Lattès maps and the interior of the bifurcation locus

We study the phenomenon of robust bifurcations in the space of holo-morphic maps of P 2 ( C ) . We prove that any Lattès example of suﬃciently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattès map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in C 2 with a well-oriented complex curve. Then we show that any Lattès map of suﬃ-ciently high degree can be perturbed so that the perturbed map exhibits this geometry.


Context
In the article [15], Mañé, Sad and Sullivan, and independently Lyubich in [14], introduced a relevant notion of stability for holomorphic families (f λ ) λ∈Λ of rational mappings of degree d on the Riemann sphere P 1 (C), parameterized by a complex manifold Λ.The family (f λ ) λ∈Λ is J-stable in a connected open subset Ω ⊂ Λ if in Ω the dynamics is structurally stable on the Julia set J. It can be shown that this is equivalent to the fact that periodic repelling points stay repelling points inside the given family.The bifurcation set is the complementary of the locus of stability.A remarkable fact is that the J-stability locus is dense in Λ for every such family.Moreover, parameters with preperiodic critical points are dense in the bifurcation locus.
In higher dimension, less is known.We will only discuss the 2-dimensional case in this paper.The research in this field mostly takes inspiration from two different types of maps with different behaviour : polynomial automorphisms of C 2 and holomorphic endomorphisms of P 2 (C).Knowledge about bifurcations of polynomial automorphisms is growing quickly.Let us quote the work of Dujardin and Lyubich ([10]) which introduces a satisfactory notion of stability and shows that homoclinic tangencies, which are the 2-dimensional counterpart of preperiodic critical points, are dense in the bifurcation locus.
From now on, we are interested in the case of holomorphic endomorphisms of P 2 (C).The natural generalization of the one-dimensional theory was designed by Berteloot, Bianchi and Dupont in [3].Their notion of stability is as follows : let (f λ ) λ∈Λ be a holomorphic family of holomorphic maps of degree d on P 2 (C) where Λ is simply connected.Then the following assertions are equivalent: 1.The function on Λ defined by the sum of Lyapunov exponents of the equilibrium measure µ f λ : λ → χ1(λ) + • • • + χ k (λ) is pluriharmonic on Λ. 2. The sets (J * (f λ )) λ∈Λ move holomorphically in a weak sense, where J * (f λ ) is the support of the measure µ f λ .3. There is no (classical) Misiurewicz bifurcation in Λ. 4. Repelling periodic points contained in J * (f λ ) move holomorphically over Λ.If these conditions are satisfied, we say that (f λ ) λ∈Λ is J * -stable.If (f λ ) λ∈Λ is not J * -stable at a parameter λ0, we will say that a bifurcation occurs at λ0.
A major difference with the one-dimensional case is the existence of open sets of bifurcations.Recently, several works have shown the existence of persistent bifurcations near well-chosen maps.By [3], to obtain open subsets in the bifurcation locus, it is enough to create a persistent intersection between the postcritical set and a hyperbolic repeller contained in J * .Dujardin gives in [9] two mechanisms leading to such persistent intersections.The first one is based on topological considerations and the second uses the notion of blender, which is a hyperbolic set with very special fractal properties.Both enable to get persistent bifurcations near maps of the form (z, w) → (p(z), w d + κ).The results of Dujardin have been improved by Taflin in [18].Taflin shows that if p and q are two polynomials of degree bounded by d such that p is a polynomial corresponding to a bifurcation in the space of polynomials of degree d, then the map (p, q) can be approximated by polynomial skew products having an iterate with a blender and then by open sets of bifurcations.Note that the idea of blender arised in the work of Bonatti and Diaz on real diffeomorphisms ( [6]) and already appeared in holomorphic dynamics in the work of the author ( [5]).
Lattès maps are holomorphic endomorphisms of P 2 (C) which are semi-conjugate to an affine map on some complex torus T (see [11] for a classification and [4] for a characterisation of Lattès maps in terms of the maximal entropy measure).It is natural to be interested in these maps in the context of bifurcation theory because their Julia set is equal to the whole projective space P 2 (C).This property seems to have a great potential to create persistent intersection between the postcritical set and the Julia set even after perturbation.Berteloot and Bianchi proved in [2] that the Hausdorff dimension of the bifurcation locus near a Lattès map is equal to that of the parameter space.

Main result
Dujardin asked in [9] if it was possible to find open sets of bifurcations near any Lattès map.In this article we give a partial answer to this question.Here is our main result : Theorem.For every two-dimensional complex torus T, there is an integer d (depending on the torus T) such that every Lattès map defined on P 2 (C) of degree d > d induced by an affine map on T is in the closure of the interior of the bifurcation locus in Hol d .
Let us point out the scarcity of tori which are associated to some Lattès example on P 2 (C) (the classification is discussed in section 3).We also remark that the degree d is unknown (the situation here is similar to Buzzard's article [7]).Moreover, d depends on the torus T (see subsection 1.3).This is due to the necessity of making only holomorphic perturbations.As a consequence of the theorem we get : Corollary.For every Lattès map L of degree d, there is an integer n(L) such that for every n ≥ n(L), the iterate L n is in the closure of the interior of the bifurcation locus in Hol d n .
The Theorem also implies that there are no open subsets of Lattès maps in the family of endomorphisms of P 2 (C) (if one does not need to iterate).Indeed, for such an open set of Lattès maps, the Lyapunov exponents would be minimal (see [4]) and the sum of Lyapunov exponents would be pluriharmonic, but the Theorem implies that this set intersects open sets of bifurcations where the sum of Lyapunov exponents is not pluriharmonic (by [3]).

Outline of proof
To prove this result, we create persistent intersections between the postcritical set and a hyperbolic repeller contained in the Julia set.Our proof has two main parts : first, we create a toy-model which allows to obtain intersections between the limit set of some particular type of IFS, called correcting IFS, and a quasi-line that is "well-oriented".Then, in a second time, we perturb the Lattès map to create both the correcting IFS and the well-oriented curve inside the postcritical set.This construction exhibits properties somehow similar to the blenders of Bonatti-Diaz ( [6]), with the difference that the covering property holds at the level of the tangent maps of the IFS (see also the notion of parablenders appeared in the work of Berger ([1])).
In a first part, we develop an intersection principle (see Proposition 2.1.6).A grid of balls G in C 2 is the union of a finite number of balls regularly located at N 4 vertices of a lattice defined by a R-basis of C 2 .If we consider a line C, a pigeonhole argument ensures that if C is well oriented and G has a sufficient number of balls N = N (r) (where r is the relative size of a ball compared to the mesh of the grid) then C intersects a ball of G.We consider a class of IFS such that each inverse branch is very close to a homothety.When we iterate them, a drift can appear : the iterates become less and less conformal.Our class of IFS (called correcting IFS) is designed so that they have the property of correcting themselves from the drift.A linear correction principle is given in Proposition 2.2.2.In subsections 2.3 and 2.4, we treat the case of a curve close to a line and an IFS close to be linear.Our interest in such IFS is that any welloriented quasi-line C intersects the limit set of a correcting IFS.To prove this result, which is Proposition 2.4.1, we ensure that at each step the quasi-line C intersects a grid of ball G j which is dynamically defined with the inverse branches of the IFS.Then we use inductively the intersection and the correction principles to ensure that at the next step, C intersects a grid of balls G j+1 with bounded drift.The intersection of the grids G j is in the limit set, so we produce an intersection between C and the limit set of the IFS.Since the property of being correcting is open, this intersection is persistent.
In the second part, we make three successive perturbations of a Lattès map L, denoted by L , L and L , in such a way that L has a robust bifurcation.We work in homogenous coordinates and do explicit perturbations of the following form : where R1 and R2 are rational maps.An important technical point (Proposition 3.2.1) is that we can choose the coordinates so that P3 splits.Then if R1 and R2 are well chosen the degree does not change.The first perturbation L (Propositions 4.4.4 and 4.4.5) is intended to create a correcting IFS in a ball B in C 2 .Another important technical point is that we can find some critical point c which is preperiodic, with associated periodic point pc such that both the preperiod nc and the period npc of the preperiodic critical orbit are bounded independently of L (see Proposition 3.3.1).Then we want to create a well-oriented quasi-line inside the postcritical set which intersects B. The second perturbation L in Lemma 4.5.10 ensures that the postcritical set at pc is not singular.The third and last perturbation L is given in Lemma 4.5.11.It is intended to control the differential at pc.This allows us to fix the orientation of the postcritical set at pc and then we use the linear dynamics of the Lattès map L on the torus T in order to propagate this geometric property up to B (see Proposition 4.5.3).Note that the periodic point need not lie in B. At this stage we have both a correcting IFS and a well-oriented quasi-line so we are in position to conclude in section 5.
In particular, let us point out that the bound d on the degree is fixed in 4.2.12,4.2.13 and 4.2.14 : d = max(d 1 , d 2 , d 3 ).Here d 1 is fixed to ensure that there are sufficiently many inverse branches in the IFS to apply Proposition 2.4.1.d 2 is intended to make the first perturbation possible in Proposition 4.4.4.(section 2 plays an important role in the determination of d 2 ).Similarly, d 3 is fixed to allow the second and third perturbations in Lemmas 4.5.10 and 4.5.11along the periodic orbit (whose length is bounded in subsection 3.3 and important to fix d 3 ).It is also interesting to remark that the bound d 2 comes from an interpolation.This has some similarities with the article [7] where Buzzard uses a Runge approximation with polynomial automorphisms of sufficiently high degree in order to prove the existence of Newhouse phenomenon in the complex setting.In particular, d 1 depends on the torus (the number of inverse branches depends on the size of a ball B depending on T) and it is also the case for d 2 (which depends on the integer i(T) defined in Proposition 3.2.1).
In section 2, we develop the theory of intersection between a quasi-line and the limit set of a correcting IFS : the intersection principle and the correction principle are respectively stated in subsections 2.1 and 2.2 and we prove the intersection result in subsection 2.4.In section 3, we provide background on Lattès maps and prove a few properties which will be useful later.Some complications arise from Lattès maps whose linear part is not the identity.In section 4, we develop the perturbative argument.After giving some preliminaries (subsection 4.1) and fixing many constants (subsections 4.2 and 4.3), we create a correcting IFS in subsection 4.4.In subsection 4.5, we create a well oriented curve inside the postcritical set.Finally, we conclude in section 5 by applying the formalism of subsection 2.4 to the perturbed map L .

Acknowledgments :
The author would like to thank his PhD advisor, Romain Dujardin.This research was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002.
2 Intersecting a curve and the limit set of an IFS

Linear model
In this section, we will work with an IFS, whose maps are small perturbations of homotheties of the form 1 a • Id with a ∈ R * and |a| > 1.This IFS will be obtained by perturbating a Lattès map and its limit set will have persistent intersections with a curve.
an integer N and r ∈ (0, 1), by a grid of balls we mean the union of the balls of radius r. min 1≤i≤4 ||ui|| centered at the points o + iu1 + ju2 + ku3 + klu4 where −N ≤ i, j, k, l ≤ N are integers.We will denote it by G = (u, o, N, r).The middle part of G is the set {o In the following, the parameter r will be bounded from below and we will let max 1≤i≤4 ||ui|| → 0 so that the radius of the balls r.min 1≤i≤4 ||ui|| will tend to 0. The integer N will be taken sufficiently large to satisfy some conditions depending on the degree of the Lattès map.Herebelow the notions of "opening" and "slope" are relative to the standard euclidean structure of C 2 .Notation 2.1.2.For a non zero vector w ∈ C 2 and θ > 0, we will denote C w,θ the cone of opening θ centered at w. Notation 2.1.3.For any quadruple of non zero vectors w1, w2, w3, w4 in C 2 , we will denote w = (w1, w2, w3, w4) its projection onto P(R 8 ).For any matrix U ∈ GL2(C), we simply denote by U • the induced action on P(R 8 ).Definition 2.1.4.The middle part of a ball (resp.the 3 4 -part) is the ball of same center and 1  2 times its radius (resp. 3 4 times its radius).Definition 2.1.5.A holomorphic curve C is a (ε, w)-quasi-line if C is a graph upon a disk in C • w of slope bounded by ε relative to the projection onto w.A (ε, w)-quasidiameter of a ball B is a (ε, w)-quasi-line C intersecting the ball of same center as B and of radius 1  10 times the radius of B.Here is our "intersection principle" : Proposition 2.1.6(Intersection Principle).For every u ∈ (C 2 ) 4 , r > 0, η > 0 and w0 ∈ C 2 , there exists a neighborhood N (u) of u in P(R 8 ), there exists θ > 0, N (r) > 0 and a vector w ∈ C 2 with ||w − w0|| < η such that the following property (P) holds : (P) For every grid of balls G = (v, o, N, r) such that v ∈ N (u) and N > N (r), for every (θ, w)-quasi-line of direction in C w,2θ intersecting the middle part of the grid of balls G, there is a non empty intersection between the (θ, w)-quasi-line and the middle part of one of the balls of the grid.
Moreover, property (P) stays true for w sufficiently close to w.
Lemma 2.1.7.There is a non empty intersection between any line of direction in C w,2θ intersecting the middle part of the grid of balls G and the middle part of one of the balls of the grid of balls if θ is sufficiently small.Proof.We divide each mesh of the lattice into m 4 hypercubes.To each of these hypercubes, we can assign the quadruple of integers given by the coordinates of a given corner.Taking new coordinates by making a translation if necessary, we can suppose that the union of the middle parts of the balls of the lattice contains the union of the hypercubes whose four coordinates are all equal to 0 modulo m.Let us take a point x0 of the line inside the middle part of the lattice, and for every k ∈ N, we denote : Then, we have that : Since N > 10βm 5 = N (r), the previous relations imply there exists some xn which intersects some hypercube of integer coordinates congruent to (0, 0, 0, 0) inside the grid of balls.This implies that the line intersects the middle part of one of the balls of the grid.This intersection persists for any line of direction in C w,2θ and for any v in a small neighborhood N (u) of u.Then, the result stays true if we take (θ, w)-quasi-lines for θ sufficiently small since property (P) is open for the C 1 topology and w sufficiently close to w.
The following corollary gives the same conclusion as the previous result but this time with more than one possible direction for the quadruple of vectors of the lattice.
Corollary 2.1.8.For every finite subgroup M ⊂ Mat2(C), for every u ∈ (C 2 ) 4 , there exists a neighborhood N (u) of u in P(R 8 ) such that for every r > 0, there exists θ > 0, N (r) > 0 and a vector w ∈ C 2 such that the following property (P) holds : (P) For every U ∈ M, for every grid of balls and N > N (r), for every (θ, w)-quasi-line of direction in C w,2θ intersecting the middle part of G, there is a non empty intersection between the (θ, w)-quasi-line and the middle part of one of the balls of the grid.
Moreover, this proposition remains true for w sufficiently close to w.
Proof.We just have to apply ord(M) times Proposition 2.1.6.
In the following, x will be a real positive parameter.We remind that in a first reading it is advised to assume that U = I2.The following proposition is the "linear correction principle" we discussed in the introduction.Proposition 2.2.2 (Linear correction principle).For every finite subgroup M ⊂ Mat2(C), there exists an integer n > 0, (n + 1) balls V 0 , V 1 , ..., V n ⊂ Mat2(C) such that for every 0 < x < 1, there exists a neighborhood Ux of I2 in GL2(C), two open sets U x ⊂ U x ⊂ GL2(C) which are union of balls U x = 1≤p≤n (U x ) p and U x = 1≤p≤n (U x ) p such that : (U x ) p ⊂ (U x ) p for each 1 ≤ p ≤ n with the following properties : and j ∈ N, then there exist two integers 1 ≤ p, p ≤ n such that M ∈ (U x ) p with the property that for every M0 ∈ (x•V 0 ) and for every M p ∈ (x• V p ), we have : Proof.We consider the vector space Mat2(C) R 8 .Let us consider a covering of the sphere of center 0 of radius r (which will be chosen later) S(0, r) by n balls B(Xi, 1 20 r) of radius 1  20 r.The following geometrical lemma is trivial : Lemma 2.2.3.For every 1 ≤ p ≤ n, X ∈ B(Xi, 1  10 r), we have : Increasing the number n of open sets (U 1 ) p if necessary, we can suppose that for every U ∈ M and for each p ≤ n, there exists p ≤ n such that Proof.The Taylor formula gives us that at 0 at the first order in X : Then, if r is sufficiently small, Lemma 2.2.3 implies that for every X ∈ B(Xi, 1 10 r) : This means that for every Now, it is clear it is possible to take sufficiently small balls V 0 , V 1 , ..., V n centered at 0, −X1, ..., −Xn such that : -If M ∈ U1, then for every M0 ∈ V 0 , we have : The previous lemma implies that if M ∈ (U 1 ) p and M0 ∈ V 0 are such that M (I2 + M0) ∈ (U 1 ) p , then for every Mp ∈ V p , we have that : Then, properties (i) and (ii) are verified for x = 1 and U = I2.For each 0 < x < 1, let us take the balls and let us apply the homothety of factor x of center I2 to the sets U1, U 1 , U 1 , (U 1 ) p and (U 1 ) p to get the sets Ux, U x , U x , (U x ) p and (U x ) p such that properties (i) and (ii) are verified for x < 1 and U = I2.

Let us now suppose that
are still true by reducing V 0 a finite number of times if necessary.Let us take p ≤ n and p ≤ n such that (U 1 ) p • U = U • (U 1 ) p and M p ∈ V p .Then : This implies that for every M0 ∈ (x • V 0 ) and for every M p ∈ (x • V p ), we have Let us point out the following obvious result for later reference.Remind that N (u) was defined in Proposition 2.1.6and U x comes from Proposition 2.2.2.

Quasi-linear model
Here we slightly perturb the linear maps we used before but we show we can keep results on persistent intersections.Let us recall that the integer n was defined in Proposition 2.2.2.Let us remind that M ⊂ Mat2(C) is a finite subgroup.
where V p was defined in Proposition 2.2.2).The modulus |a| is called the contraction factor of f .Let f be a smooth map defined on an open subset V of C 2 .We say that f is quasi-linear of type (x, p) if the differential Dfo is linear of type (x, p) for every o ∈ V, this is : f = 1 a (A + h) with h smooth and D ho ∈ x • V p for every o ∈ V (A and a depend only on f but not on o).
The following can be seen as a consequence of Proposition 2.2.2 in the quasi-linear setting.Remember that x(u) > 0 was defined in Proposition 2.2.5.
Proposition 2.3.2.Let M ⊂ Mat2(C) be a finite subgroup of unitary matrices.Reducing x(u) if necessary, for every grid of balls G = (u, o, N, r), for every quasilinear map f of type |a| times the radius of a ball of G.
If ||f || C 2 = 0, the image of each ball of G under f still contains a ball of G but this time by the Taylor formula there is an additive term smaller than size(G) in the differential of f .Then the relative size r is such that :

Intersecting a curve and the limit set of an IFS in C 2
In this subsection, we give an abstract condition ensuring the existence of an intersection (robust by construction) between a holomorphic curve in C 2 and the limit set of an IFS.This will be the model for robust bifurcations near Lattès maps.Remind that n was defined in Proposition 2.2.2, N , w, θ in Proposition 2.1.6.Remind that the middle part and the 3  4 -part of a ball were defined in Definition 2.1.4,the middle part and the hull of a grid of balls were defined in Definition 2.1.1.In the following, for a holomorphic map G defined on a (closed) ball B ⊂ C 2 , we will denote Proposition 2.4.1.Let (G1, ..., Gq) be a IFS given by q maps defined on a ball B ⊂ C 2 of radius R > 0 satisfying the following properties : 1. 1≤j≤q Gj(B) contains a grid of balls G 1 = (u 1 , o 1 , nG, r 1 ) with q = (2nG + 1) 4  such that each Gj(B) contains a ball of G 1 2. the contraction factor of the IFS (G1, ..., Gq) is |a| ≥ 2 3. there exist (n + 1) balls B0, B1, ..., Bn ⊂ B of radius larger than ν • R (with 0 < ν < 1), such that the 3 4 -parts of B0, B1, ..., Bn are included in the hull of G 1 , and satisfying the following property : for each 1 ≤ j ≤ q such that Gj(B) ⊂ Bp, Gj = 1 a (A + hj) is quasi-linear of type (x, p) with x < x(u 1 ) and a,A do not depend on j.Moreover, 1≤j≤q Gj(Bp) contains a grid of balls Then C intersects the limit set of the IFS (G1, ..., Gq).Definition 2.4.2.We say that (G1, ..., Gq) is a correcting IFS when the conditions 1, 2, 3, 4 and 5 are satisfied.Proposition 2.4.1 will be the immediate consequence of the following lemma : Lemma 2.4.3.There exist (n + 2) sequences of grids (G j ) j≥1 = (u j , o j , nG, r j ) j≥1 and (Γ j p ) j≥1 = (u j , o j p , nG, s j ) j≥1 with 0 ≤ p ≤ n such that we have the following properties : 1.For every j > 1, G j is included inside a ball of G j−1 and for every j > 1, there are i1, ..., ij−1 ≤ q such that : 4. For every j > 1, there exists 1 ≤ pj ≤ n such that the quasi-line C intersects the middle part of a ball of Γ j p j Proof.The proof of this lemma is based on an induction procedure.We begin by an initialisation called Case 0 where we pick the grids of balls at the first level G 1 and Γ 1 p for 0 ≤ p ≤ n.We intersect for the first time the quasi-line C with a ball and we construct the grids at the second level.Case 0 is somewhat different from the rest of the demonstration because we do not not control the initial position of C.Then, Case 1 has to be thought as the most frequent situation : C intersects a grid of balls whose geometry is good enough, and we can intersect C with a new grid whose geometry is very close to the previous one.Then, it may happen a time when the geometry of this grid is too deformed.Then we apply a "correction" (Cases 2 and 3), which leads back to Case 1.
By Corollary 2.1.8,C intersects in its middle part a ball of Γ 1 0 : indeed, Γ 1 0 is a grid of balls such that u 1 ∈ N (u 1 ), we have ) (beware that the matrix A corresponds to the matrix denoted by U in Corollary 2.1.8).Then it intersects the ball Gi 1 (B) of G 1 which contains this ball of Γ 1 0 .According to Proposition 2.3.2,there exists a grid of balls Let us now suppose by induction that the (n + 2) sequences of grids of balls satisfying ( 1),( 2),( 3) and ( 4) are constructed up to step j with the additional properties that C intersects in its middle part a ball of Γ j−1 p j−1 and that the following property is verified : we have : Let us construct the grids of balls at the next step.The proof is inductive, at each step of the proof we are in one of the three cases we are going to discuss, which differ by two parameters.We have a quasi-line intersecting a grid of balls and we have to make a different choice to intersect a ball corresponding to one of the (n + 1) types of differentials we introduced earlier.Note that after Case 0, we will necessarily be in Case 1.
Case 1 : Then it is possible to take an union of balls of Γ j 0 included in B j−1 0 which form a grid of balls Γ of basis u j , relative size s j and with ( 1 10 ν • nG) 4 balls.By construction, we can take Γ such that C intersects the middle part of Γ .Since u j ∈ A j−1 • N (u 1 ), s j > ν•r 1 10 and 1 10 ν • nG > N ( ν•r 1 10 ) we have according to Corollary 2.1.8that C intersects in its middle part a ball of Γ j 0 .This ball is included inside (Gi 1 • ... • Gi j )(B) with Gi j quasi-linear of type 0.
In particular, C intersects the hull of a new grid of balls According to Propositions 2.2.2, 2.3.2 and Property (Q), G j+1 is a grid of balls The grids of balls G j+1 and Γ j+1 p (for 0 ≤ p ≤ n) satisfy ( 1),( 2),( 3),(4).In particular, C intersects in its middle part a ball of Γ j 0 . Since Then, after Case 1 and according to Proposition 2.2.2,only two cases can occur.If In this case, we are going to "correct" the next grids in a procedure given by Cases 2 and 3.
By construction, C intersects in its middle part a ball of the grid of balls Γ j−1 0 .We have according to Proposition 2.2.5 that u j ∈ A j−1 • N (u 1 ).Then, using the same argument as in Case 1, we have according to Corollary 2.1.8that C intersects in its middle part a ball of Γ j p j included inside (Gi 1 • ... • Gi j−1 )(B) where Gi j−1 is quasilinear of type 0 and pj = p is chosen according to Proposition 2.2.2.In particular, C intersects the hull of a new grid of balls p ) and r j+1 , s j+1 satisfy the inequalities of property 3.The grids of balls G j+1 and Γ j+1 p (for 0 ≤ p ≤ n) satisfy ( 1),( 2),( 3), (4).
100 we have for every 0 ≤ p ≤ n, for every j ≥ 1 the following bounds : p and pj = p is chosen according to Proposition 2.2.2 (see Proposition 2.2.2 for the definition of p ), we have for every i such that (Gi After Case 2, it follows from Proposition 2.2.2 that necessarily the two conditions of the following Case 3 are satisfied. Induction shows that pj−1 had been chosen to get special composition properties (see Case 2, beware that the number denoted here by pj−1 corresponds to the number denoted by "pj" in Case 2), let us pick pj = 0.By construction, C intersects in its middle part a ball of the grid of balls Γ j−1 p j−1 .Once again : u j ∈ A j−1 • N (u 1 ) and we have according to Corollary 2.1.8that C intersects in its middle part the ball of Γ j 0 included inside (Gi 1 • ... • Gi j )(B) with Gi j quasi-linear of type pj−1.In particular, C intersects the hull of a new grid of balls G j+1 ⊂ (Gi 1 • ... • Gi j )(G 1 ).Once again, we can construct grids of balls G j+1 and Γ j+1 p (for 0 ≤ p ≤ n) which satisfy (1),( 2),( 3),( 4) but this time with In particular, C intersects in its middle part a ball of Γ j p j .Moreover, Proposition 2.2.2 still insures that (Q) is verified.
• Ux, we are now in Case 1 once again.
3 Properties of Lattès maps where T is a complex torus of dimension 2, Π is a ramified covering of the projective space P 2 (C) by the torus T and L is an affine map.Proposition 3.1.2.The periodic points of any Lattès map are dense in P 2 (C).The Julia set of any Lattès map is equal to P 2 (C).Notation 3.1.3.In the following, for every τ ∈ C such that Im(τ ) > 0, we will denote L(τ ) the lattice in C given by : L(τ . We also let ξ = e i 2π 6 .
The following proposition can be found in [13].
Proposition 3.1.4.If an affine map on a torus T induces a Lattès map L on P 2 (C), then the torus T is of the form C 2 /Λ where Λ is one of the six following lattices and the projection Π : T → P 2 (C) is given (in some affine chart for Cases 1,2,3,4) by the following formulas : where e1 = ℘( 1 2 ) and (x1, y1) is the function of (x, y) given by : In the following, we will denote π the projection from C 2 to T 2 = C 2 /Λ.Definition 3.1.5.A product in the sense of Ueda is a holomorphic map on P 2 (C) such that there exists a Lattès example L on P 1 (C) such that we have : where η is the map between P 1 (C) × P 1 (C) and P 2 (C) which is just the projectivization of (x, y) → (x + y, xy), given by : Such a map L is semi-conjugate to an affine map on the complex torus T and is a Lattès map.
Lattès maps corresponding to Cases 1,2,3 and 4 of Proposition 3.1.4are products in the sense of Ueda.The following technical result was shown in [11] (Theorems 4.2 and 4.4).It will be used in the proof of Proposition 3.3.1.Proposition 3.1.6.For any Lattès map L on P 2 (C), one of the following is true : 1. either one map in {L, L 2 , L 3 } is a product in the sense of Ueda 2. either one of the maps L k in {L, L 2 , L 3 , L 6 } is preserving an algebraic web associated to a smooth cubic (see [8] for this notion) The following is an easy consequence of Propositions 3.1 to 3.6 of [11].
Proposition 3.1.7.Let Λ, Π be one of the lattices and associated coverings defined in Proposition 3.1.4.There exists a finite group of unitary matrices GLattès = GLattès(Λ, Π) of finite order such that every Lattès map has its linear part of the form aA where a ∈ C * , |a| ≥ 1 and A ∈ GLattès.
Remark.Here, the scaling factor a takes discrete values.Moreover, arbitrarily large values of |a| can be obtained (it can be easily seen by taking the composition of a Lattès map with itself ).The equality of the two topological degrees gives : where d is the algebraic degree of L.
Since according to the previous result, there are only finitely many possible linear parts A for a Lattès map (up to multiplication by the factor a) which are all of finite order, we can define the following integer.Definition 3.1.8.We denote by ord Lattès the product of all the orders of the possible linear parts A for a Lattès map.
It can be found in [11] that ordLattès is equal to 6 2 • 8 2 • 12 • 24.In a first reading, we encourage the reader to consider only the case where the linear part of the Lattès map is equal to Id.In the other cases, the dynamical ideas are the same but with a few additional technicalities from algebra.In particular, it is sufficient in order to prove in some cases the corollary of the main result (see subsection 1.2).

An algebraic property of Lattès maps
The goal of this subsection is to prove the following result.Proposition 3.2.1.For every torus T, there exists an integer i = i(T 2 ) such that for any k > 0, there exists an integer d k > 0 such that for any Lattès map L of algebraic degree d > d k , coming from an affine map on T, there exists a homogenous change of coordinates ϕ on P 2 (C) such that : ϕ −1 • L • ϕ is a holomorphic endomorphism of P 2 (C) of the form [P 1 : P 2 : P 3] where the polynomial P 3 is a product of irreducible factors P 3,j such that at least k factors P 3,j are of degree bounded by i. Definition 3.2.2.Let v be a vector of C 2 which belongs to a lattice Λ and v0 ∈ C 2 .We suppose that the action of Λ upon C • v by translation is cocompact.Let T 2 = C 2 /Λ and π : C 2 → T 2 be the natural projection.Then, then we say that π(v0 +C•v) is a compact line of the torus T of direction v.It is compact and π(v0 The family of compact lines of the torus T of direction v is the family of all the compact lines of the torus of direction v obtained by varying v0. Let us point out the fact that v ∈ Λ is not sufficient to conclude that the action of Λ upon C • v by translation is cocompact.Proposition 3.2.3.Let Λ, Π be one of the lattices and associated coverings defined in Proposition 3.1.4.Let v be a vector of C 2 which belongs to Λ such that the action of Λ upon C • v by translation is cocompact.The family of images under Π of compact lines of direction v on the torus T is a family of algebraic curves of P 2 (C) of degree bounded by i = i(v, T 2 ).
Proof.Let F be the family of images of compact lines of direction v on the torus T under Π.The family F is a holomorphic compact family of compact curves so that by the GAGA principle it is an algebraic family of curves and in particular their degree is bounded by some i = i(v, T 2 ) Proposition 3.2.4.Let Λ, Π be one of the lattices and associated coverings defined in Proposition 3.1.4.Then, there exists a line δ in P 2 (C) such that Π −1 (δ) contains at least one compact line D of T.
Proof.In each case, the following compact lines are convenient for δ and we give the preimage compact lines D. The first four cases cover the case of a product in the sense of Ueda.
Cases 2, 3 and 4: δ = {Y = 0}.The proof is similar to Case 1 with respectively ℘ (x)℘ (y) = 0, ℘ 2 (x)℘ 2 (y) = 0 and (℘ (x)) 2 (℘ (y)) 2 = 0.In all the cases, the preimage of δ by Π contains a compact line of the torus.Proposition 3.2.5.If an affine map L of linear part aA on a torus T induces a Lattès map L on P 2 (C) and D is the preimage under Π of the compact line δ given by Proposition 3.2.4,then the preimage of D under L is a finite union of compact lines of the torus.Moreover, the number of possible directions is finite.For each k > 0, there exists d k > 0 such that for every Lattès map L of algebraic degree greater than d k induced by an affine map L on T, there exist at least k distinct irreducible components of L −1 (δ) of degree bounded by i.
Proof.From Proposition 3.1.7,we know that the linear part of L is of the form aA with A ∈ GLattès.We denote by L C 2 an affine map on C 2 which induces the affine map L on T. The linear part of L C 2 is aA.We know that D is a compact line of the torus T of direction v (for some vector v of C 2 ) and the preimage of D under the natural projection π : C 2 → T is an union of lines of C 2 of direction v. Since D is a compact line of T, by definition, the action of Λ on C • v is cocompact.This is equivalent to the existence of two complex numbers ω1 and ω2 which are not R-colinear and such that ω1v ∈ Λ, ω2v ∈ Λ.We fix ω1 and ω2.We have aA • Λ ⊂ Λ because aA is the linear part of L. Then a 2 A 2 • Λ ⊂ Λ, . . ., a ord(A)−1 A ord(A)−1 • Λ ⊂ Λ (here ord(A) is the order of A, we know that A is of finite order because it belongs to the finite group GLattès).But a ord(A)−1 A ord(A)−1 = a ord(A) (aA) −1 .Then (aA) −1 (a ord(A) ω1v) ∈ Λ and (aA) −1 (a ord(A) ω2v) ∈ Λ.For every line ∆ of C 2 of direction v, the preimage of ∆ under L C 2 is a line of C 2 of direction (aA) −1 (v).The two complex numbers a ord(A) ω1 and a ord(A) ω2 are not R-colinear and satisfy a ord(A) ω1 • (aA the preimage of D under L is an union of compact lines of T which all have the same direction.GLattès is finite (see Proposition 3.1.7)and so the possible number of directions is finite.Let D be a preimage of D under L.
Their images under Π are irreducible components of degree bounded by i = i(v, T) by Proposition 3.2.3.If |a| > a k , at least k (this term k is not optimal and we get it by projection of the previous 100k lines) of them are distinct preimages of δ under L.But |a| > a k if deg(L) is superior to some value d k,A (see the remark after Proposition 3.1.7).Then, it suffices to take for d k the maximal value of d k,A when varying A in GLattès (see Proposition 3.1.7).
Proof of Proposition 3.2.1.Let δ be a line in P 2 (C) as in Proposition 3.2.4.The result is a consequence of Proposition 3.2.5 because after a suitable change of coordinates, we can take δ = {Z = 0}.Then {P 3 = 0} contains at least the k irreducible components of degree bounded by i which are preimages of δ by L. Proposition 3.3.1.There exists an integer K > 0 such that for every Lattès map L defined on P 2 (C), there exists a point c in the critical set of L which is sent after nc iterations on a periodic point pc of period npc such that : Proof.Let us start with the case of one dimensional Lattès maps.Lemma 3.3.2.Let L be a one-dimensional Lattès map.There exists a critical point c of L which is sent after ñc ≤ 12 iterations on a periodic point pc of period ñpc ≤ 12.
Proof.The Lattes map L, according to Lemma 3.4 of [16], is such that the postcritical set PL of L is entirely included inside the set of critical values of the covering Θ of P 1 (C) by the complex torus T 1 .This implies that every critical point of L is sent after one iteration inside the set of the critical values of Θ.Moreover, let us bound from above the number of critical values.This number cr is bounded from above by the number of critical points (counted with multiplicity).Still according to [16], Θ can only be a covering of orders ord(Θ) = 2, 3, 4 or 6.The Riemann-Hürwitz formula gives us that : χ(T 1 ) = ord(Θ)χ(P 1 (C)) − cr which implies cr = 2 • ord(Θ).In particular, this means that the image of every critical point c of L is sent after ñc ≤ 12 iterations on a periodic point pc of period ñpc ≤ 12 .
Lemma 3.3.3.Let L be a product in the sense of Ueda.There exists a point c in the critical set of L which is sent after nc ≤ 12 iterations on a periodic point pc of period npc ≤ 24 • ord Lattès which is a multiple of ord Lattès .In particular, we have : nc + npc ≤ 12 + 24 • ord Lattès .
Proof.We take a critical point c of L given by the previous lemma.We take a periodic point p of L of period 2 • ordLattès (it can be found in [12] that such a point actually exists because any rational map on P 1 (C) of degree greater than 2 has a point of strict period 2 • ordLattès > 4 ).Then the point c = η(c, p) is a critical point of L (remind that η was defined in Definition 3.1.5).It is sent after nc ≤ 12 iterations on a periodic point η(pc, Lnc (p)).The period of pc is ñpc ≤ 12 and p is of period 2 • ordLattès.This implies that in P 1 (C) × P 1 (C), the periodic point (pc, Lnc (p)) for ( L, L) is of period a multiple of 2 • ordLattès bounded by 24 • ordLattès.Since the map η is a two-covering, in P 2 (C), the periodic point η(pc, Lnc (p)) for L is of period npc which is a multiple of ordLattès bounded by 24 • ordLattès.Then nc + npc ≤ 12 + 24 • ordLattès.
Let us now prove Proposition 3.3.1.According to Proposition 3.1.6,we have : 1. Either one map in {L, L 2 , L 3 } is a product in the sense of Ueda.In this first case, the previous lemma shows that one of the maps in {L, L 2 , L 3 } has a point of its critical set which is sent after at most 12 iterations onto a periodic point of period a multiple of ordLattès bounded by 24 • ordLattès.This implies that there exists a critical point of L which is sent after nc iterations onto a periodic point of period npc which is a multiple of ordLattès with nc+npc ≤ 3•(12+24•ordLattès).
2. One of the maps L k in {L, L 2 , L 3 , L 6 } is preserving an algebraic web associated to a smooth cubic.This implies (see the remark after Theorem A in [8]) that the critical set of L k is sent after one iteration into the set of critical values of Π which is a curve PC.In this second case, we have that L k (PC) ⊂ PC and L k induces by restriction a map on PC.Taking the normalization of PC if necessary, we can suppose that PC is regular.There are two possibilities.Either PC is isomorphic to P 1 (C) and L k induces a rational map so it has a periodic point of period ordLattès(again, it can be found in [12] that such a point actually exists).
Either PC is isomorphic to a complex torus and L k induces a multiplication on this torus which also has a periodic point of period ordLattès.In both cases, we see that L has a critical point which is sent after at most 6 iterations on a point of period a multiple of ordLattès bounded by 6 • ordLattès.
4 Perturbations of Lattès maps

Some useful lemmas
In this subsection, we prove two lemmas about complex analysis.The constants which are involved in these lemmas will be fixed in the two next subsections.The following first lemma will be used in 4.2.8, 4.3.20 and in the proof of Lemma 4.4.8.Lemma 4.1.1.For every m > 0, for every ball B, for every 1 > ψ1 > 0, 1 > ψ2 > 0, there exist constants ρ = ρ(m, B) > 0, σ = σ(m, B) > 0 such that for every rational function h of degree equal to m, there exists a ball B ⊂ B ⊂ C 2 of radius larger than ρ such that : The lemma will be a consequence of the following lemma.
Lemma 4.1.2.For every m > 0, for every ball B, there exist constants ρ = ρ(m, B) > 0, τ = τ (m, B) > 0 such that for every rational function h of degree m, there exists a ball B ⊂ B ⊂ C 2 of radius larger than ρ such that : Proof.Let us denote Rnorm the set of rational maps of degree m which can be written h = h 1 h 2 where h1 and h2 are two polynomials whose coefficients (aij) and (bij) are such that : max(aij) = max(bij) = 1.Rnorm is a compact set.For a given h ∈ Rnorm, since h = 0, there exists The constants ρ h and τ h can be chosen locally constant for rational functions in Rnorm near h.Since Rnorm is compact, if we choose ρ = ρ(n, B) the minimum of the ρ h and τ = τ (n, B) the minimum of the τ h for a finite covering of Rnorm, we have : for every rational map h ∈ Rnorm of degree m, there exists a ball B ⊂ B ⊂ C 2 of radius larger than ρ such that : Since every rational map h of degree m can be written h = C ste • h with h ∈ Rnorm, the result is true for every rational map of degree m.
The following interpolation result will be used in 4.2.7,4.2.10,4.3.21(and thus in the proof of Lemma 4.4.6) and in the proof of Lemma 4.4.7.Remind that n was defined in Proposition 2.2.2.Lemma 4.1.3.Let us take n balls V1, ..., Vn ⊂ Mat2(C).There exists an integer d = d(V1, ..., Vn) and two real numbers 1 > ψ1 > 0 and 1 > ψ2 > 0 such that for every ξ > 0, there exists a constant ν = ν(n, ξ) > 0 such that : for every ball B ⊂ C 2 of radius bounded by 1, for every θ0 ∈ R, there exist a polynomial map H = H(V1, ..., Vn, B, θ0) of C 2 of degree d and (n + 1) balls B0, ..., Bn ⊂ B of radius greater than ν • rad(B) such that on each Bj : Proof.We call ṽ1, ..., ṽn the centers of the balls V1, ..., Vn ⊂ Mat2(C).Let us take the ball B = B(0, 1).For a given θ0 ∈ [0, 2π], there exists H having its differentials at n points pi ∈ B(0, 1) satisfying H(pi) = 0 and DHp i = e −i.θ 0 • ṽj by interpolation.Taking sufficiently small balls B1, ..., Bn of radius ν around the points pi, this gives the result for a given θ0 ∈ [0, 2π] and t = 1.Moreover, since the required condition are open, H can be taken uniform on a small interval of values of θ and a small interval (1 − ψ1, 1 + ψ1) of values of t.Then d, ν and ψ1 can be taken locally constant in θ.Since [0, 2π] is compact, we take the maximal value of d and the minimal values of ν and ψ1 on a finite covering of [0, 2π] by intervals where d, ν and ψ1 can be taken constant on each interval of the covering.In particular, since this covering is finite, there exists ψ2 > 0 such that for each θ0 ∈ [0, 2π], it is possible to find constant H, m, ν, ψ1 for every θ ∈ (θ0 − ψ2, θ0 + ψ2).This gives us the result for the fixed ball B(0, 1).Then, the result follows for any ball B(γ, r) with r ≤ 1 by taking the map

Fixing the constants relative to the torus T and the matrix of the linear part A
In the two next subsections, we fix some notation and define a certain number of constants and objects in the following specified order.As a guide for the reader objets denoted in roman letters are relative to P 2 (C), and gothic letters are relative to the torus.
Let us point out that this bound on the radius is still independent of L.

Creating a correcting IFS
We often forget the εi and just denote L , L , L for simplicity when there is no risk of confusion.Notation 4.4.2.We consider the q = q(d) preimages of Π(B) under L included inside Π(B) and the corresponding local inverses (gj) 1≤j≤q of L. We denote by (Gj) 1≤j≤q the corresponding maps on B. For further perturbations L , L , L of L, we consider the analogous objects and we call them (g j ) 1≤j≤q , (g j ) 1≤j≤q , (g j ) 1≤j≤q and (G j ) 1≤j≤q , (G j ) 1≤j≤q , (G j ) 1≤j≤q .
In the following, we will see that (G j ) 1≤j≤q is a correcting IFS.Notation 4.4.3.In the following, we will consider the continuation p(L ) (resp.p(L ), p(L )) of the periodic point pc.This one is well defined according to the implicit function Theorem since pc is repelling.In fact, for the successive perturbations that we will consider, we will always have p(L ) = p(L ) = p(L ) = pc.P 3 ) is defined by : P 1 (z1, z2) = P1(z1, z2) + ε1h(z1, z2)P3(z1, z2)H1(z1, z2) P 2 (z1, z2) = P2(z1, z2) + ε1h(z1, z2)P3(z1, z2)H2(z1, z2) where h was defined in 4.3.18,H in 4.3.21and ε1 ∈ D is such that : 1.For every ε1 ∈ D, L = L ε 1 extends to a holomorphic map of P 2 (C) of the same degree as L and (L ε 1 )ε 1 is a holomorphic family of holomorphic maps of P 2 (C) 2. p(L ) = pc is periodic for L and is in the forward orbit of c : pc = (L ε 1 ) nc (c) and Let first remark that since P3 admits at least nH = (E( m+2K) i ) + 1) factors of degree bounded by i, the degrees of hP3H1 and hP3H2 are bounded by deg(P1) = deg(P2).Since the property of being a holomorphic mapping is open, L is a holomorphic mapping for sufficiently small values of ε1.For simplicity we will suppose that this is true for ε1 ∈ D after rescaling if necessary.Since ε1 is just a linear factor, (L ε 1 )ε 1 is a holomorphic family of holomorphic maps of P 2 (C).Thus item 1 is proven.Item 2 is a consequence of the quadratic terms Q 2 j in h (see 4.3.18).
Proposition 4.4.5.Let L be a Lattès map of degree d > d coming from an affine map on T, of linear part aA.We are working in the chart {[z1, z2, z3] : z3 = 0} defined in 4.2.3.In this chart, L = ( P 1 P 3 , P 2 P 3 ).Let L as in Proposition 4.4.4.Then there exists t > 0 such that for every 0 ≤ p ≤ n, for every real 0 < ε1 < 1, there exists a ball Bp ⊂ B ⊂ C 2 of radius rad(Bp) ≥ ν • rad(B) and a neighborhood Xε 1 of L in Hol d such that for every L ∈ Xε 1 , if j is such that G j (B) ⊂ Bp then G j is quasi-linear of type (tε1, p) (remind that the notion of type was defined in Definition 2.3.1).
Proof.In the following, we omit the index j on gj, g j , Gj and G j and we take 0 < ε1 < 1.Let us remind we work in the chart : [z1, z2, z3] → ( z 1 z 3 , z 2 z 3 ) on P 2 (C).We first show the result for L .We have for every p ∈ B ∩ G (B) : Then we have : • DΠp and : Lemma 4.4.6.For any p ∈ B ∩ G (B) we have : Proof.This is due to the fact that H has been taken in 4.3.21so that − h(Π(p)) Then we have :

The three previous lemmas imply that on
. Then by continuity, for a given ε1 (and then a given L ), there exists a neighborhood Xε 1 of L in Hol d such that for every sufficiently small perturbation L ∈ Xε 1 of L , if j is such that G j (B) ⊂ Bp then G j is quasi-linear of type (tε1, p).The proof of Proposition 4.4.5 is complete.

Well oriented postcritical set
Notation 4.5.1.We fix pc a point of Π −1 (pc) (remind that the periodic point pc was defined in Proposition 3.3.1) .Notation 4.5.2.We denote by PCrit(L) the postcritical set of L, this is the set PCrit(L) = n≥0 (L) n (Crit(L)) where Crit(L) is the critical set of L. The notation will be the same for perturbations L , L , L .
We pick a vector w1 and a value θ satisfying Property (P) of Corollary 2.1.8.Still according to Corollary 2.1.8,there exists an open set of admissible values for w1 so we choose to take it in the following way.The map L npc•ord(A) is an affine map on the torus T of linear part with |a| ≥ 2 (see 3.2.12).Points with dense forward orbit for L npc•ord(A) are dense in T.Moreover, since npc divides npc • ord(A), pc is a fixed point of L npc•ord(A) .We pick w1 such that pc + π(w1) is a point of dense forward orbit for L npc•ord(A) (remind that π : C 2 → T is the natural projection).Since the linear part of L npc•ord(A) is a npc•ord(A) • I2 and pc is a fixed point of L npc•ord(A) , we have that the whole forward orbit of pc + π(w1) under L npc•ord(A) is contained in the line going through pc and pc + π(w1).In particular, this line is dense in the torus T. We pick w2 such that (w1, w2) is a basis of C 2 and π(w2) is not tangent to Π −1 (PCrit(L)) at pc.
Here is the main result of this subsection : whose restriction to B is a (θ, w1)-quasi-diameter (remind that this notion was defined in Definition 2.1.5).
The following lemma is well known.Lemma 4.5.4.Let L be a linear automorphism of C 2 and Γ ⊂ C 2 a complex submanifold through 0 such that : 1. the eigenvalues λ, µ of L are such that |λ| > |µ| > 1.Let w λ and wµ be the respective eigenvectors.
Proof.We can take w λ = e1 and wµ = e2.The eigenvector wµ of µ is transverse to Γ at 0. Then locally Γ is a graph γ over a small disk Dγ ⊂ D : k converges uniformly to 0 on Dγ.Then, for every θ > 0, there exists k such that Proof.Since GLattès is a finite group, we have that A is of finite order.In particular, A ord(G Lattès ) = I2.Then R(A) = 0 where R(X) = X ord(G Lattès ) − 1 has simple roots.Then A is diagonalizable.
The proof will be based on the following well known result (Theorem 6.2.3 in [17]).
The following lemma will be used to compare ϕ f and Π : Lemma 4.5.9.Let F be an invertible map in a neighborhood of 0 with a repelling fixed point at 0. Let us denote the eigenvalues of DF0 by λ, µ.Let us suppose that ϕ1 and ϕ2 are two holomorphic maps conjugating 2 and j ∈ {1, 2}.We have that χ commutes with Diag λ,µ .Then : In particular, since λ, µ = 1 this implies that χ 1 k = χ 2 k = 0 for every |k| > 1.

D((L
Proof.We first make an invertible linear change of coordinates so that in the new coordinates In {x3 = 0}, the critical set of L is the set {Jac(P ) = 0} where Jac(P ) is equal to If this is the case, there is nothing to do and we can take L = L .Let us suppose this is not so.We distinguish two cases.
First case : we suppose that ∂P 2 ∂x 2 = 0.For every ε2 ∈ C we consider the following perturbation of L defined by L ε 1 ,ε 2 = L = ( ∂x 2 .Then, it is non zero for ε2 ∈ D * .This implies that the critical set is not singular at c. Then there is a component of the critical set at c which is not singular.Since DLc, • • • ,DL L npc −1 (pc) are not singular, there is a component of the postcritical set at pc which is not singular.Thus item 2 is true.
Second case : we suppose that ∂P 2 ∂x 2 = 0.For every ε2 ∈ C we consider the following perturbation of L defined by L ε 1 ,ε 2 = L = ( ∂x 1 = 0, this implies that we still have Jac(P )(c) = 0 and the point c is still critical.The only second order partial derivative which depends on ε2 is : ∂x 1 ∂x 2 (c)+γ1(c)• ε2 with γ1(c) = 0. Then the map ε2 → ∂ ∂x 1 (Jac(P ))(c) is a polynomial of degree 2 in ε2 of non zero coefficient of degree 2 equal to (γ1(c)) 2 .Then, rescaling if necessary, it is non zero for ε2 ∈ D * .As in case 1, we conclude that item 2 is satisfied.This concludes the proof of the proposition.
Remind that w1 and w2 were defined just at the beginning of this subsection.The notation p(L ) was introduced in Notation 4.4.3.In the following lemma, we perturb the periodic orbit pc in such a way that we can choose the two eigenvalues at this periodic point .

3. 1 Definitions
Definition 3.1.1.A Lattès map is a holomorphic endomorphisms of P 2 (C) of degree d ≥ 2 which is semi-conjugate to an affine map on the torus.For such a map, we have the following commutative diagram :

Notation 4 . 4 . 1 .
In the following we construct three holomorphic families of holomorphic maps of P 2 (C) which are successive perturbations of L : L
∂P 2 ∂x 1 .The critical set at c is not singular if the gradient of the map (x1, x2) → ∂P 2 ∂x 1 is non zero at c, in particular if :