Julie Déserti
email: deserti@math.univ-paris-diderot.fr

Martin Leguil
email: martin.leguil@imj-prg.fr

DYNAMICS OF A FAMILY OF POLYNOMIAL AUTOMORPHISMS OF C 3 , A PHASE TRANSITION

The polynomial automorphisms of the affine plane have been studied a lot: if f is such an automorphism, then either f preserves a rational fibration, has an uncountable centralizer and its first dynamical degree equals 1, or f preserves no rational curves, has a countable centralizer and its first dynamical degree is > 1. In higher dimensions there is no such description. In this article we study a family (Ψ α ) α of polynomial automorphisms of C 3 . We show that the first dynamical degree of Ψ α is > 1, that Ψ α preserves a unique rational fibration and has an uncountable centralizer. We then describe the dynamics of the family (Ψ α ) α , in particular the speed of points escaping to infinity. We also observe different behaviors according to the value of the parameter α.

INTRODUCTION

Hénon gave families of quadratic automorphisms of the plane which provide examples of dynamical systems with very complicated dynamics ( [START_REF] Hénon | Numerical study of quadratic area-preserving mappings[END_REF][START_REF] Hénon | A two-dimensional mapping with a strange attractor[END_REF][START_REF] Benedicks | The dynamics of the Hénon map[END_REF][START_REF] Devaney | Shift automorphisms in the Hénon mapping[END_REF]). In [START_REF] Friedland | Dynamical properties of plane polynomial automorphisms[END_REF] Friedland and Milnor proved that if f belongs to the group Aut(C 2 ) of polynomial automorphisms of C 2 , then f is Aut(C 2 )-conjugate either to an elementary automorphism (elementary in the sense that they preserve a rational fibration), or to an automorphism of Hénon type, i.e. to g 1 g 2 . . . g k , g j : (z 0 , z 1 ) → (z 1 , P j (z 1 ) -δ j z 0 ),

δ j ∈ C * , P j ∈ C[z 1 ], deg P j ≥ 2.
The topological entropy allows to measure chaotic behaviors. In dimension 1 the topological entropy of a rational fraction coincides with the logarithm of its degree; but the algebraic degree of a polynomial automorphism of C 2 is not invariant under conjugacy so [START_REF] Russakovskii | Value distribution for sequences of rational mappings and complex dynamics[END_REF][START_REF] Friedland | Entropy of algebraic maps[END_REF] introduce the first dynamical degree. The topological entropy is equal to the logarithm of the first dynamical degree ( [START_REF] Smillie | The entropy of polynomial diffeomorphisms of C 2 . Ergodic Theory Dynam[END_REF][START_REF] Bedford | Polynomial diffeomorphisms of C 2 . III. Ergodicity, exponents and entropy of the equilibrium measure[END_REF]). If f is of Hénon type, then the first dynamical degree of f is equal to its algebraic degree ≥ 2; on the other hand if f is conjugate to an elementary automorphism, then its first dynamical degree is 1 (see [START_REF] Friedland | Dynamical properties of plane polynomial automorphisms[END_REF]). Another criterion to measure chaos is the size of the centralizer of an element. The group Aut(C 2 ) has a structure of amalgamated product ( [START_REF] Jung | Über ganze birationale Transformationen der Ebene[END_REF]) hence according to [START_REF] Serre | Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass[END_REF] this group acts non trivially on a tree. Using this action Lamy proved that a polynomial automorphism is of Hénon type if and only if its centralizer is countable ( [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]).

The group Aut(C 3 ) and the dynamics of its elements are much less-known. In this article we study the properties of the family of polynomial automorphisms of C 3 given by Ψ α : (z 0 , z 1 , z 2 ) → (z 0 + z 1 + z q 0 z d 2 , z 0 , αz 2 ) where α denotes a nonzero complex number with modulus ≤ 1, q an integer ≥ 2, and d an integer ≥ 1.

The automorphism Ψ α can be seen as a skew product over the map z 2 → αz 2 , and whose dynamics in the fibers is given by automorphisms of Hénon type. More precisely, if z 2 ∈ C, let us denote ψ z 2 : (z 0 , z 1 ) → (z 0 + z 1 + z q 0 z d 2 , z 0 ); then Ψ α (z 0 , z 1 , z 2 ) = (ψ z 2 (z 0 , z 1 ), αz 2 ), and for every n ≥ 1, we have Ψ n α (z 0 , z 1 , z 2 ) = ((ψ z 2 ) n (z 0 , z 1 ), α n z 2 ), where

(ψ z 2 ) n = ψ α n-1 z 2 • • • • • ψ αz 2 • ψ z 2 .
(1.1)

If α = 0, we also define the map φ α := α l ψ 1 where l := d/(q -1). We will see later on that Ψ α is semiconjugate to φ α . The family of automorphisms {Ψ α } α satisfies the following properties:

Proposition A. Take 0 < |α| ≤ 1. Then -the first dynamical degree of the automorphism Ψ α (resp. Ψ -1 α ) is q ≥ 2; -the centralizer of Ψ α is uncountable; -if 0 < |α| ≤ 1, then Ψ α preserves a unique rational fibration, {z 2 = cst}.

We then focus on the dynamics of Ψ α , 0 < |α| ≤ 1. Let us introduce a definition. We denote by ϕ := 1+ √ 5 2 the golden ratio. We say that the forward orbit of p goes to infinity with Fibonacci speed if the sequence (Ψ n α (p)ϕ -n ) n≥0 converges and lim n→+∞ Ψ n α (p)ϕ -n = p ′ = 0 C 3 . In particular this implies

Ψ n α (p) ∼ p ′ ϕ n .
The hypersurface {z 2 = 0} is fixed by Ψ α , and the induced map on it is a linear Anosov diffeomorphism. We see that for any p ∈ {z 2 = 0}, either its forward orbit goes to 0 C 3 exponentially fast, or it escapes to infinity with Fibonacci speed. Concerning points escaping to infinity with maximal speed, we prove:

Theorem B. Fix 0 < |α| ≤ 1. For any point p ∈ C 3 , the limit lim

n→+∞ log + ||Ψ n α (p)|| q n
exists. The function

G + Ψ α (p) = lim n→+∞ log + ||Ψ n α (p)|| q n
is plurisubharmonic, Hölder continuous, and satisfies

G + Ψ α • Ψ α = q • G + Ψ α . Set l := 2 max d q-1 , 1 ; then 1 ≤ lim sup p →+∞ G + Ψ α (p) log p ≤ l.
Moreover, the set E := p ∈ C 3 | G + Ψ α (p) > 0 of points escaping to infinity with maximal speed is open, connected, and has infinite Lebesgue measure on any complex line where G + Ψ α is not identically zero. In particular, the set p ∈ C 3 | lim n→+∞ Ψ n α (p) = +∞ is of infinite measure. We also exhibit an explicit open set Ω ⊂ E.

Theorem C. Assume 0 < |α| < 1. Then Ψ α has a unique periodic point at finite distance, 0 C 3 = (0, 0, 0), which is a saddle point of index 2. The fixed hypersurface {z 2 = 0} attracts any other point. Moreover, the set K + Ψ α of points with bounded forward orbit is exactly the stable manifold W s Ψ α (0 C 3 ), and the latter can be characterized analytically. The set J + Ψ α := ∂K + Ψ α thus corresponds to W s Ψ α (0 C 3 ).

We observe a phase transition in the dynamics of the family {Ψ α } 0<|α|≤1 for the value |α| = ϕ (1-q)/d : Theorem D. Assume 0 < |α| < ϕ (1-q)/d . The set V := p ∈ C 3 | G + Ψ α (p) = 0 is a closed neighborhood of the hyperplane {z 2 = 0}. It consists in the disjoint union Ω ′′ ⊔W s Ψ α (0 C 3 ), where Ω ′′ has non-empty interior and the forward orbit of any point p ∈ Ω ′′ goes to infinity with Fibonacci speed.

Note that we also define an analytic function g whose domain of definition is equal to V , and which parametrizes the stable manifold in the sense that W s Ψ α (0 C 3 ) coincides with the zero set Z of g. Theorem E. Assume now ϕ (1-q)/d < |α| < 1. For any p ∈ C 3 , exactly one of the following cases occurs:

-either p ∈ W s Ψ α (0 C 3 ) and its forward orbit converges to 0 C 3 exponentially fast; -or p ∈ {z 2 = 0} W s Ψ α (0 C 3 ) and it goes to infinity with Fibonacci speed; -or the speed explodes:

G + Ψ α (p) > 0.
In particular, contrary to the previous situation where the set Ω ′′ has non-empty interior, we see here that Fibonacci speed does not occur outside the hypersurface {z 2 = 0}. Remark 1.1. We stress the fact that for any 0 < |α| < 1, the forward orbit of a point under Ψ α is bounded if and only if it goes to 0 C 3 . Moreover, we see that in the case the orbit is unbounded, it has to escape to infinity. This rigidity phenomenon is related to the properties of the automorphism of Hénon type φ α to which Ψ α is semi-conjugate, and which possesses an attractor at infinity such that the positive iterates of any point whose forward orbit is not bounded escape to it.

Theorem F. Assume |α| = 1. We define K Ψ α to be the set of points p ∈ C 3 whose orbit (Ψ n α (p)) n∈Z is bounded. Similarly to what we did above, we define the Green function G - Ψ α . Then for any point p ∈ C 3 , exactly one of the following assertions is satisfied:

-either the orbit of p is bounded, i.e. p ∈ K Ψ α ; -or p ∈ {z 2 = 0} {0 C 3 } and either its forward orbit or its backward orbit escapes to infinity with Fibonacci speed; -or G + Ψ α (p) > 0 or G - Ψ α (p) > 0; in this case, either its forward orbit or its backward orbit escapes to infinity with maximal speed. We define the associate Green currents T ± Ψ α := dd c G ± Ψ α , and we set µ

Ψ α := T + Ψ α ∧ T - Ψ α ∧ dz 2 ∧ dz 2 .
The measure µ Ψ α is invariant by Ψ α and supported on the Julia set J Ψ α := ∂K Ψ α . For any p 2 = 0, the set

C p 2 := C 2 × p 2 e ix | x ∈ R is invariant under Ψ α . Define J p 2 := J Ψ α ∩ C p 2 ;
it is also invariant and we show that when α is not a root of unity, (Ψ α J p 2 , µ Ψ α ) is ergodic.

Acknowledgement. The first author would like to thank Artur Avila for helpful and fruitful discussions.

INVARIANT FIBRATIONS AND DEGREE GROWTHS

2.1. Invariant fibrations. Let us come back to dimension 2 for a while. As recalled in §1 if f is a polynomial automorphism of C 2 , then up to conjugacy either f is an elementary automorphism or f is of Hénon type. In the first case f preserves a rational fibration, whereas in the second one f does not preserve any rational curve ( [START_REF] Brunella | Minimal models of foliated algebraic surfaces[END_REF]). This gives a geometric criterion to distinguish maps of Hénon type and elementary maps. What about dimension n ≥ 3? Contrary to the 2-dimensional case we will see, as soon as n = 3, that "no invariant rational curve" does not mean "first dynamical degree > 1".

Assume that 0 < |α| ≤ 1. Let Φ α be the automorphism of C 3 given by Φ α = (α l (z 0 + z 1 + z q 0 ), α l z 0 , αz 2 ) where l := d q-1 . It is possible to show that Ψ α is conjugate to Φ α through the birational map of P 3 C given in the affine chart

z 3 = 1 by θ = (z 0 z l 2 , z 1 z l 2 , z 2 ), that is θ • Ψ α = Φ α • θ.
The advantage is that the action of Φ α in the fibers is independent of the base point. Moreover, it has a lot of good properties; in particular, we will see that it is algebraically stable ( §2.2). Nevertheless θ is birational so we might loose some information ( §6).

Proposition 2.1. For any 0 < |α| ≤ 1, the polynomial automorphism Φ α preserves a unique rational fibration, the fibration given by {z 2 = cst}.

Corollary 2.2. For any 0 < |α| ≤ 1, the polynomial automorphism Ψ α preserves a unique rational fibration, the fibration given by {z 2 = cst}.

Proof of Proposition 2.1. Note that Φ α = (φ α (z 0 , z 1 ), αz 2 ) where φ α ∈ Aut(C 2 ) is the automorphism of Hénon type given by φ α : (z 0 , z 1 ) → α l (z 1 + z 0 + z q 0 , z 0 ). Since φ α does not preserve rational curves ( [START_REF] Brunella | Minimal models of foliated algebraic surfaces[END_REF]) the only invariant rational fibration is {z 2 = cst}.

2.2.

Degrees and degree growths. The results in this part hold for any α = 0. As we saw in §1 the first dynamical degree is an important invariant; in this section we will thus compute λ(Ψ ±1 α ) and λ(Φ ±1 α ). Let us first mention a big difference between dimension 2 and higher dimensions: if f belongs to Aut(C 2 ) then deg f = deg f -1 . This equality does not necessarily hold in higher dimension; nevertheless if f belongs to Aut(C n ), then deg f ≤ (deg f -1 ) n-1 and deg f -1 ≤ (deg f ) n-1 (see [START_REF] Sibony | Dynamique des applications rationnelles de P k . In Dynamique et géométrie complexes[END_REF]).

Lemma 2.3. We have for any

n ≥ 0 both deg(Ψ n α ) = q n + d × q n -1 q -1 and deg(Ψ -n α ) = deg(Ψ n α ).
Proof. Let us denote by (P

(n) α ) n≥-1 the sequence of polynomials where      P (-1) α (z 0 , z 1 , z 2 ) := z 1 P (0) α (z 0 , z 1 , z 2 ) := z 0 ∀ n ≥ 0 P (n+1) α := P (n) α + P (n-1) α + (P (n) α ) q (α n z 2 ) d .
In particular, for every n ≥ 0,

Ψ n α (z 0 , z 1 , z 2 ) = P (n) α (z 0 , z 1 , z 2 ), P (n-1) α (z 0 , z 1 , z 2 ), α n z 2 .
Since the degree of the third component does not change, and the second component is just the first one at time n -1, the growth of the degree is supported by the first component, that is deg(

Ψ n α ) = deg(P (n) α ).
Let us then show the result by induction on n. The result is true for n = 0. If it holds for n ≥ 0, then we have deg(P

(n+1) α ) = deg (P (n) α ) q (α n z 2 ) d = q q n + d × q n -1 q -1 + d = q n+1 + d × q n+1 -1 q -1 .
Since the degree is not invariant under conjugacy, [START_REF] Russakovskii | Value distribution for sequences of rational mappings and complex dynamics[END_REF][START_REF] Friedland | Entropy of algebraic maps[END_REF] introduce the first dynamical degree. If f is a polynomial automorphism of C 3 , the first dynamical degree of f is defined by

λ( f ) = lim n→+∞ (deg f n ) 1/n .

It satisfies the following inequalities

: 1 ≤ λ( f ) ≤ deg f . Corollary 2.4. Since q n ≤ deg(Ψ n α ) ≤ (d + 1)q n , it follows that λ(Ψ α ) = λ(Ψ -1 α ) = q.
To any polynomial automorphism f = ( f 0 , f 1 , f 2 ) of C 3 of degree d one can associate a birational self map of P 3 C as follows

(z 0 : z 1 : z 2 : z 3 ) z d 3 f 0 z 0 z 3 , z 1 z 3 , z 2 z 3 : z d 3 f 1 z 0 z 3 , z 1 z 3 , z 2 z 3 : z d 3 f 2 z 0 z 3 , z 1 z 3 , z 2 z 3 : z d 3 ;
we still denote it by f . If g = (g 0 : g 1 : g 2 : g 3 ) is a birational self map of P 3 C , the indeterminacy set Ind(g) of g is the set where g is not defined, that is

Ind(g) = m ∈ P 3 C | g 0 (m) = g 1 (m) = g 2 (m) = g 3 (m) = 0 .
Remark that if we look at a birational map of P 3 C that comes from a polynomial automorphism of C 3 then its indeterminacy set is contained in {z 3 = 0}.

A polynomial automorphism of C 3 is algebraically stable if for every n ∈ N,

f n {z 3 = 0} Ind( f n ) ⊂ Ind( f ).
Let us recall the following result.

Proposition 2.5 ([9]). The map f is algebraically stable if and only if deg(

f n ) = (deg( f )) n for every n ≥ 1.
Lemma 2.3 and Proposition 2.5 imply that Ψ α is not algebraically stable, as well as Ψ -1 α . It can also be seen directly from the definition. Indeed the map

Ψ α = (z 0 + z 1 )z q+d-1 3 + z q 0 z d 2 : z 0 z q+d-1 3 : αz 2 z q+d-1 3 : z q+d 3
sends z 3 = 0 onto (1 : 0 : 0 : 0) and Ind(Ψ α ) = {z 0 = 0, z 3 = 0} ∪ {z 2 = 0, z 3 = 0}. Similarly, we see that

Ind(Ψ -1 α ) = {z 1 = 0, z 3 = 0} ∪ {z 2 = 0, z 3 = 0},
and Ψ -1 α sends z 3 = 0 onto (0 : 1 : 0 : 0) ∈ Ind(Ψ -1 α ). On the other hand Φ α ({z 0 = 0, z 3 = 0}) = (1 : 0 : 0 : 0) does not belong to Ind(Φ α ) and (1 : 0 : 0 : 0) is a fixed point of Φ α hence Φ n α ({z 0 = 0, z 3 = 0}) = (1 : 0 : 0 : 0) for any n ≥ 1. In particular, Φ α is algebraically stable. We have for every n ≥ 0, deg(Φ n α ) = q n . Notice that for n ≥ 3, there exist examples of maps f ∈ Aut(C 3 ) which are algebraically stable but whose inverse f -1 is not algebraically stable (let us mention the following example due to Guedj: f = (z 2 0 + λz 1 + az 2 , λ -1 z 2 0 + z 1 , z 0 ) with a and λ in C * ). Yet this is not the case for Φ α . Indeed,

Φ -1 α (z 0 : z 1 : z 2 : z 3 ) = z 1 z q-1 3 α l : - z 1 z q-1 3 α l + z 0 z q-1 3 α l - z q 1
α lq :

z 2 z q-1 3 α : z q 3 so Φ -1 α ({z 1 = 0, z 3 = 0}) = (0 : 1 : 0 : 0) does not belong to Ind(Φ -1 α ) = {z 1 = 0, z 3 = 0} and is fixed by Φ -1
α . Hence Φ -1 α is also algebraically stable. As a result one can state:

Proposition 2.6. For any integer n ≥ 1 the following equalities hold

deg(Φ n α ) = deg(Φ -n α ) = q n , λ(Φ α ) = λ(Φ -1 α ) = q.

CENTRALIZERS

If G is a group and f an element of G, we denote by Cent( f , G) the centralizer of f in G, that is

Cent( f , G) := g ∈ G | f g = g f .
The description of centralizers of discrete dynamical systems is an important problem in real and complex dynamics: Julia ( [START_REF] Julia | Mémoire sur la permutabilité des fractions rationnelles[END_REF]) and Ritt ([19]) showed that the centralizer of a rational function f of P 1 is in general the set of iterates of f (we then say that the centralizer of f is trivial) except for some very special f . Later Smale asked if the centralizer of a generic diffeomorphism of a compact manifold is trivial ( [START_REF] Smale | Differentiable dynamical systems[END_REF]). Since then a lot of mathematicians have looked at this question in different contexts; for instance as recalled in §1 Lamy has proved that the centralizer of a polynomial automorphism of C 2 of Hénon type is in general trivial ( [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]).

Fix α with 0 < |α| ≤ 1. We would like to describe Cent(Φ α , Aut(C 3 )). Of course it contains Φ n α | n ∈ Z but also the following one-parameter family

(ηz 0 , ηz 1 , νz 2 ) | ν ∈ C * , η a (q -1)-th root of unity .
We show that the centralizer is essentially reduced to the iterates of Φ α and such maps. Since the automorphism φ α = α l (z 1 + z 0 + z q 0 , z 0 ) is of Hénon type, it follows from a result of Lamy [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF] that Cent(φ α , Aut(C 2 )) ≃ Z ⋊ Z n for some n ∈ N.

Let f ∈ Cent(Φ α , Aut(C 3 )); we write f = ( f 0 , f 1 , f 2 ). Lemma 3.1. We have ∂ f 2 ∂z 0 = 0, ∂ f 2 ∂z 1 = 0.
Therefore, the last component f 2 only depends on z 2 , and in fact it is a homothety:

f 2 (z 0 , z 1 , z 2 ) = f 2 (z 2 ) = µz 2 , µ ∈ C * .
Proof. If we focus on the third coordinate in relation

Φ α • f = f • Φ α , we get α f 2 = f 2 • Φ α , that is, for every (z 0 , z 1 , z 2 ) ∈ C 3 , α f 2 (z 0 , z 1 , z 2 ) = f 2 (α l z 0 + α l z 1 + α l z q 0 , α l z 0 , αz 2 ).
Taking the derivatives in the different coordinates, we obtain:

     α ∂ f 2 ∂z 0 = α l (1 + qz q-1 0 ) ∂ f 2 ∂z 0 • Φ α + α l ∂ f 2 ∂z 1 • Φ α , α ∂ f 2 ∂z 1 = α l ∂ f 2 ∂z 0 • Φ α , α ∂ f 2 ∂z 2 = α ∂ f 2 ∂z 2 • Φ α . (3.1)
Let us consider the first coordinate, and assume that ∂ f 2 ∂z 0 = 0; we will get a contradiction by looking at highest-order terms in z 0 . Since

f 2 ∈ C[z 1 , z 2 ][z 0 ], we can write f 2 (z 0 , z 1 , z 2 ) = ∑ k≤k 0 R k (z 1 , z 2 )z k 0 ,
where the R k are polynomials and k 0 is the degree in z 0 of f 2 . From our hypothesis, k 0 ≥ 1. We also look at the expansion of R k 0 = 0:

R k 0 (z 1 , z 2 ) = ∑ m≤m 0 Q m (z 2 )z m 1 , Q m 0 = 0.
For the three terms, we look at the term of highest order in z 0 :

     α ∂ f 2 ∂z 0 (z 0 , z 1 , z 2 ) = αk 0 R k 0 (z 1 , z 2 )z k 0 -1 0 + . . . α l (1 + qz q-1 0 ) ∂ f 2 ∂z 0 • Φ α (z 0 , z 1 , z 2 ) = qk 0 α l(k 0 +m 0 ) Q m 0 (αz 2 )z qk 0 +m 0 -1 0 + . . . α l ∂ f 2 ∂z 1 • Φ α (z 0 , z 1 , z 2 ) = m 0 α l(k 0 +m 0 ) Q m 0 (αz 2 )z qk 0 +m 0 -1 0 + . . .
Since we assume k 0 ≥ 1, and q > 1, we have

qk 0 + m 0 -1 > k 0 -1 so the coefficient of the term in z qk 0 +m 0 -1 0 must vanish. But this coefficient is (qk 0 + m 0 )α l(k 0 +m 0 ) Q m 0 (αz 2 ) = 0, a contradiction. Hence ∂ f 2 ∂z 0 = 0, and it follows from the second equation of (3.1) that ∂ f 2 ∂z 1 = 0 as well. Therefore, f 2 = f 2 (z 2 ).
Now, since f ∈ Aut(C 3 ), we know that f 2 is of degree at most 1. The map f commutes with Φ α , so it must preserve its fixed point 0 C 3 , and we conclude that f 2 :

z 2 → µz 2 for some µ ∈ C * . Recall that φ α : (z 0 , z 1 ) → α l (z 1 + z 0 + z q 0 , z 0 ). Let us denote f := ( f 0 , f 1 )
. By projecting the commutation relation on the first two coordinates, we get

φ α • f = f • Φ α . (3.2) Lemma 3.2.
The map f only depends on the first two coordinates.

Proof. We rewrite (3.2) as the following system:

α l f 0 + α l f 1 + α l f q 0 = f 0 • Φ α α l f 0 = f 1 • Φ α . (3.3)
We then get:

α l f 0 • Φ α + α 2l f 0 + α l f q 0 • Φ α = f 0 • Φ 2 α .
Let d 0 be the degree of

f 0 ∈ C[z 0 , z 1 ][z 2 ]
. Since Φ α does not change the degree in z 2 , we obtain

deg(α l f 0 • Φ α ) = deg(α 2l f 0 ) = deg( f 0 • Φ 2 α ) = d 0 , deg(α l f q 0 • Φ α ) = d q 0 , but q > 1
, which implies that d 0 = 0: f 0 does not depend on z 2 . Using the second equation of (3.3), we see that f 1 does not depend on z 2 either. Therefore, Equation (3.2) can be rewritten:

φ α • f = f • φ α .
But φ α is a Hénon automorphism, so according to [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF] one has:

Corollary 3.3. The map f belongs to the countable set Cent(φ α , Aut(C 2 )) ≃ Z ⋊ Z n , n ∈ N.
We have seen that for any f = (

f 0 , f 1 , f 2 ) ∈ Cent(Φ α , Aut(C 3 )), ( f 0 , f 1 )
depends only on (z 0 , z 1 ) and belongs to Cent(φ α , Aut(C 2 )), and that f 2 depends only on z 2 and is a homothety. We conclude:

Proposition 3.4. The centralizer of Φ α in Aut(C 3 ) is uncountable. More precisely Cent(Φ α , Aut(C 3 )) = Cent(φ α , Aut(C 2 )) × z 2 → µz 2 | µ ∈ C * ≃ (Z ⋊ Z n ) × C * , n ∈ N. Corollary 3.5. The centralizer of Ψ α in Aut(C 3 ) is uncountable. 4. DYNAMICS ON THE INVARIANT HYPERSURFACE z 2 = 0
The following holds for any α = 0. Let us recall that the Fibonacci sequence is the sequence (F n ) n defined by: F 0 = 0, F 1 = 1 and for all n ≥ 2

F n = F n-1 + F n-2 .
The hypersurface {z 2 = 0} is invariant, and when |α| < 1, it attracts every point p ∈ C 3 . On restriction to this hypersurface, the growth is given by the Fibonacci numbers (F n ) n :

Ψ n α z 2 =0 = (F n+1 z 0 + F n z 1 , F n z 0 + F n-1 z 1 ), n ≥ 1. (4.1) Since Ψ -1 α (z 0 , z 1 , z 2 ) = z 1 , -z 1 + z 0 -z q 1 z d 2 α d , z 2 α ,
similarly, we have

Ψ -n α z 2 =0 : (z 0 , z 1 ) → (-1) n (F n-1 z 0 -F n z 1 , -F n z 0 + F n+1 z 1 ), n ≥ 1. (4.2)
Moreover, it is easy to see that any periodic point of Ψ α belongs to the hypersurface {z 2 = 0}. In fact, Ψ α has a unique fixed point at finite distance, 0 C 3 = (0, 0, 0), and has no periodic point of period larger than 1.

Let

ϕ := 1+ √ 5 2
be the golden ratio and ϕ ′ := -1/ϕ. Since

F n = ϕ n -(ϕ ′ ) n √ 5 = ϕ n √ 5 + o(1),
we deduce from (4.1) that any point (ϕ ′ z, z, 0) with z ∈ C converges to 0 C 3 when we iterate Ψ α , while any other point of the form (βz, z, 0) with z = 0 and β = ϕ ′ goes to infinity. Likewise, we see from (4.2) that any point (ϕz, z, 0) with z ∈ C converges to 0 C 3 when we iterate Ψ -1 α , while any other point of the form (βz, z, 0) with z = 0 and β = ϕ goes to infinity. Furthermore, in both cases, the speed of the convergence is exponential since it is in O(|ϕ| -n ) with |ϕ| > 1. In other terms, the linear map Ψ α z 2 =0 : (z 0 , z 1 ) → (z 0 + z 1 , z 0 ) is hyperbolic, with a unique fixed point 0 C 2 = (0, 0) of saddle type, and whose stable, respectively unstable manifolds correspond to the following lines:

W s Ψ α z 2 =0 (0 C 2 ) = ∆ ϕ ′ := ϕ ′ z, z z ∈ C , W u Ψ α z 2 =0 (0 C 2 ) = ∆ ϕ := (ϕz, z) z ∈ C .
Moreover, ϕ and ϕ ′ are just the eigenvalues of Ψ α z 2 =0 , and ∆ ϕ , ∆ ϕ ′ the corresponding eigenspaces.

POINTS WITH BOUNDED FORWARD ORBIT, DESCRIPTION OF THE STABLE MANIFOLD

W s Ψ α (0 C 3 )
When 0 < |α| < 1, we remark that 0 C 3 is a hyperbolic fixed point of saddle type. The tangent space at 0 C 3 can be written as

T 0 C 3 (C 3 ) = E s Ψ α (0 C 3 ) ⊕ E u Ψ α (0 C 3 )
, where the stable, respectively unstable spaces are given by

E s Ψ α (0 C 3 ) = ∆ ϕ ′ × {0} ⊕ {0 C 2} × C, E u Ψ α (0 C 3 ) = ∆ ϕ × {0}.

These spaces integrate to stable and unstable manifolds

W s Ψ α (0 C 3 ) := p ∈ C 3 | lim n→+∞ Ψ n α (p) = 0 C 3 , W u Ψ α (0 C 3 ) := p ∈ C 3 | lim n→+∞ Ψ -n α (p) = 0 C 3 which are invariant by the dynamics; furthermore, W u Ψ α (0 C 3 ) = ∆ ϕ × {0}, while W s Ψ α (0 C 3 ) is biholomorphic to C 2 (see [22]). Note that ∆ ϕ ′ × {0} ∪ {0 C 2 } × C ⊂ W s Ψ α (0 C 3 ), but it is easy to see 1 that W s Ψ α (0 C 3 ) = ∆ ϕ ′ × C. 1. Indeed if p = (p 0 , p 1 , p 2 ) satisfies p 2 = 0 and p 1 = -ϕp 0 , we see that P (0) α (p) + ϕP (1) α (p) = p 0 (1 + ϕ -ϕ 2 ) + ϕp q 0 p d 2 = ϕp q 0 p d 2 = 0 hence ∆ ϕ ′ × C is not left invariant by Ψ α .
In the next statement, we introduce a series that encodes the growth of forward iterates of a point. Lemma 5.1. Let p = (p 0 , p 1 , p 2 ) ∈ C 3 . For every n ≥ 0 and any α ∈ C, we have

P (n+1) α (p) + ϕ -1 P (n) α (p) = ϕ n ϕp 0 + p 1 + p d 2 n ∑ j=0 P ( j) α (p) q ϕ -j α jd . (5.1)
Proof. For n ≥ 0 we have the following set of equalities:

P (n+1) α (p) = P (n) α (p) + P (n-1) α (p) + (P (n) α (p)) q (α n p 2 ) d (× ϕ) P (n) α (p) = P (n-1) α (p) + P (n-2) α (p) + (P (n-1) α (p)) q (α n-1 p 2 ) d (× ϕ 2 ) P (n-1) α (p) = P (n-2) α (p) + P (n-3) α (p) + (P (n-2) α (p)) q (α n-2 p 2 ) d . . . = . . . + . . . + . . . (× ϕ n-1 ) P (2) α (p) = P (1) α (p) + p 0 + (P (1) α (p)) q (αp 2 ) d (× ϕ n ) P (1) α (p) = p 0 + p 1 + (P (0) α (p)) q (p 2 ) d
Summing up, and because ϕ 2 -ϕ -1 = 0, we obtain

P (n+1) α (p) + ϕ -1 P (n) α (p) = ϕ n ϕp 0 + p 1 + p d 2 n ∑ j=0 P ( j) α (p) q ϕ -j α jd .
For every n ≥ -1, we define the polynomial

g n ∈ C[z] = C[z 0 , z 1 , z 2 ] by g n (z) := ϕz 0 + z 1 + z d 2 n ∑ j=0 P ( j) α (z) q ϕ -j α jd = P (n+1) α (z) + ϕ -1 P (n) α (z) ϕ -n .
We also introduce the power series

g(z) := ϕz 0 + z 1 + z d 2 +∞ ∑ j=0 P ( j) α (z) q ϕ -j α jd = ϕz 0 + z 1 + +∞ ∑ j=-1 ϕ -( j+1) P ( j+2) α (z) -P ( j+1) α (z) -P ( j) α (z) .
Let us denote by D its domain of definition, that is the set of p ∈ C 3 such that the series ∑ j P ( j) α (p) q ϕ -j α jd converges, and let

Z := p ∈ D | g(p)
= 0 be the set of its zeroes. It is easy to check that both D and Z are invariant by the dynamics, that is

Ψ α (D) ⊂ D and Ψ α (Z) ⊂ Z. Moreover, if p ∈ D, we denote by r n (p) := ∑ j≥n+1 P ( j) α (p)
q ϕ -j α jd the tail of the corresponding series.

Corollary 5.2. Suppose 0 < |α| ≤ 1. Let K + Ψ α denote the set of points p = (p 0 , p 1 , p 2 ) whose forward orbit {Ψ n α (p), n ≥ 0} is bounded. This is equivalent to the fact that the sequence

|P (n) α (p)| n≥0 is bounded. If p ∈ K + Ψ α , then for every n ≥ 0, we have |g n (p)| = O(ϕ -n ). (5.2)
In particular we deduce that

K + Ψ α ⊂ Z, and |r n (p)| = O(ϕ -n ).
(5.3)

Proof. It follows immediately from Lemma 5.1. Indeed under our assumptions we have:

|g n (p)| ≤ (|P (n+1) α (p)| + ϕ -1 |P (n) α (p)|)ϕ -n = O(ϕ -n ).
This implies p ∈ Z. Then we also have

g n (p) = g(p) -p d 2 r n (p) = -p d 2 r n (p) and |r n (p)| = O(ϕ -n
). Remark 5.3. We can see (5.3) as a codimension one condition that points with bounded forward orbit have to satisfy. Also, we see from (5.2) that locally, such points are close to the analytic manifold Z n := p ∈ C 3 | g n (p) = 0 for n ≥ 0 big. If p = (p 0 , p 1 , 0), we recover from (5.3) that p has bounded forward orbit if and only if it belongs to the stable manifold W s

Ψ α (0 C 3 ) ∩ {z 2 = 0} = ∆ ϕ ′ × {0}.
When 0 < |α| < 1, we have the following analytic characterization of the stable manifold W s Ψ α (0 C 3 ).

Proposition 5.4. Assume 0 < |α| < 1. The point p = (p 0 , p 1 , p 2 ) ∈ C 3 belongs to the stable manifold W s Ψ α (0 C 3 ) if and only if the following properties hold:

-p ∈ Z;

-the series ∑ j |r j (p)|ϕ j is convergent.

Equivalently, p ∈ W s Ψ α (0 C 3 ) if and only if ∑ j |g j (p)|ϕ j converges.

Proof. If p belongs to the stable manifold W s Ψ α (0 C 3 ), then its forward orbit is bounded and Corollary 5.2 tells us that p ∈ Z. Moreover we have

|r n (p)| = O ∑ j≥n+1 ϕ -j α jd = O(ϕ -n |α| nd ), hence |r n (p)|ϕ n = O(|α| nd
) and the series ∑ j |r j (p)|ϕ j converges.

For the other implication, we get from Lemma 5.1 that for every j ≥ 0,

P ( j+1) α (p) + ϕ -1 P ( j) α (p) = -p d 2 ϕ j r j (p).
(5.4)

Now let n ≥ 0. Write equations (5.4) for j = 0, . . . , n and combine them to obtain

P (n+1) α (p) = (-1) n+1 ϕ n+1 p 0 + p d 2 n ∑ j=0 (-1) n+ j+1 r j (p)ϕ j ϕ j-n .
The first term of the right hand side goes to 0 with n; we split the sum as follows:

n ∑ j=0 (-1) n+ j+1 r j (p)ϕ j ϕ j-n = ⌊n/2⌋ ∑ j=0 (-1) n+ j+1 r j (p)ϕ j ϕ j-n + n ∑ j=⌊n/2⌋+1
(-1) n+ j+1 r j (p)ϕ j ϕ j-n .

We get

⌊n/2⌋ ∑ j=0 (-1) n+ j+1 r j (p)ϕ j ϕ j-n ≤ ϕ ⌊n/2⌋-n +∞ ∑ j=0 |r j (p)|ϕ j
hence it goes to 0 with respect to n. For the remaining term, we estimate

n ∑ j=⌊n/2⌋+1 (-1) n+ j+1 r j (p)ϕ j ϕ j-n ≤ +∞ ∑ j=⌊n/2⌋+1
|r j (p)|ϕ j , which goes to 0 as well. We conclude that lim

n→+∞ P (n) α (p) = 0, hence p ∈ W s Ψ α (0 C 3 ).
The other equivalence follows from the fact that for p ∈ Z, we have g n (p) = -p d 2 r n (p).

Corollary 5.5. Assume 0 < |α| < 1. Then the forward orbit of a point p = (p 0 , p 1 , p 2 ) is bounded if and only if p belongs to the stable manifold

W s Ψ α (0 C 3 ); in other terms, K + Ψ α = W s Ψ α (0 C 3 ). Proof. Let p ∈ K + Ψ α .
We have already seen in Corollary 5.2 that p ∈ Z. Moreover, if we denote r n (p) :=

∑ j≥n+1 (P ( j) α (p)) q ϕ -j α jd , then ∑ j |r j (p)|ϕ j is convergent since |r j (p)| = O(ϕ -j |α| jd ). Then, Proposition 5.4 tells us that p ∈ W s Ψ α (0 C 3 )
. The other implication is straightforward.

Lemma 5.6. Assume now |α| = 1. We have seen that p ∈ K +

Ψ α implies that p ∈ Z and |r n (p)| = O(ϕ -n ). Conversely, if p ∈ Z and ∑ j |r j (p)|ϕ j is convergent, then p ∈ K + Ψ α .
Proof. Assume that p ∈ Z and that ∑ j |r j (p)|ϕ j converges. As previously, we have for every n ≥ 0: 1) .

P (n) α (p) = (-1) n ϕ n p 0 + p d 2 n-1 ∑ j=0 (-1) n+ j r j (p)ϕ j ϕ j-(n-
From our assumption we deduce that |P

(n) α (p)| ≤ |p 0 | + |p 2 | d +∞ ∑ j=0 |r j (p)|ϕ j , hence |P (n) α (p)| = O(1).

BIRATIONAL CONJUGACY, DYNAMICAL PROPERTIES OF AUTOMORPHISMS OF HÉNON TYPE

Assume 0 < |α| ≤ 1. We recall here some facts concerning the dynamics of the Hénon automorphism φ α , and so on the dynamics of Φ α = (φ α , αz 2 ). Denote by F + φ α the largest open set on which (φ n α ) n is locally equicontinuous, by K + φ α the set of points p ∈ C 2 such that (φ n α (p)) n≥0 is bounded and by J + φ α its topological boundary. The automorphism φ α is regular, that is, Ind(φ α ) ∩ Ind(φ -1 α ) = / 0. In particular, it is algebraically stable, hence the following limit exists and defines a Green function:

G + φ α (z 0 , z 1 ) := lim n→+∞ log + ||φ n α (z 0 , z 1 )|| q n .

It satisfies the invariance property G

+ φ α • φ α = q • G + φ α .
We define the associate current T + φ α = dd c G + φ α , where

d c = i(∂-∂) 2π .
Of course there are similar objects

F - φ α , K - φ α , J - φ α , G - φ α and T - φ α associated to the inverse map φ -1 α ; we also set K φ α := K + φ α ∩ K - φ α . One inherits a probability measure µ φ α = T + φ α ∧ T - φ α which is invariant by φ α . Set φ := φ 1 . If |α| > 1, then φ α = α l φ = ((α -1 ) l φ -1 ) -1
, where α -1 < 1 and φ -1 is of the same form as φ, so that we can restrict ourselves to the case where |α| ≤ 1. Besides, according to [START_REF] Bedford | Polynomial diffeomorphisms of C 2 . II. Stable manifolds and recurrence[END_REF][START_REF] Fornaess | Complex dynamics in higher dimensions[END_REF][START_REF] Bedford | Polynomial diffeomorphisms of C 2 . III. Ergodicity, exponents and entropy of the equilibrium measure[END_REF][START_REF] Bedford | Distribution of periodic points of polynomial diffeomorphisms of C 2[END_REF][START_REF] Bedford | Polynomial diffeomorphisms of C 2 . IV. The measure of maximal entropy and laminar currents[END_REF] the following properties hold: for 0 < α ≤ 1, -the function G + φ α is Hölder continuous; -we have the following characterization of points with bounded orbit:

K ± φ α = p ∈ C 2 | G ± φ α (p) = 0 ; (6.1)
this tells us that points either have bounded foward orbit, or escape to infinity with maximal speed; -let p be a saddle point of φ α , then J + φ α is the closure of the stable manifold W s φ α (p); -the support of T + φ α coincides with the boundary of K + φ α , that is J + φ α ; -the current T + φ α is extremal among positive closed currents in C 2 and is -up to a multiplicative constant -the unique positive closed current supported on K + φ α ; -the measure µ φ α has support in the compact set ∂K φ α , is mixing, maximises entropy and is well approximated by Dirac masses at saddle points. One introduces analogous objects for the automorphism Φ α . In particular, Φ α is algebraically stable so we can also define the Green function

G + Φ α := lim n→+∞ log + ||Φ n α || q n .
In fact, for any

(z 0 , z 1 , z 2 ) ∈ C 3 , G + Φ α (z 0 , z 1 , z 2 ) = G + φ α (z 0 , z 1 ) because if z 2 = 0, lim n→+∞ log |α n z 2 | q n
= 0. We introduce the holomorphic map

h : C 3 → C 2 , h : (z 0 , z 1 , z 2 ) → (z 0 z l 2 , z 1 z l 2 ). It follows from the previous remark that G + Φ α • θ = G + φ α • h where θ = (z 0 z l 2 , z 1 z l 2 , z 2 )
conjugates Φ α to Ψ α (i.e. θΨ α = Φ α θ). Moreover, for 0 < |α| < 1, one gets that

K + Φ α = K + φ α × C, K - Φ α = K - φ α × {0}, K Φ α = K φ α × {0},
while for |α| = 1,

K + Φ α = K + φ α × C, K - Φ α = K - φ α × {C}, K Φ α = K φ α × {C}.
When f is an algebraically stable polynomial automorphism of C 3 one can inductively define the analytic sets X j ( f ) by

X 1 ( f ) = f {z 3 = 0} Ind( f ) X j+1 ( f ) = f X j ( f ) Ind( f ) ∀ j ≥ 1
The sequence (X j ( f )) j is decreasing, X j ( f ) is non-empty since f is algebraically stable, so it is stationary. Denote by X( f ) the corresponding limit set. An algebraically stable polynomial automorphism f of C 3 is weakly regular if X( f ) ∩ Ind( f ) = / 0. For instance elements of H are weakly regular. A weakly regular automorphism is algebraically stable. Moreover X( f ) is an attracting set for f ; in other words there exists an open neighborhood V of X( f

) such that f (V ) ⋐ V and +∞ j=1 f j (V ) = X( f ).
Both Φ α and Φ -1 α are weakly regular; furthermore X(Φ α ) = (1 : 0 : 0 : 0) and X(Φ -1 α ) = (0 : 1 : 0 : 0). Therefore (1 : 0 : 0 : 0) (resp. (0 : 1 : 0 : 0)) is an attracting point for Φ α (resp. Φ -1 α ). From [12, Corollary 1.8] the basin of attraction of (1 : 0 : 0 : 0) is biholomorphic to C 3 .

As recalled above, the automorphisms φ ± α are regular, hence weakly regular, and similarly to Φ ± α , they possess attractors X(φ + α ) = (1 : 0 : 0) and X(φ - α ) = (0 : 1 : 0) whose basins are biholomorphic to C 2 . We do not inherit these properties for Ψ α . Indeed remark that K Φ α , X(Φ α ) and X(Φ -1 α ) are contained in {z 2 = 0}; but {z 2 = 0} is contracted by θ -1 (recall that θΨ α = Φ α θ) onto {z 2 = z 3 = 0} and {z 2 = z 3 = 0} = Ind(Ψ α ).

DEFINITION OF A GREEN FUNCTION FOR Ψ α

In this part, we assume 0 < |α| ≤ 1. As for φ α and Φ α , and despite the fact that Ψ α is not algebraically stable, we will see that it is possible to define a Green function for the automorphism Ψ α which has almost as good properties. In particular, we will see that this function carries a lot of information about the dynamics of the automorphism Ψ α .

Let p = (p 0 , p 1 , p 2 ) ∈ C 3 , and define

C = C(p 2 ) := 3 max(1, |p 2 | d ) > 0.
We remark that for n ≥ 0, we have

|α n p 2 | d ≤ C, hence max( Ψ n+1 α (p) , 1) ≤ C max( Ψ n α (p) , 1) q .
We deduce that for every n ≥ 0, log

+ Ψ n+1 α (p) q n+1 - log + Ψ n α (p) q n ≤ log(C) q n+1 , hence lim n→+∞ log + ||Ψ n α (p) q n = lim n→+∞ log + max(|P (n) α (p)|, |P (n-1) α (p)|) q n =: G + Ψ α (p)
exists. By construction, the function

G + Ψ α satisfies G + Ψ α • Ψ α = q • G + Ψ α .
We note that on restriction to the hypersurface {z 2 = 0},

G + Ψ α {z 2 =0} = 0.
Indeed, if p = (p 0 , p 1 , 0), then log + |P (n) α (p)| = O(n) since we have seen that the forward iterates of p grow at most with Fibonacci speed.

For every n ≥ 0, we have θ

• Ψ n α = Φ n α • θ.
In particular, the following limits exist and satisfy:

lim n→+∞ log + ||θ • Ψ n α || q n = lim n→+∞ log + ||Φ n α • θ|| q n = G + Φ α • θ. (7.1)
Define the open set U := (z 0 , z 1 , z 2 ) ∈ C 3 | z 2 = 0 and let p = (p 0 , p 1 , p 2 ) ∈ U. For every n ≥ 0, recall

that θ • Ψ n α (p) = θ P (n) α (p), P (n-1) α (p), α n p 2 = P (n) α (p)(α n p 2 ) l , P (n-1) α (p)(α n p 2 ) l , α n p 2 .
For j ∈ {n -1, n}, we have

log + |P ( j) α (p)| -l log + |α n p 2 | ≤ log + |P ( j) α (p)(α n p 2 ) l | ≤ log + |P ( j) α (p)| + l log + |α n p 2 |, so that log + ||θ • Ψ n α (p)|| q n = log + ||Ψ n α (p)|| q n + o(1).
We deduce from (7.1) that

G + Ψ α (p) = lim n→+∞ log + ||Ψ n α (p)|| q n = lim n→+∞ log + ||θ • Ψ n α (p)|| q n = G + Φ α • θ(p) = G + φ α • h(p). (7.2)
Now if p = (p 0 , p 1 , 0), then we have seen that

G + Ψ α (p) = 0. Note that h(p) = 0 C 2 and θ(p) = 0 C 3 ; therefore, G + Φ α • θ(p) = G + φ α • h(p) = 0.
We conclude that (7.2) holds for any point p ∈ C 3 .

The function G + Ψ α is not ≡ -∞, it is upper semicontinuous and satisfies the sub-mean value property (since G + Φ α does and θ : C 3 → C 3 is holomorphic); in other terms, G + Ψ α is plurisubharmonic. Moreover, we know that G + φ α is Hölder continuous, and h is holomorphic, hence G + Ψ α is Hölder continuous as well. We have shown: Proposition 7.1. For any point p ∈ C 3 , the limit

lim n→+∞ log + ||Ψ n α (p)|| q n =: G + Ψ α (p) exists; the function G + Ψ α = G + Φ α • θ = G + φ α • h is plurisubharmonic, Hölder continuous, and satisfies G + Ψ α • Ψ α = q • G + Ψ α .
We can then define the positive current T + Ψ α := dd c G + Ψ α . The maps θ U and h U are submersions, and

T + Ψ α U = (θ U ) * (T + Φ α U ) = (h U ) * (T + φ α U
). We also have

Ψ * α (T + Ψ α ) = q • T + Ψ α .
Remark 7.2. We observe that contrary to the case of φ α , the set K + Ψ α of points whose forward orbit is bounded is stricly contained in p ∈ C 3 | G + Ψ α (p) = 0 ; indeed, we have seen that the latter always contains {z 2 = 0} ⊂ K + Ψ α .

ANALYSIS OF THE DYNAMICS OF THE AUTOMORPHISM Ψ α

In this section, we further analyze the dynamics of the automorphism Ψ α , distinguishing between the value of 0 < |α| ≤ 1. In particular, we want to describe what happens outside the invariant hypersurface {z 2 = 0}, where we have seen that the dynamics corresponds to the one of a linear Anosov diffeomorphism.

We see that a transition occurs for |α| = ϕ (1-q)/d . Indeed, when |α| < ϕ (1-q)/d , we observe different behaviors in the escape speed outside {z 2 = 0} according to the choice of the starting point p = (p 0 , p 1 , p 2 ): Fibonacci, or bigger than η q n for some η > 1. On the contrary, for |α| > ϕ (1-q)/d , we see that it is impossible to escape to infinity with Fibonacci speed, while the second case persists.

Let us say a few words about the critical value ϕ (1-q)/d where the transition happens. We define the cocycle A : C 3 → GL 2 (C) by:

A(z 0 , z 1 , z 2 ) := 1 + z q-1 0 z d 2 1 1 0 , and if M ∈ M 2 (C) and v = (v 0 , v 1 ) ∈ C 2 , we set M • v := vM T .
Recall that for every z 2 ∈ C, we consider ψ z 2 = (z 0 + z 1 + z q 0 z d 2 , z 0 ). We remark that for every p = (p 0 , p 1 , p 2 ) ∈ C 3 , ψ p 2 (p 0 , p 1 ) = A(p) • (p 0 , p 1 ). As usual we denote A 0 (p) := Id and for n ≥ 1,

A n (p) := A(Ψ n-1 α (p)) • A(Ψ n-2 α (p)) . . . A(Ψ α (p)) • A(p).
In particular, for every n ≥ 0,

Ψ n α (p) = (A n (p) • (p 0 , p 1 ), α n p 2 ); equivalently, (P (n) α (p), P (n-1) α (p)) = A n (p) • (p 0 , p 1 ). Note that A(Ψ n α (p)) = 1 + (P (n) α (p)) q-1 α nd p d 2 1 1 0 . -If lim n→+∞ P (n) α (p) q-1 α nd = 0, then lim n→+∞ A(Ψ n α (p)) = 1 1 1 0
, whose largest eigenvalue is ϕ. Then the growth will be exactly Fibonacci unless the initial point belongs to W s Ψ α (0 C 3 ). But then

ϕ q-1 |α| d n = O |P (n) α (p)| q-1 |α| nd = o(1)
,

and necessarily |α| < ϕ (1-q)/d . -If |α| > ϕ (1-q)/d , Fibonacci growth is impossible; indeed if |P (n) α (p)| ≥ Cϕ n with C > 0, then |P (n) α (p)| q-1 |α| nd ≥ C q-1 ϕ q-1 |α| d n but here η := ϕ q-1 |α| d > 1 so that |P (n+1) α (p)| C q-1 η n |P (n)
α (p)| and the growth is much more in fact. We will also see that this transition reflects an analogous change in the dynamics of the Hénon automorphism φ α : for |α| < ϕ (1-q)/d , the point 0 C 2 is a sink of φ α , while for |α| > ϕ (1-q)/d , the point 0 C 2 becomes a saddle fixed point.

The following general lemma will be useful in the analysis that follows.

Lemma 8.1. Assume that 0 < |α| ≤ 1, and that p = (p 0 , p 1 , p 2 ) satisfies:

|P (n) α (p)| = O([(1 -ε)ϕ] n ).
Then with our previous notations, p ∈ Z.

Proof. This follows again from Lemma 5.1. Indeed for every n ≥ 0,

|g n (p)| ≤ |P (n+1) α (p)| + ϕ -1 |P (n) α (p)| ϕ -n = O (1 -ε) n hence g(p) = lim n→+∞ g n (p) = 0.
8.1. Points escaping to infinity with maximal speed. The results of this subsection hold for any 0 < |α| ≤ 1. We start by exhibiting an explicit non-empty open set of points escaping to infinity very fast; then we state some facts concerning the set of points going to infinity with maximal speed, and show how they can be derived from the properties of the Green function G + Ψ α . Set γ := ln |α| ln(ϕ) . We choose M ≥ 0 sufficiently large so that M(q -1) + dγ > 0 (this is possible since by hypothesis, q -1 > 0).

Proposition 8.2. We define the open set

Ω := p = (p 0 , p 1 , p 2 ) ∈ C 3 | |p 0 | > |p 1 | > 0 and |p 1 | q-1 |p 2 | d > 2 + ϕ M .
Then for any point p ∈ Ω, we have G + Ψ α (p) > 0; moreover the sequence (|P

(n) α (p)|) n is increasing.
The proof splits in two lemmas that we are going to detail now.

(p)| = |p 1 | ≥ |p 1 |ϕ -M . -|P (0) α (p)| = |p 0 | ≥ |p 1 |.
-Take n ≥ 0 and assume that |P

|P (n+1) α (p)| ≥ |P (n) α (p)|(|P (n) α (p)| q-1 |α n p 2 | d -2) ≥ |P (n) α (p)|(|p 1 | q-1 |p 2 | d ϕ (M(q-1)+dγ)n -2) ≥ |p 1 |ϕ Mn (|p 1 | q-1 |p 2 | d -2) ≥ |p 1 |ϕ M(n+1) because M(q -1) + dγ > 0 and |p 1 | q-1 |p 2 | d -2 > ϕ M . Since |p 1 | q-1 |p 2 | d -2 ≥ 1, the previous inequalities also show that |P (n+1) α (p)| ≥ |P (n)
α (p)|, which concludes the induction. Lemma 8.4. Recall that γ := ln |α| ln(ϕ) and that M ≥ 0 is chosen such that M(q -1) + dγ > 0. Take p ∈ C 3 such that the sequence (|P (n) α (p)|) n≥0 is increasing, and assume that there exists n 0 ≥ 1 such that for every n ≥ n 0 , the following inequality holds 2 :

|P (n) α (p)| ≥ ϕ Mn .
Then the escape speed is much bigger in fact: there exist n 1 ≥ n 0 and η > 1 such that for every n ≥ n 1 ,

|P (n) α (p)| ≥ η q n .
In terms of the Green function introduced above, we then get

G + Ψ α (p) > 0. Proof. Since (|P (n) α (p)|) n≥0
is increasing, we have for every n ≥ 0,

|P (n) α (p)| q (|α| n |p 2 |) d = |P (n+1) α (p) -P (n) α (p) -P (n-1) α (p)| ≤ 3|P (n+1) α (p)|. (8.2) Set x n := ln |P (n) α (p)|.
From our hypotheses, we know that for every n ≥ n 0 , x n ≥ Mn ln ϕ. Since M(q -1) + dγ > 0, we can take ε > 0 small such that we still have M(q -1 -ε) + dγ > 0. Let n ′ 0 ≥ n 0 be chosen such that for n ≥ n ′ 0 , n(M(q -1 -ε) + dγ) lnϕ + d ln|p 2 | -ln3 ≥ 0. Thanks to (8.2), we get: for every n ≥ n ′ 0 ,

x n+1 ≥ qx n + ndγ lnϕ + d ln|p 2 | -ln 3 ≥ (1 + ε)x n + (n(M(q -1 -ε) + dγ) lnϕ + d ln|p 2 | -ln 3) ≥ (1 + ε)x n .
We then obtain: for every n ≥ n ′ 0 ,

x n ≥ (1 + ε) n-n ′ 0 x n ′ 0 ≥ (1 + ε) n-n ′ 0 Mn ′ 0 ln ϕ. As a result there exists n 1 ≥ n ′ 0 such that for n ≥ n 1 , x n n 2 ≥ (q/2) n-n ′ 0 Mn ′ 0 ln ϕ n 2 ≥ -(ndγ ln ϕ + d ln|p 2 | -ln3).
We can then refine the previous inequalities: for n ≥ n 1 ,

x n+1 ≥ qx n + ndγ lnϕ + d ln|p 2 | -ln 3 ≥ q 1 - 1 n 2 x n . Let C := ∏ n≥n 1 1 -1 n 2 x n 1 q -n 1 > 0; for every n ≥ n 1 , we have x n ≥ Cq n , hence |P (n) α (p)| ≥ η q n
, where η := e C > 1.

2. In particular, this is satisfied for points p ∈ Ω as we have seen in Lemma 8.3.

Remark 8.5. We have seen that the automorphism φ α possesses an attractor at infinity X(φ α ) = (1 : 0 : 0) whose basin is biholomorphic to C Proposition 8.6. Set l := 2 max(l, 1). We have

1 ≤ lim sup p →+∞ G + Ψ α (p) log p ≤ l.
The set

E := p ∈ C 3 | G + Ψ α (p)
> 0 of points escaping to infinity with maximal speed is open, connected and of infinite measure on any complex line where G + Ψ α is not identically zero. In particular, the set

p ∈ C 3 | lim n→+∞ Ψ n α (p) = +∞
of points whose forward orbit goes to infinity is of infinite measure.

Proof. The openness of E follows directly from the continuity of G + Ψ α , shown in Proposition 7.1.

The proof of the fact that E has infinite measure follows arguments given by Guedj-Sibony [START_REF] Guedj | Dynamics of polynomial automorphisms of C k[END_REF]. Since φ α is algebraically stable, we know from Proposition 1.3 in [START_REF] Guedj | Dynamics of polynomial automorphisms of C k[END_REF] that lim sup

p →+∞ G + φα ( p) log p = 1. Therefore lim sup p →+∞ G + Ψ α (p) log p = lim sup p=(p 0 ,p 1 ,p 2 ) →+∞ G + φ α • h(p) log h(p) × l log |p 2 | + log (p 0 , p 1 ) log p ≤ l lim sup p →+∞ G + φ α ( p) log p = l.
For the other inequality, we remark that lim sup

p →+∞ G + Ψ α (p) log p ≥ lim sup p=(p 0 ,p 1 ,1) →+∞ G + φ α • h(p) log h(p) × log (p 0 , p 1 ) log (p 0 , p 1 , 1) = lim sup p →+∞ G + φ α ( p) log p = 1.
Assume that p ∈ C 3 satisfies G + Ψ α (p) > 0, and for some v = 0 C 3 , consider the line

L := p + tv | t ∈ C . Denote by m(r) the Lebesgue measure of the set e ix , x ∈ R | G + Ψ α (p + re ix v) > 0 .
From what precedes, we know that there exists C > 0 such that for every r ≥ 0,

G + Ψ α (p + re ix v) ≤ l log + (r) + C.
By the sub-mean value property,

0 < G + Ψ α (p) ≤ 1 2π 2π 0 G + Ψ α (p + re ix v)dx ≤ 1 2π ( l log + (r) + C)m(r). Therefore, m(r) ≥ 2πG + Ψα (p) l log + (r)+C
, and integrating over r, we get that the set of points p in L such that G + Ψ α (p) > 0 has infinite measure. The proof of connectivity is also based on the slow growth of G + Ψ α and follows from similar arguments (see [START_REF] Heins | On a notion of convexity connected with a method of Carleman[END_REF]). 8.2. General remarks when 0 < |α| < 1. In this case, 0 C 3 is a hyperbolic fixed point of Ψ α of saddle type, and Corollary 5.5 tells us that the set K + Ψ α of points with bounded forward orbit is exactly the stable manifold

W s Ψ α (0 C 3 ). The positive Julia set J + Ψ α thus corresponds to ∂K + Ψ α = W s Ψ α (0 C 3 ). Let p = (p 0 , p 1 , p 2 ) ∈ C 3 . For n ≥ 0, θ • Ψ n α (p) = P (n) α (p)(α n p 2 ) l , P (n-1) α (p)(α n p 2 ) l , α n p 2 = φ n α • h(p), α n p 2 . (8.3)
From (8.3), we get that h(W s Ψ α (0 3 We deduce that

C 3 )) = W s φ α (0 C 2 ) = K + φ α .
J + φ α = ∂W s φ α (0 C 2 ). Besides, we know that K + φ α = p ∈ C 2 | G + φ α ( p) = 0
, and this set is closed by continuity of G + φ α . In particular, W s φ α (0 C 2 ) is closed. For any p ∈ C 2 , there are two possible behaviors: either p ∈ W s φ α (0 C 2 ) and then its forward iterates converge to 0 C 2 exponentially fast, or they go to infinity with maximal speed. 8.3. Analysis of the dynamics in the case where 0 < |α| < ϕ (1-q)/d . We show that under this assumption, we can construct a set of points with non-empty interior for which the escape speed is much smaller, in fact Fibonacci.

From our hypothesis on α, we can take ε > 0 small enough so that η := ((1 + ε)ϕ) q |α| d < ϕ.

Proposition 8.7. Assume 0 < |α| < ϕ (1-q)/d . We consider the following open neighborhood of the hypersurface {z 2 = 0}:

Ω ′ := p = (p 0 , p 1 , p 2 ) ∈ C 3 | (|p 0 | + |p 1 |) q-1 |p 2 | d < ϕε .
If p ∈ Ω ′ , there are two possible behaviors: -either p belongs to the stable manifold W s Ψ α (0 C 3 ) and then its forward iterates converge to 0 C 3 exponentially fast; -or p goes to infinity with Fibonacci speed:

P (n) α (p)ϕ -n
n≥0 converges and we have

lim n→+∞ P (n) α (p)ϕ -n ∈ C * .
Remark 8.8. The last result tells us that if we start close enough to {z 2 = 0} ⊂ Ω ′ , the dynamics is similar to the one we observe on restriction to this invariant hypersurface: either the starting point belongs to W s Ψ α (0 C 3 ) and in this case its forward orbit converges to 0 C 3 with exponential speed, or the iterates escape to infinity with speed exactly Fibonacci. We remark that when |α| is small, we can choose ε > 0 reasonably large, so that the set Ω ′ becomes larger and larger. This is coherent with the fact that the smaller |α| is, the faster we converge to the hypersurface {z 2 = 0}.

We start by showing that the speed cannot be more than Fibonacci. Lemma 8.9. Any point p ∈ Ω ′ grows at most with Fibonacci speed, that is, there exists C = C(p 0 , p 1 ) > 0 such that for any n ≥ 0, |P

α (p)| ≤ Cϕ n . Proof. Let C = C(p 0 , p 1 ) := |p 0 | + |p 1 | (n)
and ε > 0 be chosen as explained above. We first show that for every n ≥ 0, |P

α (p)| ≤ C((1 + ε)ϕ) n . (n)
The result is clearly true for n = 0, and for n = 1 we have

|P (1) α (p)| ≤ |p 0 | + |p 1 | + |p 0 | q |p 2 | d ≤ C(1 + C q-1 |p 2 | d ) ≤ C(1 + ε)ϕ. 3. Indeed, if p ∈ W s Ψα (0 C 3 ), then h(p) ∈ W s φα (0 C 2 ); conversely, if (p 0 , p 1 ) ∈ K + φα , then (p 0 , p 1 ) = h(p 0 , p 1 ,1) and (p 0 , p 1 ,1) ∈ K + Ψα = W s Ψα (0 C 3 ).
Suppose that it holds for n -1 and n, that is

|P (n-1) α (p)| ≤ C((1 + ε)ϕ) n-1 , |P (n) α (p)| ≤ C((1 + ε)ϕ) n .
We then have

|P (n+1) α (p)| ≤ |P (n) α (p)| + |P (n-1) α (p)| + |(P (n) α (p)) q (α n p 2 ) d | ≤ C (1 + ε)ϕ n + C (1 + ε)ϕ n-1 + C q |p 2 | d ((1 + ε)ϕ) q |α| d n ≤ C(1 + ε) n ϕ n+1 + Cϕεη n ≤ C(1 + ε) n ϕ n+1 + Cε(1 + ε) n ϕ n+1 = C (1 + ε)ϕ n+1 ,
which concludes the induction. Using this fact, we obtain a good control on the non-linear term: for any n ≥ 0,

|(P (n) α (p)) q (α n p 2 ) d | ≤ C q |p 2 | d ((1 + ε)ϕ) q |α| d n = C 0 η n ,
where C 0 = C 0 (p 0 , p 1 ) := Cϕε. For every n ≥ 0, we have

|P (n+1) α (p)| ≤ |P (n) α (p)| + |P (n-1) α (p)| + |(P (n) α (p)) q (α n p 2 ) d | ≤ |P (n) α (p)| + |P (n-1) α (p)| + C 0 η n .
Thanks to the same trick as in the proof of Lemma 5.1, we obtain:

|P (n+1) α (p)| + (ϕ -1)|P (n) α (p)| ≤ ϕ n ϕ|p 0 | + |p 1 | + C 0 n ∑ j=0 η ϕ j . Since η < ϕ, we can set C = C(p 0 , p 1 ) := ϕ -1 ϕ|p 0 | + |p 1 | + C 0 +∞ ∑ j=0 η ϕ j
. We then get: for every n ≥ 0,

|P (n) α (p)| ≤ Cϕ n .
The proof of Proposition 8.7 is the combination of Lemma 8.9 and of the next result.

Lemma 8.10. Assume 0 < |α| < ϕ (1-q)/d and take p ∈ C 3 W s Ψ α (0 C 3 ) with speed less than Fibonacci, i.e., there exists C > 0 such that for every n ≥ 0, |P (n) α (p)| ≤ Cϕ n . Then p goes to infinity with speed exactly Fibonacci:

P (n) α (p)ϕ -n
n≥0 converges and we have

lim n→+∞ P (n) α (p)ϕ -n ∈ C * . Proof. Take p ∈ C 3 W s Ψ α (0 C 3 ) such that for every n ≥ 0, |P (n) α (p)| ≤ Cϕ n , C > 0.
We first show that p goes to infinity with speed at least Fibonacci too, i.e., there exists C ′ = C ′ (p 0 , p 1 ) > 0 such that for every n ≥ 0, |P

(n) α (p)| ≥ C ′ ϕ n . Recall that if (z 0 , z 1 , z 2 ) ∈ C 3 , we denote A(z 0 , z 1 , z 2 ) := 1 + z q-1 0 z d 2 1 1 0 ,
and that for n ≥ 0,

P (n) α (p), P (n-1) α (p) = A n (p) • (p 0 , p 1 )
, where

A n (p) := A(Ψ n-1 α (p)) • A(Ψ n-2 α (p)) . . . A(Ψ α (p)) • A(p).
Note that for every j ≥ 0,

A(Ψ j α (p)) = 1 + (P ( j) α (p)) q-1 α jd p d 2 1 1 0 . (8.4) Since |P ( j)
α (p)| ≤ Cϕ j , we see that

|(P ( j) α (p)) q-1 α jd p d 2 | ≤ C q-1 |p 2 | d (ϕ q-1 |α| d ) j ≤ νη j , (8.5)
where η := ϕ q-1 |α| d < 1 and

ν := C q-1 |p 2 | d ≥ 0. Set M 0 := 1 1 1 0 . If ε > 0, let us consider B ε (M 0 ) := M ∈ M 2 (C) | M -M 0 ∞ < ε .
We see from (8.4) and (8.5) that for every j ≥ 0, A(Ψ j α (p)) ∈ B νη j (M 0 ). For ε > 0 small, every matrix M ∈ B ε (M 0 ) is hyperbolic with eigenvalues close to ϕ and ϕ ′ ; moreover we can choose a family of cones (C ε ) ε>0 around ∆ ϕ satisfying the following: there exist C 0 ,C 1 > 0 such that for each M ∈ B ε (M 0 ), every vector v ∈ C ε will be expanded by a factor close to ϕ:

(1 -C 0 ε)ϕ v ≤ M • v ≤ (1 + C 1 ε)ϕ v . Since p ∈ W s Ψ α (0 C 3 )
, the iterates of (p 0 , p 1 ) are expanded and accumulate on the unstable space ∆ ϕ = (ϕz, z) | z ∈ C of M 0 . In fact the angle ∠(∆ ϕ , A j (p) • (p 0 , p 1 )) decreases exponentially fast, and we can assume that for some n 0 ≥ 0 and for every j ≥ n 0 , A(Ψ j α (p)) ∈ B νη j (M 0 ) maps C νη j to C νη j+1 . We deduce that for every n ≥ n 0 ,

n-1 ∏ j=n 0 (1 -C 0 νη j ) A n 0 (p) • (p 0 , p 1 ) ≤ A n (p) • (p 0 , p 1 ) ϕ n-n 0 ≤ n-1 ∏ j=n 0 (1 + C 1 νη j ) A n 0 (p) • (p 0 , p 1 ) . Let C ′ := ϕ n 0 +∞ ∏ j=n 0
(1 -C 0 νη j ) A n 0 (p) • (p 0 , p 1 ) > 0. We have thus obtained: for every n ≥ 0,

(P (n) α (p), P (n-1) α (p)) ≥ C ′ ϕ n .
Now, we note that p belongs to D, the domain of definition of the series g introduced earlier. Indeed, for any j ≥ 0, |P

α (p)| q ϕ -j |α| jd ≤ C q (ϕ q-1 |α| d ) j and ϕ q-1 |α| d < 1. We have shown that the sequence (P α (p))ϕ -n . We have seen that p ∈ D, and then, (g n (p)) n converges. From (8.6) we deduce that for every n ≥ 0,

g n (p) = P (n+1) α (p) P (n) α (p) + ϕ -1 P (n) α (p)ϕ -n ∼ ϕ + ϕ -1 P (n) α (p)ϕ -n .
This implies that (P Recall that we take ε > 0 small enough so that ((1 + ε)ϕ) q |α| d < ϕ, and that

Ω ′ := (p 0 , p 1 , p 2 ) ∈ C 3 | (|p 0 | + |p 1 |) q-1 |p 2 | d < ϕε .
We deduce from Proposition 8.7 that

G + Ψ α Ω ′ = 0. (8.7)
Indeed, the forward iterates of any point in Ω ′ grow at most with Fibonacci speed as we have seen. In particular, the set p ∈ C 3 | G Ψ + α (p) = 0 has non-empty interior. Set δ := (ϕε) 1/(q-1) and define the open ball

B δ := p = (p 0 , p 1 ) ∈ C 2 | p 1 = |p 0 | + |p 1 | < δ .
Recall that l = d q-1 and that h :

(z 0 , z 1 , z 2 ) → (z 0 z l 2 , z 1 z l 2 ). Remark that h(Ω ′ ) ⊂ B δ . Indeed, if p = (p 0 , p 1 , p 2 ) ∈ Ω ′ , then h(p) 1 = |p 0 p l 2 | + |p 1 p l 2 | = (|p 0 | + |p 1 |) q-1 |p 2 | d 1/(q-1) < (ϕε) 1/(q-1) = δ. Conversely, if (p 0 , p 1 ) ∈ B δ , then (p 0 , p 1 ) = h(p 0 , p 1 , 1) with (p 0 , p 1 , 1) ∈ Ω ′ , so that h(Ω ′ ) = B δ in fact. Since G + Ψ α = G + φ α • h, we deduce from (8.7) that G + φ α B δ = 0.
But as we have seen,

K + φ α = (p 0 , p 1 ) ∈ C 2 | G + φ a (p 0 , p 1 ) = 0 .
We conclude that for |α| < ϕ (1-q)/d , any point (p 0 , p 1 ) ∈ B δ has bounded forward orbit under φ α . Actually we can say more. Recall that φ α = α l (z 0 + z 1 + z q 0 , z 0 ). We see that 0 C 2 is a sink of φ α ; indeed, the largest eigenvalue of the Jacobian is α l ϕ, which is stricly smaller than 1 from the assumption we made on α. For any p ∈ C 3 and n ≥ 0,

θ • Ψ n α (p) = P (n) α (p)(α n p 2 ) l , P (n-1) α (p)(α n p 2 ) l , α n p 2 = φ n α • h(p), α n p 2 . (8.8)
If p ∈ Ω ′ , we know from Proposition 8.7 that there exists C > 0 such that for any n ≥ 0, |P 1) , and ϕ q-1 |α| d < 1, so we deduce from (8.8) and the equality B δ = h(Ω ′ ) that any point in B δ goes to 0 C 2 by forward iteration of φ α ; equivalently, the basin of attraction W s φ α (0 C 2 ) of the sink 0 C 2 contains the ball B δ . Recall also that it is a general fact that for a sink p of φ α , W s φ α (p) is biholomorphic to C 2 .

α (p)| ≤ Cϕ n . But then, |P (n) α (p)(α n p 2 ) l | ≤ C|p 2 | l (ϕ q-1 |α| d ) n/(q- (n)
In the following, we will see how the previous results enable us to give a description of the dynamics of Ψ α in the case where 0 < |α| < ϕ (1-q)/d . Let p ∈ C 3 . We have shown previously that K + Ψ α = W s Ψ α (0 C 3 ), so assume that the forward orbit of p under Ψ α is not bounded. There are two possibilities:

-either G + Ψ α (p) > 0 and the iterates of p go to infinity with maximal speed; in this case, we also have

G + φ α (h(p)) = G + Ψ α (p) > 0; -or G + Ψ α (p) = 0 and G + φ α (h(p)) = G + Ψ α (p) = 0
too. But then we know from the general properties of φ α that h(p) ∈ K + φ α . From §8.2, we also have

K + φ α = W s φ α (0 C 2 ). It follows from (8.8) that lim n→+∞ |P (n) α (p)(α n p 2 ) l | exists and vanishes. Since l = d/(q-1), we deduce that lim n→+∞ |P (n) α (p)| q-1 |α n p 2 | d = 0, hence lim n→+∞ A(Ψ n α (p)) = M 0 = 1 1 1 0
where A is the cocycle introduced earlier. Reasoning as before, and since by assumption p ∈ W s Ψ α (0 C 3 ), we conclude accordingly that p escapes to infinity with Fibonacci speed. We have thus shown: Proposition 8.11. When 0 < |α| < ϕ (1-q)/d , the point 0 C 3 is a saddle fixed point of Ψ α of index 2, and

J + Ψ α := ∂K + Ψ α = W s Ψ α (0 C 3 ). Moreover, p ∈ C 3 | G + Ψ α (p) = 0 = h -1 (W s φ α (0 C 2 )) = Ω ′′ ⊔W s Ψ α (0 C 3 ),
where Ω ′′ has non-empty interior (it contains the set Ω ′ W s Ψ α (0 C 3 )), and the forward orbit of points that belong to it goes to infinity with Fibonacci speed. Moreover, W s φ α (0 C 2 ) is biholomorphic to C 2 , and

K + φ α = W s φ α (0 C 2 ) = p ∈ C 2 | G + φ α ( p) = 0 .
We summarize this as follows:

h {z 2 = 0} → {0 C 2 }; K + Ψ α = W s Ψ α (0 C 3 ) → K + φ α = W s φ α (0 C 2 ); Ω ′′ → W s φ α (0 C 2 ); p ∈ C 3 | G + Ψ α (p) > 0 → p ∈ C 2 | G + φ α ( p) > 0 .
(8.9)

Thanks to the last statement, we now give an alternative description of the stable manifold W s Ψ α (0 C 3 ) in terms of the set Z of zeros of the series g introduced previously.

Proposition 8.12. Assume 0

< |α| < ϕ (1-q)/d . Set V := p ∈ C 3 | G + Ψ α (p) = 0 = Ω ′′ ⊔W s Ψ α (0 C 3 ). Then
V coincides with the domain of definition D of the series g introduced earlier. Moreover, we have the following parametrization of the stable manifold:

W s Ψ α (0 C 3 ) = Z = p ∈ V | g(p) = 0 = n≥0 Ψ -n α (Ω ′ ∩ Z).
Proof. Let p ∈ V . We know from Proposition 8.11 that there exists C > 0 such that for any j ≥ 0, |P ( j) α (p)| ≤ Cϕ j . Then p belongs to the domain of definition of g since in this case, |P

( j) α (p)| q ϕ -j |α| jd ≤ C q (ϕ q-1 |α| d ) j and ϕ q-1 |α| d < 1. It is also clear that if G + Ψ α (p) > 0, then p ∈ D.
Recall that for any n ≥ 0,

g n (z) = ϕz 0 + z 1 + z d 2 n ∑ j=0 P ( j) α (z) q ϕ -j α jd = P (n+1) α (z) + ϕ -1 P (n) α (z) ϕ -n . (8.10) If p ∈ Ω ′′ = V W s Ψ α (0 C 3 ), then
P (n) α (p) + ϕ -1 P (n) α (p)ϕ -n = ϕ + ϕ -1 lim n→+∞ P (n) α (p)ϕ -n .
But we also know that lim n→+∞ |P (n) α (p)ϕ -n | > 0, so that g(p) = lim n→+∞ g n (p) = 0 and p ∈ Z. This shows that if p ∈ Z then p ∈ W s Ψ α (0 C 3 ); the other implication is always true.

The last point follows from the fact that for Ω ′ ⊂ V , we have Ω ′ ∩W s Ψ α (0 C 3 ) = Ω ′ ∩Z. Moreover, Ω ′ contains a neighborhood of 0 C 3 so the orbit of any point p ∈ W s Ψ α (0 C 3 ) will eventually reach Ω ′ . To conclude, we note that by invariance of the stable manifold, we have

W s Ψ α (0 C 3 ) = ∪ n≥0 Ψ -n α (Ω ′ ∩W s Ψ α (0 C 3 )).
8.4. Analysis of the dynamics in the case where ϕ (1-q)/d < |α| < 1. Thanks to previous results, we show the following intermediate result concerning the dynamics of Ψ α in this case.

Proposition 8.13. Assume ϕ (1-q)/d < |α| < 1. For p ∈ U = {z 2 = 0} c , we show the following trichotomy:

-either p belongs to the stable manifold W s Ψ α (0 C 3 ); in this case, its forward iterates converge to 0 C 3 with exponential speed; -or there exist ε > 0 and n 0 ≥ 0 such that for n ≥ n 0 ,

|P (n) α (p)| ≤ ((1 -ε)ϕ) n ; in this case p ∈ Z. Furthermore, lim sup n→+∞ |P (n) α (p)| q-1 |α| nd > 0;
-or the orbit of p escapes to infinity very fast:

G + Ψ α (p) > 0. Moreover the sequence (|P (n) α (p)|) n is increasing after a certain time. Proof. Take 0 < ε < 1 -ϕ -1 |α| d/(1-q) . Note that µ := |α| d ((1 -ε)ϕ) q-1 > 1. Let p = (p 0 , p 1 , p 2 ) ∈ W s
Ψ α (0 C 3 ); according to Corollary 5.5 its forward orbit is unbounded. Suppose that there exists n 0 ≥ 0 such that for every n ≥ n 0 , |P

(n) α (p)| ≤ ((1 -ε)ϕ) n . From Lemma 8.1, we know that p ∈ Z. Moreover, if lim sup n→+∞ |P (n) α (p)| q-1 |α| nd = 0, then with our previous notations, lim n→+∞ A(Ψ n α (p)) = 1 1 1 0
and the growth is at least Fibonacci, which is excluded. We are then in the second case of Proposition 8.13.

Let us handle the remaining case. In particular, we can take n 0 ≥ 0 as big as we want such that |P

(n 0 ) α (p)| > ((1 -ε)ϕ) n 0 .
Note that since by assumption |α| > ϕ (1-q)/d , and we have (q -1) + dγ > 0 (with the notations of Lemma 8.4). In particular, M = 1 satisfies the hypotheses of this lemma. Take n 0 ≥ 0 sufficiently large such that |P Remark 8.14. Denote by S the set of points corresponding to the second case described in Proposition 8.13. A priori, points in S might exhibit a rather complicated dynamics: their forward orbit is not bounded, still, it could happen that it does not escape to infinity. We will see that in fact this behavior does not occur: S = / 0. This is related to the properties of the Hénon map φ α to which Ψ α is semi-conjugate: φ α possesses an attractor at infinity which attracts any point whose forward orbit is not bounded.

(n 0 ) α (p)| > ((1 -ε)ϕ) n 0 and µ n 0 |p 2 | d ≥ 2 + ϕ. We can always assume that |P (n 0 ) α (p)| ≥ |P (n 0 -1) α (p)|.
Let us see how the previous result enables us to conclude the analysis of the dynamics of Ψ α when ϕ (1-q)/d < |α| < 1. Note that in this case, 0 C 2 becomes a saddle point for φ α . We have seen in §8.2 that W s φ α (0 C 2 ) = K + φ α is closed. Therefore, we recover the fact recalled above in the particular case of the saddle fixed point 0 C 2 , and which asserts that J + Since G + Ψ α = G φ α • h, we deduce that h(S ) ⊂ p ∈ C 2 | G + φ α ( p) = 0 = W s φ α (0 C 2 ). Assume that S is nonempty and take p ∈ S. From (8.11), and because by definition S ⊂ U, we see that lim When ϕ (1-q)/d < |α| < 1, the automorphism Ψ α shares a certain number of properties with the Hénon automorphism φ α . The point 0 C 3 is a fixed point of Ψ α of saddle type, and J + Ψ α := ∂K + Ψ α = W s Ψ α (0 C 3 ). Moreover, it follows from the previous discussion that

h {z 2 = 0} → {0 C 2 }; K + Ψ α = W s Ψ α (0 C 3 ) → K + φ α = W s φ α (0 C 2 ); p ∈ C 3 | G + Ψ α (p) > 0 → p ∈ C 2 | G + φ α ( p) > 0 .
(8.12)

In this situation, we see that the set Ω ′′ introduced in the case where 0 < |α| < ϕ (1-q)/d shrinks to the hyperplane {z 2 = 0} which is contracted by h; in particular, it has empty interior.

8.5.

A few words on the case where |α| = 1. Note that in this case, the point 0 C 3 is still fixed by Ψ α but it is no longer hyperbolic. We show the following trichotomy: Proposition 8.16. Let p ∈ U = {z 2 = 0} c . We have three possibilities:

-either p ∈ K + Ψ α , that is, its forward orbit is bounded;

-or p ∈ Z K + Ψ α ; in particular, |P Proof. Assume that p ∈ K + Ψ α and p ∈ D. This implies |P (n) α (p)| = o(ϕ n/q ). From Lemma 8.1, and since q > 1, we deduce that p ∈ Z and we are in the second case.

Let us then assume that p ∈ D and fix ε > 0 such that (1 -ε)ϕ > 1. From Lemma 8.1, we see that for every n 0 ≥ 0, it is possible to find n ≥ n 0 such that |P (n) α (p)| ≥ ((1 -ε)ϕ) n . Arguing as in the proof of Proposition 8.13, we see that the assumptions of Lemma 8.4 are satisfied after a certain time, and we conclude that we are in the third case described above. Remark 8.17. Note that for any p ∈ C 3 , either its forward orbit escapes to infinity with maximal speed (this corresponds to the third case), or p ∈ Z. We denote by S ′ the set of points corresponding to the second case described in Proposition 8.16. We will show later that in fact S ′ = / 0.

When α n = 1 for some n ≥ 1, we see that the dynamics of Ψ α is essentially given by the one of the Hénon automorphism φ = φ 1 , so we assume in the following that α is not a root of unity.

Reasoning as before, (8.11) tells us that h(K

+ Ψ α ) = K + φ α = p ∈ C 2 | G + φ α ( p) = 0 , but now, h(W s Ψ α (0 C 3 )) W s φ α (0 C 2 ). Again, K + Ψ α = p ∈ C 3 | G + Ψ α (
p) = 0 since there are points in {z 2 = 0} escaping to infinity with Fibonacci speed. The point 0 C 2 is still a saddle point of φ α , hence J + φ α := ∂K + φ α = W s φ α (0 C 2 ). The map Ψ -1 α is of the same form as Ψ α , and similarly, we have h(K - Ψ α ) = K - φ α = p ∈ C 2 | G - φ α ( p) = 0 as well as J - φ α := ∂K - φ α = W u φ α (0 C 2 ). We define K Ψ α := K + Ψ α ∩ K - Ψ α ; note that K Ψ α ∩ {z 2 = 0} = (∆ ϕ × {0}) ∩ (∆ ϕ ′ × {0}) = {0 C 3 }, and that h(K Ψ α ) = K φ α . We also see that θ U maps bijectively K Ψ α ∩ U onto K Φ α ∩ U = K φ α × C * . In particular,

Lemma 8 . 3 .

 83 For any point p in Ω the escape speed is superpolynomial: for any n ≥ -1,|P (n) α (p)| ≥ |p 1 |ϕ Mn . (8.1)Moreover the sequence (|P(n) α (p)|) n is increasing.Proof. The proof is by induction on n ≥ -1. Let p ∈ Ω; we will show that (|P (n) α (p)|) n is increasing and that (8.1) holds. It follows from our assumptions that -|P (-1) α

α

  (p)| ≥ |p 1 |ϕ Mn and |P (n) α (p)| ≥ |P (n-1) α (p)|. We estimate:

2 .

 2 Then there exists C = C(α) > 0 such that the forward orbit of any point p = (p 0 , p 1 ) ∈ C 2 such that |p 0 | ≫ |p 1 | and p ≥ C is attracted by X(φ α ); in particular, p ∈ K + φ α , and thus, G + φ α ( p) > 0. If p = (p 0 , p 1 , p 2 ) ∈ C 3 satisfies |p 0 | ≫ |p 1 | and (p 0 , p 1 ) • |p 2 | l ≥ C, we see that h(p) ≥ C, hence G + φ α (h(p)) > 0, and G + Ψ α (p) > 0 as well. The definition of the set Ω in Proposition 8.2 is coherent with this observation.

  ) n accumulates on the unstable direction (ϕz, z) | z ∈ C of M 0 ; therefore, we get lim

αα

  (p)ϕ -n ) n≥0 converges. But we also know from what precedes that lim n→+∞ |P (p)ϕ -n ∈ C * , which concludes.

4

 4 Since (P(n 0 ) α (p)) q-1 (α n 0 ) d ≥ µ n 0 , we deduce P α (p)) q-1 (α n 0 ) d |p 2 | d + 1 + P ≥ µ n 0 |p 2 | d -2 ≥ ϕ.This shows that after time at most n 0 , the sequence (|P (n) α (p)|) n is increasing, moreover, there exists C 0 > 0 such that for any n ≥ 0, |P(n)α (p)| ≥ C 0 ϕ n . The assumptions of Lemma 8.4 are satisfied, and we thus get the desired estimate on the speed.

  φ α = W s φ α (0 C 2 ) = W s φ α (0 C 2 ). Let p = (p 0 , p 1 , p 2 ) ∈ C 3 . For n ≥ 0, θ • Ψ n α (p) = P (n) α (p)(α n p 2 ) l , P (n-1) α (p)(α n p 2 ) l , α n p 2 = φ n α • h(p), α n p 2 . (8.11)Recall that U := {z 2 = 0} c and that S ⊂ U denotes the set of points whose behavior is described in the second item of Proposition 8.13. From the estimate on the speed we obtained, we know thatS ⊂ p ∈ C 3 | G + Ψ α (p) = 0 .

4 .

 4 Else there exists n 1 < n 0 such that |P(n 1 ) α (p)| > ((1 -ε)ϕ) n 0 and |P (n 1 ) α (p)| ≥ |P (n 1 -1) α(p)| and we consider n 1 instead of n 0 .

α

  (p)|•|α| nl exists and vanishes. Since l = d/(q -1), this is in contradiction with the estimate lim sup n→+∞ |P (n) α (p)| q-1 • |α| nd > 0 given in Proposition 8.13. Let us rephrase what we have obtained: Proposition 8.15.

α

  (p)| = o(ϕ n/q ); -or G + Ψ α (p) > 0; moreover the sequence (|P (n) α (p)|) n is increasing after a certain time.

  as in the proof of Lemma 8.10, we see that the sequence (P

				(n) α (p)ϕ -n ) n≥0
	converges; moreover, we get from (8.10):
	lim n→+∞	n→+∞ g n (p) = lim	P α (n+1)	(p)

Since θ U is a biholomorphism, we deduce that:

Now, Proposition 8.16 implies that for any p ∈ U, either |P (n) α (p)| = O(ϕ j/q ) (this corresponds to the first and the second cases described in this proposition), or G + Ψ α (p) > 0. With the notations of Proposition 8.16, assume that S ′ = / 0 and let p ∈ S ′ ⊂ (K +

We define a Green function G - Ψ α in the same way as we did before, as well as a current T -

), and by construction, the currents

Since µ φ α has support in the compact set J φ α := ∂K φ α , we deduce from (8.13) that µ Ψ α is supported on

For every p 2 = 0, the set C p 2 := C 2 × p 2 e ix | x ∈ R is invariant both by Ψ α and Φ α . We know that (φ α , µ φ α ) is mixing (in particular, weakly mixing), and for any p 2 = 0, the restriction of

, µ Φ α ) is ergodic (see [START_REF] Bunimovich | Dynamical systems, ergodic theory and applications[END_REF] for instance). We define J p 2 := J Ψ α ∩ C p 2 ; this set is invariant, and we know that µ Ψ α J p 2 is supported on it. By (8.13), we conclude that (Ψ α J p 2 , µ Ψ α ) is ergodic too. Yet there is no hope to get mixing properties for Ψ α since by projection on the third coordinate, z 2 → αz 2 is a quasiperiodic factor of the dynamics. We have thus obtained: Proposition 8.18. For any point p ∈ C 3 , we are in exactly one of the following cases:

-either the orbit of p is bounded, i.e. p ∈ K Ψ α ; -or p ∈ {z 2 = 0} {0 C 3 }; -or G + Ψ α (p) > 0 or G - Ψ α (p) > 0. The measure µ Ψ α is invariant by Ψ α and supported on the Julia set J Ψ α = {0 C 3 } ∪ θ -1 (J φ α × C * ). Moreover, when p 2 = 0 and α is not a root of unity, (Ψ α J p 2 , µ Ψ α ) is ergodic.