we can always linearize a fixed point of the Fatou set.

Fatou sets of Hénon automorphisms.

-Let f be a Hénon automorphism. Let us denote by K ± the subset of C 2 whose positive/negative orbit is bounded:

Let us state some properties.

• The family of the iterates f n , n ≥ 0, is a normal family in the interior of K + .

Theorem 11.4.1. -Assume that n ≥ 7. Assume that j divides n and that (a, b) belongs to Γ j ∩ V n .

There exists a complex number t such that

Let us now deal with those of rank 2.

has a rotation domain on rank 2 centered at m.

There are examples where rotation domains of rank 1 and 2 coexist.

and a rotation domain of rank 1 centered at

Weyl groups ([22])

Let us recall that E 1 and E 2 are the divisors obtained by blowing up p 1 and p 2 . To simplify let us introduce some notations:

the basis e 0 , . . . , e n of Pic(S =) is geometric.

Bedford and Kim prove that they can apply Theorem 10.5.1 and deduce from it the following statement.

Theorem 11.5.1. -Let X be a rational surface obtained by blowing up P 2 (C) in a finite number of points π : X → P 2 (C) and let F be an automorphism on X which represents the standard element of the Weyl group W n , n ≥ 5. There exists an automorphism A of P 2 (C) and some complex numbers a and b such that f a,b Aπ = AπF.

Moreover they get that a representation of the standard element of the Weyl group can be obtained from f a,b,Y .

Birational maps whose exceptional locus is a line

Let us consider the birational map defined by

The sequence (deg Φ k n ) k∈N is bounded (it's easy to see in the affine chart z = 1), so Φ n is conjugate to an automorphism on some rational surface S and an iterate of Φ n is conjugate to an automorphism isotopic to the identity ( [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]). The map Φ n blows up one point P = (1 : 0 : 0) and blows down one curve ∆ = {z = 0}.

Here we will assume that n = 3 but the construction is similar for n ≥ 4 (see [START_REF] Déserti | Special automorphisms of rational surfaces with positive topological entropy[END_REF]). We first construct two infinitely near points P 1 and P 2 such that Φ 3 induces an isomorphism between Bl P 1 P 2 and Bl P 2 P 2 . Then we give "theoretical" conditions to produce automorphisms ϕ of P 2 (C) such that ϕΦ 3 is conjugate to an automorphism on a surface obtained from P 2 (C) by successive blow-ups.

, coordinates in which the exceptional divisor is given by E = {u 1 = 0} and the strict transform of ∆ is given by ∆ 1 = {v 1 = 0}. Set y = r 1 s 1 and z = s 1 ; note that (r 1 , s 1 ) are coordinates near Q = (0, 0) (r 1 ,s 1 ) , coordinates in which E = {s 1 = 0}. We have

and

; therefore P 1 is a point of indeterminacy, ∆ 1 is blown down to P 1 and E is fixed.

12.1.1.2. Let us blow up P 1 in the domain and in the range. Set

Remark that A is a point of indeterminacy. We also have

so F and ∆ 2 are blown down to A.

12.1.1.3. Now let us blow up A in the domain and in the range. Set r 2 = u 3 and

r 3 s 3 and s 2 = s 3 , then (r 3 , s 3 ) is a system of coordinates in which E 2 = {r 3 = 0} and G = {s 3 = 0}. We have

; so G is fixed and F 1 is blown down to S = (1, 0) (r 3 ,s 3 ) .

12.1.1.4. Let us blow up T in the domain and S in the range. Set r 3 = u 4 -1 and s 3 = u 4 v 4 ; in the system of coordinates (u 4 , v 4 ) we have G 1 = {v 4 = 0} and H = {u 4 = 0}. Note that (r 4 , s 4 ), where r 3 = r 4 s 4 -1 and s 3 = s 4 , is a system of coordinates in which H = {s 4 = 0}. On the one hand

so H is sent on F 2 . On the other hand

We can also set r 3 = c 4 d 4 + 1 and s 3 = d 4 ; in the system of coordinates (c 4 , d 4 ) the exceptional divisor K is given by d 4 = 0.

Note that

; thus F 2 is sent on K. We remark that

; so ∆ 4 is blown down to C = (0, 0) (c 4 ,d 4 ) . In the first (resp. second) system of coordinates the exceptional divisor M is given by {a 5 = 0} (resp. {d 5 = 0}). We have

; in particular ∆ 5 is sent on M.

Proposition 12.1.1 ([73]

). -Let P 1 (resp. P 2 ) be the point infinitely near P obtained by blowing up P 2 (C) at P, P 1 , A, T and U (resp. P, P 1 , A, S and U ′ ).

The map Φ 3 induces an isomorphism between Bl P 1 P 2 and Bl P 2 P 2 .

The different components are swapped as follows

12.1.2. Second step: gluing conditions. -The gluing conditions reduce to the following problem: if u is a germ of biholomorphism in a neighborhood of P, find the conditions on u in order that u( P 2 ) = P 1 .

Proposition 12.1.2 ([73]

). -Let u(y, z) = ∑ (i, j)∈N 2 m i, j y i z j , ∑ (i, j)∈N 2 n i, j y i z j be a germ of biholomorphism at P.

We can verify that ψ

The sequence of blow-ups is a little bit different; let us describe it. Denote by ∆ the line x = 0.

• First we blow up R in the domain and in the range and denote by E the exceptional divisor.

We can show that C 1 = {u 1 + v 1 = 0} is sent on E, E is blown down to Q = (0 : 1 : 0) and

1 is a point of indeterminacy. • Next we blow up P in the domain and Q in the range and denote by F (resp. G) the exceptional divisor associated with P (resp. Q). We can verify that F is sent on C ′ 2 , E 1 is blown down to T = G ∩ ∆ 2 and ∆ ′ 2 is blown down to T. • Then we blow up S in the domain and T in the range and denote by H (resp. K) the exceptional divisor obtained by blowing up S (resp. T ). We can show that H is sent on K; E 2 , ∆ ′ 3 are blown down to a point V on K and there is a point of indeterminacy U on H. • We will now blow up U in the domain and V in the range; let L (resp. M) be the exceptional divisor obtained by blowing up U (resp. V ). There is a point of indeterminacy Y on L, L is sent on G 2 , E 3 on M and ∆ ′ 4 is blown down to a point Z of M. • Finally we blow up Y in the domain and Z in the range. We have: ∆ ′ 5 is sent on Ω and N on ∆ ′′ 5 , where Ω (resp. N) is the exceptional divisor obtained by blowing up Z (resp. Y ).

Proposition 12.2.1. -Let P 1 (resp. P 2 ) denote the point infinitely near R (resp. Q) obtained by blowing up R, S, U and Y (resp. Q, T, V and Z). The map ψ induces an isomorphism between Bl P 1 ,P P 2 and Bl P 2 ,R P 2 . The different components are swapped as follows:

The following statement gives the gluing conditions.

Proposition 12.2.2. -Let u(x, z) = ∑ (i, j)∈N 2 m i, j x i z j , ∑ (i, j)∈N 2 n i, j x i z j be a germ of biholomorphism at Q. Then u can be lifted to a germ of biholomorphism between Bl P 2 P 2 and Bl P 1 P 2 if and only if • m 0,0 = n 0,0 = 0;

• n 0,1 = 0;

• n 0,2 + n 1,0 + m 2 0,1 = 0; • n 0,3 + n 1,1 + 2m 0,1 (m 0,2 + m 1,0 ) = 0.

Let ϕ be an automorphism of P 2 . We will adjust ϕ such that (ϕψ) k ϕ sends P 2 onto P 1 and R onto P. As we have to blow up P 2 at least ten times to have nonzero entropy, k must be larger than two, P 1 , ϕ P 2 , ϕψϕ P 2 , (ϕψ) 2 ϕ P 2 , . . . , (ϕψ) k-1 ϕ P 2 must all have distinct supports and (ϕψ) k ϕ P 2 = P 1 . We provide such matrices for k = 3; then by Proposition 12.2.2 we have the following statement.