Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of low degree in C^{3}

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Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of

low degree in C ³

1 Introduction 1

1.1 Background . . . . 1 1.2 Results and outline . . . . 3

2 Preliminaries 4

2.1 One complex dimension . . . . 4 2.2 Two complex dimensions . . . . 6 2.3 Three complex dimensions . . . . 7 3 Structure of the basic set : unperturbed case 9 3.1 Generalities and main result in the unperturbed case . . . . 9 3.2 Proof of Proposition 3.1.7 . . . . 12 4 Structure of the basic set : perturbed case 14 4.1 Generalities and main result in the perturbed case . . . . 14 4.2 Proof of Proposition 4.1.8 . . . . 16 4.3 Central curves . . . . 19

5 First heteroclinic tangency 20

5.1 Perturbation . . . . 20 5.2 Motion of the point of tangency . . . . 22

6 Persistent homoclinic tangencies 23

6.1 Persistent heteroclinic tangencies . . . . 23 6.2 Persistent homoclinic tangencies . . . . 25

7 Proof of the main result 26

A From homoclinic tangencies to sinks 27

References 29

1 Introduction

1.1 Background

Hyperbolic systems such as the horseshoe introduced by Smale were orig-

inally supposed to be dense in the set of parameters in the 1960’s. This was

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quickly discovered to be false in general for diffeomorphisms of manifolds of dimension greater than 2 (see [1]). The discovery in the seventies of the so- called Newhouse phenomenon, i.e. the existence of residual sets of C

²

diffemor- phisms of compact surfaces with infinitely many sinks (periodic attractors) in [15] showed it was false in dimension 2 too. In a subsequent work, Newhouse showed (see [16]) that such sets appear in fact close to any diffeomorphism with an homoclinic tangency. The technical core of the proof is the reduction to a line of tangency between the stable and unstable foliations where two Cantor sets must have persistent intersection. This gives persistent homoclinic tangencies between the stable and unstable foliations, ultimately leading to infinitely many sinks. Indeed, it is a well known fact that a sink is created in the unfolding of a generic homoclinic tangency.

Palis and Viana showed in [17] an analogous result for real diffeomorphisms in higher dimensions. More precisely, they proved that near any smooth diffeomorphism of R

³

exhibiting a homoclinic tangency associated to a sectionally dissipative saddle, there is a residual subset of an open set of diffeomorphisms such that each of its elements displays infinitely many coexisting sinks.

In the complex setting, this reduction is not possible anymore and to get persistent homoclinic tangencies and then residual sets of diffeomorphisms dis- playing infinitely many sinks, we have to intersect two Cantor sets in the plane.

This was done by Buzzard who proved in [7] that there exists d > 0 such that there exists an automorphism G ∈ Aut

d

( C

²

) and a neighborhood N ⊂ Aut

d

( C

²

) of G such that N has persistent homoclinic tangencies. Then, there is a residual subset of Aut

d

( C

²

) of automorphisms with infinitely many sinks. In fact, after extending the stable and unstable foliations of a basic set, there is still a complex disk of tangency where intersections with the two foliations are two Cantor sets in two (real) dimensions. Buzzard gives an elegant criterion (see [6]) which generates the intersection of two such Cantor sets, hence leading to persistent homoclinic tangencies. More precisely, he gets persistent intersections between a "spiralic" Cantor set and a second Cantor set with high topological dimension. In his article, Buzzard uses a Runge approximation argument to get a polynomial automorphism, which implies that the degree d remains unknown and is supposedly very high.

In the article [4], Bonatti and Diaz introduce a type of horseshoe they called

blender horseshoe. The important property of such hyperbolic sets lies in the

fractal configuration of one of their stable/unstable manifold which implies per-

sistent intersection between any well oriented graph and this foliation. In some

sense, the foliation behaves just as it had greater Hausdorff dimension than ev-

ery individual manifold of the foliation. They find how to get robust homoclinic

tangencies for some C

^r

-diffeomorphism of R

³

with an homoclinic tangency by

some geometric intersection procedure using the properties which define the

blender in [5] . In the article [9] , one can find real polynomial maps of degree

2 with a blender. Other studies using blenders include [8], [2] and [3].

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1.2 Results and outline

In this article, we adapt a complex blender. More precisely, we solve here the degree problem in dimension 3 by introducing a kind of 3-dimensional central- stable complex blender. Here is our main result :

Main Theorem. There exists f ∈ Aut

5

( C

³

) such that for every g ∈ Aut( C

³

) sufficiently close to f , g admits a tangency between the stable and unstable laminations of some hyperbolic set.

Notice that in the previous result, g is not assumed to be polynomial.

Corollary 1. For each d ≥ 5, there exists an open set V ⊂ Aut

d

( C

³

) in which the automorphisms having a homoclinic tangency are dense.

Corollary 2. For each d ≥ 5, there exists an open set V ⊂ Aut

d

( C

³

) in which the automorphisms having infinitely many sinks are dense.

Let us remark that there are classes of interesting polynomial automorphisms of C

³

called regular and semi-regular automorphisms which have received much attention due to their interesting dynamical properties (e.g [13],[18]). It is possible to choose the automorphisms to be regular or semi-regular in the above results because the condition of being regular or semi-regular is dense for the Zariski topology.

Let us present quickly our method to prove this result. We consider an automorphism of C

³

which is a perturbation of the following map f

1

:

f

₁

: (z

₁

, z

₂

, z

₃

) → (p(z

₁

) + b.z

₂

, z

₁

, λ.z

₁

+ µ.z

₃

) (1)

where p is a polynomial and the coefficients b, λ, µ are complex numbers. Then, we prove that f

₁

has a hyperbolic set with the property that the stable lamination of dimension 2 can be fixed with enough freedom while the 1-dimensional unstable lamination presents one direction where the lamination is very thick.

Moreover, this hyperbolic set has a fixed point which is sectionally dissipative. Perturbating the automorphism f

1

creates an initial heteroclinic tangency.

Then, we show that the two laminations are such that there are robust hetero- clinc tangencies. Then we get robust homoclinic tangencies. By a classical Baire category argument, this gives a subset of the set of automorphisms of limited degree 5 in which automorphisms with infinitely many sinks are dense.

In section 2, we precise the automorphism. We give first a complex polynomial p, then a 2-dimensional Henon automorphism h using p and finally the 3-dimensional automorphism f

1

by adding to h a third linear component.

In section 3 and 4, we describe the basic set : we prove propositions which precise the transversal shape of the unstable foliation.

In section 5, then we perturb f

₁

to get a heteroclinic tangency with good ori- entation of the two foliations. This is done by increasing the degree of p and adding a small term in the first component of f

₁

.

In section 6, we prove that these heteroclinic tangencies are robust and we con- clude the proof of the main result in section 7.

For the convenience of the reader we recall how to create sinks from homoclinic

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tangencies in the sectionally dissipative case.

Acknowledgments : This research was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002.

2 Preliminaries

2.1 One complex dimension

We denote by D ⊂ C the unit disk.

Lemma 2.1.1. For r =

₁₀¹

and r

⁰

=

₁₀₀¹

, there exists a polynomial p of degree 4 such that :

(i) p

⁻¹

( D ) ⊂⊂ D and p

⁻¹

( D ) admits 3 components D

1

, D

2

, D

3

(ii) For D

₁

and D

₂

we have : D

₁

⊂ B(1 − r, r

⁰

), D

₂

⊂ B(r − 1, r

⁰

), p

_|D_i

is a biholomorphism and |p

⁰

| > 100 on D

_i

for i ∈ {1, 2}

(iii) p admits a fixed point α

_p

∈ D

₃

such 1 < |p

⁰

(α

_p

)| < 1.1

Proof. Let us start with a real polynomial z

²

+ c where c = 0.249. We modify it by adding a term of degree 4 : p(z) = z

²

+ 0.249 −

_R^d₂

.z

⁴

on a disk of radius R. Then, it suffices to rescale to get the result on D . We want to choose R such that the following inequalities hold :

p((1 − r + 2r

⁰

)R) < −2R (2)

p((1 − r − 2r

⁰

)R) > 2R (3)

that is :

((1 − r + 2r

⁰

)R)

²

−

_R^d2

(1 − r + 2r

⁰

)

⁴

)R

⁴

+ 0.249 < −2R ((1 − r − 2r

⁰

)R)

²

−

_R^d₂

(1 − r − 2r

⁰

)

⁴

)R

⁴

+ 0.249 > 2R is equivalent to :

((1 − r + 2r

⁰

)

²

− d(1 − r + 2r

⁰

)

⁴

)R

²

+ c < −2R ((1 − r − 2r

⁰

)

²

− d(1 − r − 2r

⁰

)

⁴

)R

²

+ c > 2R

Once for all, we take d =

_(1−r)¹ ₂

. Then, if we take R sufficiently large, the two inequalities are verified.

The critical points of p are {0, c

p

= +

^√^R

2d

, −

^√^R

2d

}. Since p(0) = c ∈ B (0, R) and :

p( R

√

2d ) = R

²

2d + c − d. R

⁴

4d

²

R

²

= R

²

4d + c ' R

²

4d > R

by the Riemann-Hürwitz formula we see that p

⁻¹

(B(0, R)) admits 3 components, two of them are univalent, we call them D

₁

and D

₂

, and one is not, we call it D

₃

. In R , p

⁻¹

((−R, R)) admits 3 components I

₁

, I

₂

, I

₃

with I

₁

near −R, I

₂

near +R and 0 ∈ I

₃

(see Figure 1). Moreover, the 3 components D

₁

, D

₂

, D

₃

of p

⁻¹

(D(0, R)) in C are such that D

_i

∩ R = I

_i

. Indeed, p is real, the sets D

_i

are symmetric w.r.t. to R and simply connected by the maximum principle so the intersections D

i

∩ R must be intervals. p

_|D_i

is a biholomorphism for i ∈ {1, 2}

and we have : p

⁻¹

( D ) ⊂⊂ D .

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Then, let us consider the univalent map q = p

⁻¹_|D0 1

: B(0, 2R) → D

⁰₁

where D

⁰_i

is the component of p

⁻¹

(D(0, 2R)) such that : D

_i

⊂ D

_i⁰

. Since p((1−r+2r

⁰

)R) <

−2R and p((1 − r − 2r

⁰

)R) > 2R, there is a point z

_q

∈ B(0, R) such that :

|p

⁰

(q(z

q

)| >

^2R_4r0

=

_2r^R0

, that is |q

⁰

(z

q

)| <

^2r_R⁰

. Now, by the Koebe Theorem we know that for an univalent map g : D → C we have the following inequalities :

∀z ∈ D , 1 − |z|

(1 + |z|)

³

< |g

⁰

(z)|

|g

⁰

(0)| < 1 + |z|

(1 − |z|)

³

Figure 1 : graph of p on the real line

Applying this result to q and the ball B(0, R) ⊂ B(0, 2R) we get that

∀z ∈ B(0, R), 12.|q

⁰

(0)| > |q

⁰

(z)| >

₂₇⁴

.|q

⁰

(0)| then : ∀z ∈ B(0, R), |q

⁰

(z)| <

144.|q

⁰

(z

q

)| <

^144.2r_R ⁰

then : ∀z ∈ D

1

, |p

⁰

(z)| >

_288r^R 0

>

_r¹0

= 100 and ∀z ∈ B(0, R), |q

⁰

(z)| < r

⁰

if R > 288. This inequality implies that D

1

⊂ B(1 − r, r

⁰

).

Then, |p

⁰

| > 100 on D

1

. The corresponding result holds for D

2

.

The multiplier of the repelling fixed point z

²

+ 0.249 is smaller than 1.1 in modulus, so increasing the value of R if necessary we get the same estimate for p and we are done.

Remark 2.1.2. Let us further introduce two sets D

⁰⁰₁

and D

⁰⁰₂

such that : D

₁

⊂ D

⁰⁰₁

⊂ D

⁰₁

and D

₂

⊂ D

₂⁰⁰

⊂ D

⁰₂

, D

⁰⁰₁

and D

₂⁰⁰

are 2 components of p

⁻¹

(D(0,

^3R₂

)).

The Koebe Theorem gives us that |p

⁰

| > 100 on D

_i⁰⁰

.

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2.2 Two complex dimensions

We now perturb the polynomial p into a complex Henon map with small Jacobian −b.

h : (z

1

, z

2

) 7→ (p(z

1

) + bz

2

, z

1

)

In this subsection, we will denote by C

^u

the two-dimensional cone centered at e

₁

of opening

₁₀¹

and C

^ss

the two-dimensional cone centered at e

₂

of opening

1

10000

. In the next subsection, we will introduce analogous three dimensional cones. To simplify the notations, we will denote them too by C

^u

and C

^ss

. Definition 2.2.1. Given an automorphism F : C

²

→ C

²

, we say that T

∞

−∞

F

ⁿ

( D

²

) is a horseshoe for F if :

- F ( D

²

) ∩ D

²

is an union of two bidisks D

¹

and D

²

such that D

ⁱ

∩ ∂( D

²

) is included in ∂( D ) × D for i = 1, 2

- on F

⁻¹

( D

²

) ∩ F ( D

²

), the cone field C

^u

is F-invariant and the cone field C

^ss

is F

⁻¹

-invariant

- there exists C

F

> 1 such that on F

⁻¹

( D

²

) ∩ F( D

²

) :

∀u ∈ C

^u

, ||DF (u)|| > C

F

||u|| and ∀v ∈ C

^ss

, ||D(F

⁻¹

)v|| > C

F

||v||

In the following, we will have to take the constant b such that 0 < |b| < b

i

where b

_i

is a bound which will be reduced a finite number of times so that it will ensure some properties on f

₁

. The following result is classical. For instance, it follows from the work of Hubbard-Oberste-Vorth, see [14] or [11], we just give a justification for the constants of the cones.

Proposition 2.2.2. There exists a positive number b

₀

such that if |b| < b

₀

, then T

−∞<n<+∞

h

ⁿ

( D

²

) is a horseshoe for h

_|D2

. Besides, the fixed point α

_h

continuation of the fixed point α

_p

of p is a saddle point of expanding eigenvalue 1 < |ξ

_h

| < 1.1.

Proof. We have that |p

⁰

| > 100 on D

₁⁰⁰

∪ D

₂⁰⁰

. This implies that there exists a positive number b

₀

such that if |b| < b

₀

, then C

^u

is h-invariant and C

^ss

is h

⁻¹

-invariant with C

_h

= 50.

Definition 2.2.3. We denote h[1] by the restriction of h to D

₁⁰⁰

× D and by h[2]

the restriction of h to D

₂⁰⁰

× D (where D

⁰⁰₁

and D

₂⁰⁰

were defined in the precedent Remark). Then we inductively define h[I] for an arbitrary sequence of digits by h[Ij] = h[j] ◦ h[I] on (h[I])

⁻¹

(D

⁰⁰_j

× D ). We define : H

1

= (h[1])(D

⁰⁰₁

× D ) ∩ D

²

, H

₂

= (h[2])(D

₂⁰⁰

× D ) ∩ D

²

and H

_Ij

= (h[j])(U

_I

) ∩ D

²

.

The following Proposition is a consequence of the general study on complex horseshoes.

Proposition 2.2.4.

(i) For each finite sequence of indices I = (i

₁

, ..., i

_p

), i

_j

∈ {1, 2}, the set H

_I

is of the form : H

_I

= S

z₁∈D

{z

1

} × I

_I^z¹

where I

_I^z¹

is an open topological disk included in D .

(ii) If I is an infinite sequence I ∈ {1, 2}

^N

, the intersection H

_I

= T

p≥0

H

_I_≤p

is a piece of the intersection of the unstable manifold of one point of the set T

∞

−∞

h

ⁿ

( D × D ) with D

²

× Q (where I ≤ p denotes the finite subsequence of the

p first digits of I).

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Let us call t

_z₁_,I

= δ

₁

(I

_I^z¹

) = δ

₂

({z

₁

} × I

_I^z¹

). We call t

_n

= max

_z₁_,|I|=n

t

_z₁_,I

. We first prove that the rate of decay of t

_n

can be made arbitrary small.

Lemma 2.2.5. For all k ∈]0, 1[, there exists 0 < b

1

< b

0

such that for all

|b| < b

1

, n ∈ N we have : t

n+1

< kt

n

. Consequently : P

+∞

n=0

t

n

<

_1−k¹

Proof. Let us prove the lemma by induction. The property is obvious for I = ∅.

Let us suppose the lemma is proven for all the I until a certain rank. We bound here t

_z₁_,I∪{i_p+1_}

. The intersection of the set H

I

= h[I]( D

²

) with p(z

1

) + bz

2

= C

^st

is a curve Γ

I

such that δ

1

(Γ

I

) < k.t

n

if we take 0 < b

1

< b

0

sufficiently small. Mapping by h : (z

1

, z

2

) 7→ (p(z

1

) + b.z

2

, z

1

), we get : t

n+1

< k.t

n

.

2.3 Three complex dimensions

We consider now the 3 dimensional map f

1

introduced in (1) : f

1

: (z

1

, z

2

, z

3

) 7→ (p(z

1

) + b.z

2

, z

1

, λ.z

1

+ µ.z

3

)

In the following, we will see that the first direction is expanded by f

1

and corresponds to the direction of the unstable manifolds of a hyperbolic set we are going to describe. The second and third directions are contracted by f

1

and correspond to the direction of the stable manifolds of this hyperbolic set. We fix 3 small angular cones C

^u

, C

^ss

, C

^cs

centered in the three axis of coordinates with thin opening.

Notation. C

^u

is the cone centered at e

₁

of opening

₁₀¹

, C

^ss

is the cone centered at e

₂

of opening

₁₀₀₀₀¹

and C

^cs

is the cone centered at e

₃

of opening

₁₀₀₀₀¹

. Definition 2.3.1. Given an automorphism F : C

³

→ C

³

, we say that T

∞

−∞

F

ⁿ

( D

²

× Q) is a horseshoe for F if :

- F ( D

²

× Q) ∩ ( D

²

× Q) is an union of two tridisks D

¹

and D

²

such that D

ⁱ

∩ ∂( D

²

× Q) is included in (∂( D ) × D × Q) for i = 1, 2

- on F

⁻¹

( D

²

× Q) ∩ F( D

²

× Q), the cone field C

^u

is F-invariant and the cone fields C

^ss

, C

^cs

are F

⁻¹

-invariant

- there exists C

F

> 1 such that on F

⁻¹

( D

²

× Q) ∩ F ( D

²

× Q) :

∀u ∈ C

^u

, ||DF (u)|| > C

F

||u|| and ∀v ∈ C

^ss

∪ C

^cs

, ||D(F

⁻¹

)v|| > C

F

||v||

Definition 2.3.2. We say that a saddle periodic point q of multipliers |λ

1

| ≤

|λ

2

| < 1 < |λ

3

| is sectionally dissipative if the product of any two of its eigenvalues is less than 1 in modulus, that is, |λ

1

λ

₃

| < 1 and |λ

2

λ

₃

| < 1 .

Once for all, we fix now : µ = 9

10 i, λ = 1 10

√ 2

2 e

ⁱ^π⁴

= 1

10 (1 + i) (4)

We denote by pr

_i

the projection on the i

^th

coordinate in C

³

.

Proposition 2.3.3. Let f

1

: (z

1

, z

2

, z

3

) → (p(z

1

) + b.z

2

, z

1

, λ.z

1

+ µ.z

3

) with

|b| < b

1

. Then, T

−∞<n<+∞

f

₁ⁿ

( D

²

× Q) is a horseshoe. Moreover, f

1

has a

fixed point α

f₁

that is sectionally dissipative.

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Proof. The fact that C

^u

is f

₁

-invariant and C

^ss

is f

₁⁻¹

-invariant comes from the analogous result on h. The fact that C

^cs

is f

₁⁻¹

-invariant comes from the fact that e

3

is an eigenvector at each point for Df

₁⁻¹

of associated eigenvalue

1

µ

=

¹⁰_9i

. The existence of α

f₁

is obvious. Looking at the differential, we see that α

f₁

has three eigenvalues ξ

h

, µ,

^−b_ξ

h

so the result follows since 1 < |ξ

h

| < 1.1,

|µ| =

₁₀⁹

and

₁₀⁹

.

¹¹₁₀

< 1.

Definition 2.3.4. We denote f

1

[1] the restriction of f

1

on D

⁰⁰₁

× D ×Q and f

1

[2]

the restriction of f

1

on D

₂⁰⁰

× D × Q (where D

₁⁰⁰

and D

₂⁰⁰

were defined in Remark 2.1.2). Then, we inductively define f

1

[I] for an arbitrary sequence of digits by f

1

[Ij] = f

1

[j] ◦ f

1

[I] on (f

1

[I])

⁻¹

(D

⁰⁰_j

× D × Q). We define U

1

= (f

1

[1])(D

₁⁰⁰

× D × Q) ∩ D

³

, U

₂

= (f

₁

[2])(D

⁰⁰₂

× D × Q) ∩ D

³

and U

_Ij

= (f

₁

[j])(U

_I

) ∩ ( D

²

× Q).

In the following, we denote by D

.,z2,z3

the disk parallel to the z

₁

axis. In this subsection we gather a few properties of complex horseshoes which will be useful afterwards. In particular, horseshoes are compact, hyperbolic, transitive and locally invariant sets.

Lemma 2.3.5. For |b| < b

1

and for every sequence of digits I, f

1

[I]( D

.,z₂,z₃

) is a graph over the first coordinate z

1

of the form

f

1

[I]( D

.,z₂,z₃

) = {(z

1

, y

_I²

(z

1

), y

³_I

(z

1

)) : z

1

∈ D } with derivatives |(y

²_I

)

⁰

| <

₅₀¹

, |(y

³_I

)

⁰

| <

₅₀¹

.

Proof. We show the result by induction. We suppose that the conditions are verified for each step until I and we consider the following index I

⁰

= I∪{j}. In C

I

: z

₂

= y

_I²

(z

₁

) with the derivative of y

_I²

in modulus bounded by

¹₅

: |(y

²_I

)

⁰

(z

₁

)| <

₅₀¹

. Every curve {(z

1

, y

²_I

(z

₁

), y

_I³

(z

₁

)) : z

₁

∈ D } intersects, if |b| < b

₁

, each hypersurface p(z

₁

) + bz

₂

= C

^st

only one and one time. Then the curve f

₁

[I

⁰

]( D

.,z2,z3

) is a graph {(z

₁

, y

_Ij²

(z

₁

), y

_Ij³

(z

₁

)) : z

₁

∈ D }. The implicit function Theorem says that the derivate of this graph is

_p0(z₁)+b(y¹ ²_I)⁰(z₁)

inside this domain. Then, {z

₁

: p(z

₁

) + by

²_I

(z

₁

) ∈ D } = D

₁

∪ D

₂

with D

_i

⊂ D

⁰⁰_i

, we have the bound

|p

⁰

| ≥ 100. Moreover, |(y

_I²

)

⁰

| <

₅₀¹

and b < 1, so this gives : |(y

_Ij²

)

⁰

| <

₅₀¹

So, f

1

[I]( D

^.,z2,z₃

) is a graph {(z

1

, y

_I²

(z

1

), y

³_I

(z

1

)) : z

1

∈ D }. It remains to prove the bound on the derivative of y

³_I

(z

1

). We do this by induction. For |I| = ∅, this is obvious. Then for I

⁰

= I ∪ {j}, we see that f

1

[I

⁰

]( D

.,z₂,z₃

) is the image of a piece of f

1

[I]( D

.,z₂,z₃

) so we can write :

y

_I³0

(z

1

) = µ.y

_I³

(ζ(z

1

)) + λ.ζ (z

1

) (y

³_I0

)

⁰

(z

1

) = µ.(y

_I³

)

⁰

(ζ(z

1

)).ζ

⁰

(z

1

)) + λ.ζ

⁰

(z

1

)

where ζ(z

1

) denotes the first coordinate of the inverse image of the point of C

_I0

whose first coordinate is z

₁

. The derivative of ζ is bounded by

₇₀¹

because

|p

⁰

| > 100 on D

⁰⁰₁

∪ D

⁰⁰₂

and for small values of b, ζ

⁰

is near the derivate of the

local inverse branch of p. We infer that |(y

_I³

)

⁰

(z

₁

)| ≤ B

_n

where B

_n

satisfies

B

₀

= 0 and B

_n+1

<

₅₀¹

|µ|B

_n

+

₇₀¹

where n = |I|. It follows that for every n ≥ 0,

B

_n

<

₅₀¹

and we are done.

(9)

We finish this section by a lemma saying that horizontal graphs are not very perturbated by adding a term of degree 5, p

₅

z

⁵

, to p.

Lemma 2.3.6. For every sequence of digits I, f [I]( D

.,z₂,z₃

) = {(z

1

, y

²_f,I

(z

1

), y

³_f,I

(z

1

)) : z

1

∈ D } is a horizontal graph such that for all z

1

∈ D , p

5

7→ [(y

_f,I²

)(z

1

)](p

5

) and p

5

7→ [(y

³_f,I

)(z

1

)](p

5

) are 2-lipschitz continuous.

Proof. The lemma is a consequence of the Schwarz-Pick Lemma applied to the two holomorphic maps D 7→ D , p

5

7→ [(y

¹_f,I

)(z

1

)](p

5

) and D 7→ D , p

5

7→

[(y

_f,I²

)(z

1

)](p

5

).

A consequence of this lemma is that the horizontal graphs T

p≥0

U

_I≤p

= W ∩ ( D

²

× Q) = {(z

1

, y

_f^W

(z

1

), y

^W_f,2

(z

1

)) : z

1

∈ D } verify the same 2-Lipschitz continuous property.

3 Structure of the basic set : unperturbed case

3.1 Generalities and main result in the unperturbed case

In this section, we describe the geometry of a horseshoe in C

³

which is induced by f

1

. We describe a subset of the set K

f1

= T

∞

−∞

f

₁ⁿ

( D × D × Q), more precisely T

∞

−∞

f

₁ⁿ

((D

₁

∪ D

₂

)× D ×Q). Recall that f

₁

(z

₁

, z

₂

, z

₃

) = (p(z

₁

) + b.z

₂

, z

₁

, λ.z

₁

+ µ.z

₃

). The choice of Q instead of D in the last coordinate is a matter of convenience only, since the projection on the 3

^rd

coordinate is easier to analyse in terms of subsquares (see Figure 3).

Definition-Proposition 3.1.1. Given a set E ⊂ C

^k

and i ∈ {1, 2, 3}, we call δ

i

(E) = diam(pr

_i

(E)). Given z

1

, z

2

∈ C , we let : L

z₁,z₂

= {(z

1

, z

2

)} × C . Definition 3.1.2. Given a set E ⊂ C

²

and δ ∈ (0, 1), we say that E is of δ-product type w.r.t the 2

^nd

coordinate if there is a square S = c

S

+ `

S

.Q ⊂ C such that

pr

₁

(E) × (c

_S

+ (1 − δ).`

_S

.Q) ⊂ E ⊂ pr

₁

(E) × (c

_S

+ (1 + δ).`

_S

.Q) We call admissible square for E a square S with this property.

Proposition 3.1.3.

(i) For each finite sequence of indices I = (i

1

, ..., i

p

), i

j

∈ {1, 2}, the set U

I

is a fibration by squares over the set H

I

(see Figures 2,3) :

U

I

= [

(z₁,z₂)∈HI

(z

1

, z

2

) × Q

z₁,z₂,I

More precisely, U

I

= S

z₁∈D

{z

1

} × R

z₁,I

where R

z₁,I

is a non empty open connected subset in D × Q of the form :

R

z₁,I

= [

z₂∈pr₁(R_z₁_,I)

{z

2

} × {β

z₁,z₂,I

+ l

I

.Q} (5)

where Q

z₁,z₂,I

= {β

z₁,z₂,I

+ l

I

.Q} is a square whose length side l

I

is independent

of z

1

and depends only of the length |I| of I : l

I

= |µ|

^|I|

.

(10)

(ii) If I is an infinite sequence I ∈ {1, 2}

^N

, the intersection U

_I

= T

p≥0

U

_I_≤p

is a piece of the intersection of the unstable manifold of one point of the set T

∞

−∞

f

₁ⁿ

( D × D × Q) with D

²

× Q (where I ≤ p denotes the finite subsequence of the p first digits of I).

The result is essentially a consequence of Proposition 2.2.4 and Lemma 3.1.5 below.

Let us first note the following obvious consequence of the definition of f

1

. Lemma 3.1.4. For all z

₁

, z

₂

, f

₁

(L

_z₁_,z₂

) = L

_h(z₁_,z₂₎

.

The following lemma describes the geometry of non empty intersections of the form L

z₁,z₂

∩ ({z

1

} × R

z₁,I

) :

Lemma 3.1.5. For every z

1

, z

2

, if L

z₁,z₂

intersects {z

1

}×R

z₁,I

, then pr

₃

(L

z₁,z₂

∩ ({z

1

} × R

z₁,I

)) is a square inside the line L

z₁,z₂

. We denote `

z₁,z₂,I

the length side of the square L

z₁,z₂

∩ U

I

= L

z₁,z₂

∩ ({z

1

} × R

z₁,I

) when this intersection is non empty. Then, for all z

1

, I, `

z₁,z₂,I

is constant in z

1

, z

2

, we denote it by l

I

and its value is l

I

= l

z1,z2,I

= |µ|

^|I|

.

Proof. The third coordinate of f , for a fixed value of z

₁

, is affine in z

₃

of multiplier µ =

₁₀⁹

i. This implies : ∀I, l

_I∪{i_p+1_}

=

₁₀⁹

l

_I

and the result follows.

In the following, the projections of the sets R

z₁,I

will have a special configuration. This is why we introduce the next definition.

Definition 3.1.6. Let Q the unit square centered at the origin. For any square S whose axes are parallel to those of Q, we denote by N E, SE, SW, N W its four corners. Given such a square S of length size l

S

we denote by S

c,s,s⁰

where c ∈ {N E, SE, SW, N W }, s ∈ [

¹₂

, 1[, s

⁰

∈]0, 1 − s[ the subsquare of side length s.l

S

positioned near the corner c at a distance s

⁰

l

S

of the two sides.

A configuration NE-SW is a triple (S, S

_c,s,s⁰

, S

_c⁰_,s,s⁰

) where S

_c,s,s⁰

and S

_c⁰_,s,s⁰

are two subsquares of S such that {c, c

⁰

} = {N E, SW }, a configuration NW-SE is a triple (S, S

_c,s,s⁰

, S

_c⁰_,s,s⁰

) such that S

_c,s,s⁰

and S

_c⁰_,s,s⁰

are two subsquares of S and {c, c

⁰

} = {N W, SE} with s

⁰

< max(

₅₀¹

, 1 − s).

Proposition 3.1.7. There is a constant 0 < b

2

< b

1

such that for all |b| < b

2

, we get the following properties :

1. For every z

1

, I , R

z₁,I

is of

₁₀₀₀¹

-product type.

2. For each admissible square S

z₁,I

for R

z₁,I

, its length side satisfies

`

_z₁_,I

∈ 899

1000 l

_I

, 901 1000 l

_I

!

3. Let S

_z₁_,I

, S

_z₁_,I∪{1}

and S

_z₁_,I∪{2}

be admissible squares for R

_z₁_,I

, R

_z₁_,I∪{1}

and R

z₁,I∪{2}

respectively. There are c, c

⁰

∈ {N E, SE, SW, N W}, e, e

⁰

∈ (

₁₀₀₀⁸⁹⁹

,

₁₀₀₀⁹⁰¹

) and 0 < s

⁰₁

, s

⁰₂

<

₁₀₀¹

such that S

_z₁_,I∪{1}

= (S

z₁,I

)

_c,e,s⁰

1

and S

_z₁_,I∪{2}

= (S

z₁,I

)

_c⁰_,e⁰_,s⁰

2

and (S

z₁,I

, S

_z₁_,I∪{1}

,S

_z₁_,I∪{2}

) forms a configuration NE-SW if |I| is even and

(11)

a configuration NW-SE if |I| is odd.

4. The β

z₁,z₂,I

depend holomorphically on z

1

, z

2

and |

^∂β^z_∂z¹^,z²^,I

1

| <

₅₀¹

.

Figure 2 : U , U

1

and U

2

Figure 3 : R

z₁,I

and its two subsets in projection in the complex case

This proposition simply means that pr

₃

({z

1

}×R

z₁,I

) contains a square which

contains two subsquares included in, pr

₃

({z

1

}×R

z₁,I∩{1}

), pr

₃

({z

1

}×R

z₁,I∩{2}

),

whose lengths are at least

₁₀₀₀⁸⁹⁹

times the length of the larger one, and positioned

in opposite corners.

(12)

3.2 Proof of Proposition 3.1.7

Proof. First, let us remark that item 4 is a direct consequence of Lemma 2.3.5.

Let us prove items 1 and 2. It is a consequence of the following lemma.

Lemma 3.2.1. For all I and z

1

, there exists a constant r

I

which only depends on |I| such that if |b| < b

1

, for all z

1

and I, R

z₁,I

is of r

I

-product type and for all admissible squares S

z1,I

= c

z1,I

+ `

z1,I

.Q ∈ C for R

z1,I

we have :

pr

₁

(R

_z₁_,I

)×(c

z1,I

+(1−r

I

).`

_z₁_,I

.Q) ⊂ R

_z₁_,I

⊂ pr

₁

(R

_z₁_,I

)×(c

z1,I

+(1+r

_I

).`

_z₁_,I

.Q) Besides, we have :

∀z

1

∈ D , ∀I, r

I

≤ 1 1000

Proof. We have that : t

_z₁_,I

= δ

₂

(R

_z₁_,I

). We show Lemma 3.2.1 step by step, the result is the consequence of the convergence of an infinite series. The property is clear for the set U since U = S

z1∈D

({z

₁

} × D × Q). Let us suppose it is true for a finite sequence I and consider a value z

₁

∈ D , then z

₂

∈ I

_I^z¹

and two indices I, I

⁰

= I ∪ {i

p+1

} such that p(z

1

) + bz

2

∈ D . We study the set R

Z₁,I⁰

with the value Z

1

= p(z

1

) + bz

2

. We suppose inductively that for all z

1

such that f

₁⁻¹

(R

Z₁,I⁰

) ∩ R

z₁,I

is not empty, R

z₁,I

is of r

I

-product type and for all admissible square S

z₁,I

= c

z₁,I

+ `

z₁,I

.Q ⊂ C for R

z₁,I

we have : pr

₁

(R

z₁,I

) × (c

z₁,I

+ (1 − r

I

).`

z₁,I

.Q) ⊂ R

z₁,I

⊂ pr

₁

(R

z₁,I

) × (c

z₁,I

+ (1 + r

I

).`

z₁,I

.Q) for a constant r

I

. We denote : R = f

₁⁻¹

(R

Z₁,I⁰

) and δ

_R,I^z¹

= δ

1

(R). Let us show the following lemma.

Lemma 3.2.2. If R

z1,I

is of r

I

-product type, then R

Z1,I⁰

is of r

I⁰

-product type with : r

_I⁰

= r

_I

+ (

₅₀¹

+

_2|µ|^|λ|

)δ

_R,I^z¹

Proof. If |b| < b

1

, by Property 4, we get that there is a square S

Z₁,I⁰

of length

|µ|.(1 − r

I

−

₅₀¹

.δ

_R,I^z¹

).`

z₁,I

such that : pr

₁

(R

Z₁,I⁰

) ×S

Z₁,I⁰

⊂ R

Z₁,I

. We consider now the biggest complex disk S

⁰

included in R

Z₁,I⁰

\ (pr

₁

(R

Z₁,I⁰

) × S

Z₁,I⁰

) and

∆ the line which contains the preimage of this complex disk by f

1

. This line is a line parallel to the z

3

axis, we bound here the diameter δ

3

(S) of the preimage complex disk S = f

₁⁻¹

(S

⁰

). Let us call :

M = {(z

1

, z

₂

, z

₃

) ∈ R, pr

₂

(M ∩ {z

3

= C

^st

}) = pr

₂

(R)}

Then, δ

₃

(S ∩ (R/M)) < (r

_I

+

₅₀¹

.δ

_R,I^z¹

).`

_z₁_,I

(this is still due to Property 4) so δ

3

(f

1

(S ∩ (R/M))) ≤ |µ|.(r

I

+

₅₀¹

.δ

^z_R,I¹

).`

z₁,I

. Besides, δ

1

(M ) ≤ δ

^z_R,I¹

so δ

3

(f

1

(S ∩ M )) ≤ |λ|.δ

^z_R,I¹

.`

z₁,I

. Then the length of S

⁰

which is the length of the image of S by f

₁

is bounded by : (|µ|.(r

_I

+

₅₀¹

.δ

^z_R,I¹

) + |λ|.δ

^z_R,I¹

).`

_z₁_,I

. Then, there is a square S

_Z⁺

1,I⁰

of length (|µ|.(1 + r

I

+

₅₀¹

.δ

^z_R,I¹

) + |λ|.δ

^z_R,I¹

).`

z₁,I

such that : R

Z₁,I

⊂ pr

₁

(R

Z₁,I⁰

) × S

⁺_Z

1,I⁰

. It implies that R

Z₁,I⁰

is of r

I⁰

-product type with : pr

₁

(R

Z₁,I⁰

) × (c

Z₁,I⁰

+ (1 − r

I⁰

).`

Z₁,I⁰

.Q) ⊂ R

z₁,I⁰

⊂ pr

₁

(R

Z₁,I⁰

) × (c

Z₁,I⁰

+ (1 + r

I⁰

).`

Z₁,I⁰

.Q) with `

Z₁,I⁰

=

₁₀⁹

`

z₁,I

and : 2r

I⁰

= 2r

I

+

₅₀²

δ

_R,I^z¹

+

^|λ|_|µ|

δ

^z_R,I¹

so : r

I⁰

= r

I

+ (

₅₀¹

+

_2|µ|^|λ|

)δ

^z_R,I¹

.

The same arguments as in the proof of Lemma 2.2.5 imply that for every

k ∈]0, 1[, there exists 0 < b

₂

< b

₁

such that for all |b| < b

₂

, I, i

_p+1

, we have

(13)

the following : δ

_R,I∪{i^z¹

p+1}

< kδ

^z_R,I¹

. It thus follows from Lemma 3.2.2 that r

_I

converges as |I| → +∞ and

r

I⁰

≤ ( 1 50 + |λ|

2|µ| )( X

n≥0

k

ⁿ

)δ

0

so this can be made smaller than

₁₀₀₀¹

if δ

0

is small enough.

Property 1 of Proposition 3.1.7 is a direct consequence of the previous lemma.

Property 2 of Proposition 3.1.7 is a consequence of Property 1 and Lemma 3.1.5.

Let us now prove Property 3.

Lemma 3.2.3. For every z

₁

, if a complex line ∆ parallel to the third axis of coordinates intersects both {z

1

} × R

_z₁_,I

and {z

1

} × R

_z₁_,I∪{i_p+1_}

, then the sets S = pr

₃

(∆ ∩ ({z

₁

} × R

_z₁_,I

)) and S

⁰

= pr

₃

(∆ ∩ ({z

₁

} × R

_z₁_,I∪{i_p+1_}

)) are squares such that S

⁰

= S

_c,9

10,s⁰

for c ∈ {N E, SE, SW, N W } and s

⁰

≤

₁₀₀¹

. Proof. Let us first study the case where |I| = 1.

Lemma 3.2.4. If a complex line ∆ parallel to the third axis of coordinates intersects U

1

, then pr

₃

(U

1

∩ ∆) is the square Q

_{N E,}9

10,s

, and if ∆ intersects U

2

, then pr

₃

(U

₂

∩ ∆) is the square Q

_SW,9

10,s

with s <

₁₀₀¹

.

Proof. This follows from the choice of constants in Section 2. Indeed, the term λz

₁

=

₁₀¹

√2

2

e

ⁱ^π⁴

z

₁

in the expression of pr

₃

◦ f

₁

is just intended to push the two subsquares in the direction of each of the two NE and SW corners, the two subsquare are of length of side multiplied by the modulus of µ this is

₁₀⁹

.

The demonstration of Lemma 3.2.3 is made by induction on |I|. The induction step is simply the fact that the image by z

3

7→ λ.z

1

+ µ.z

3

of a NW (resp. SW,SE,NE) configuration, this is a subsquare positioned in the NW (resp.

SW,SE,NE) corner of a greater square, is a SW (resp. SE,NE,NW) configuration with length sides multiplied by

₁₀⁹

.

Now, for fixed z

₁

and I, it is enough to take ∆ and ∆

⁰

such that ∆ intersects R

_z₁_,I∪{1}

(resp. ∆

⁰

intersects R

_z₁_,I∪{2}

). We then get two squares which are respectively the square intersection of ∆ and R

_z₁_,I∪{1}

in ∆, and the square intersection of ∆

⁰

and R

_z₁_,I∪{2}

in ∆

⁰

. These two squares are admissible squares for R

_z₁_,I∪{1}

and R

_z₁_,I∪{2}

. Lemma 3.2.3 and Property 1 give us that every admissible square for R

z₁,I

with these two subsquares form the configuration given by Item 3.

The third Property has been shown, the proof of Proposition 3.1.7 is now com-

plete.

(14)

4 Structure of the basic set : perturbed case

4.1 Generalities and main result in the perturbed case

In this section, we analyze the structure of unstable manifolds of the horseshoe induced by an arbitrary perturbation of f

1

. Some arguments in Section 3 used the special form of f

₁

, so we need to adapt the arguments. We can take perturbations of f

₁

, not necessarily polynomial, in the space of automorphisms of C

³

.

Notation. So far we have considered an automorphism f

₁

= f

₁

(p, b) defined in (1). In the following, we will reduce constant b a finite number of times to ensure a certain number of properties. Furthermore, in Section 4, we will introduce decreasing neighborhoods ν

i

(f

1

) for i ∈ {0, 1, 2, 3}. In Section 5, we perturb f

1

to f

2

= f

2

(p, b, σ) with parameters p = p +p

5

z

⁵

and σ such that f

2

∈ ν

3

(f

1

) and ν

4

(f

2

) will denote a non empty neighborhood of f

2

with ν

4

(f

2

) ⊂ ν

3

(f

1

). We will subsequently make a number of further reductions until the final result.

Because of the structural stability of horseshoes, it is well known that Defi- nition 2.3.4 can be applied in a neighborhood ν

0

(f

1

) of f

1

= f

1

(p, b) in the space of automorphisms of C

³

. For such a map f ∈ ν

0

(f

1

), we can define f [1], f [2]

and a sequence of sets U

_I

= f [I]( D

²

× Q) as before. We have the following properties for the sets U

_I

:

Proposition 4.1.1. For every |b| < b

₂

, for every f ∈ ν

₀

(f

₁

), for every finite sequence of indices I = (i

₁

, ..., i

_p

), i

_j

∈ {1, 2}, the set U

_I

satisfies the following properties :

1. U

I

= S

z1∈D

{z

1

} × R

z₁,I

where R

z₁,I

is an open connected set with R

z₁,I

⊂⊂

D × Q.

2. If I is an infinite sequence I ∈ {1, 2}

^N

, the intersection T

p≥0

U

_I≤p

is a component of the intersection of the unstable manifold of some point in the set T

n∈Z

f

ⁿ

( D

²

× Q) with D

²

× Q.

Definition 4.1.2. A u-curve C is a curve whose tangent vectors are all in C

^u

(recall the cones C

^u

, C

^ss

, C

^cs

were defined in Subsection 2.3).

A s-plane is a plane which admits a basis of two vectors, one of them belonging to C

^ss

and the other one to C

^cs

.

A s-surface is a hypersurface whose tangent planes are all s-planes.

We have the counterpart of Lemma 2.3.5 in this context.

Lemma 4.1.3. For |b| < b

2

and f ∈ ν

0

(f

1

), for every sequence of digits I, f [I]( D

.,z₂,z₃

) is a u-curve.

Proof. Lemma 2.3.5 defines a cone field which is invariant by f which thus persists under small perturbations.

The previous Lemma 4.1.3 implies in particular that every piece of unstable manifold T

p≥0

U

I≤p

is a u-curve. The following lemma deals with the horseshoe

for f ∈ ν

0

(f

1

).

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Lemma 4.1.4. For each I, the intersection of U

_I

with a s-surface S is homeomorphic to D × Q. Moreover, there is a canonical homeomorphism Ψ

_S

given by the map S ∩ U

I

7→ D × Q, z 7→ ((pr

₂

, pr

₃

) ◦ (f [I])

⁻¹

)(z).

Proof. The property can be proven by induction. For U

_∅

= D

²

×Q, the property is obvious. Let us suppose it is true until the rank I. Let S be a s-surface intersecting U

_I⁰

= U

_I∪{i0}

with i

⁰

∈ {1, 2}. Then (f [i

⁰

])

⁻¹

(S∩( D

²

×Q))∩( D

²

×Q) is a s-surface too. Then, its intersection with U

_I

is by the induction hypothesis homeomorphic to D × Q by the map z 7→ ((pr

₂

, pr

₃

) ◦ (f [I])

⁻¹

)(z). Now, taking its image by f , we have that : S ∩ U

_I⁰

is homeomorphic to D × Q by the map z 7→ ((pr

₂

, pr

₃

) ◦ (f [I

⁰

])

⁻¹

)(z).

In particular, we have :

Lemma 4.1.5. For all z

1

, I, R

z₁,I

is homeomorphic to D × Q.

Definition 4.1.6. Given a set E ⊂ C and δ ∈ (0, 1), we say that E is δ-square shaped if there is a square S = c

_S

+ `

_S

.Q ⊂ C (`

_S

can be complex non real) such that

(c

_S

+ (1 − δ).`

_S

.Q) ⊂ E ⊂ (c

_S

+ (1 + δ).`

_S

.Q) We call admissible square for E a square S with this property.

Definition 4.1.7. Let S, S

11

, S

12

, S

21

, S

22

be squares in C . We say that S is well subdivided by S

11

, S

12

, S

21

, S

22

if there exist squares S

1

, S

2

such that S

1

, S

2

⊂ S, S

11

, S

12

⊂ S

1

and S

21

, S

22

⊂ S

2

and a linear isomorphism ψ of C such that (ψ(S), ψ(S

1

), ψ(S

2

)) is a configuration NE-SW, and (ψ(S

1

), ψ(S

11

), ψ(S

12

)), (ψ(S

2

), ψ(S

21

), ψ(S

22

)) are configurations NW-SE (see Definition 3.1.6).

Let δ ∈ (0,

₁₀₀¹

). Given five δ-square shaped sets (E

_i

)

_1≤i≤5

, we say that E

₁

is well subdivided by the other sets if there are 5 admissible squares (S

_i

)

_1≤i≤5

respectively for E

_i

such that S

₁

is well subdivided by the other squares.

The following Proposition is an analogous of Proposition 3.1.7.

Proposition 4.1.8. For every |b| < b

₂

, there exists a neighborhood ν

₃

(f

₁

) ⊂ ν

₀

(f

₁

) of f

₁

= f

₁

(p, b) such that for every f ∈ ν

₃

(f

₁

), we have :

1. For every z

1

, I , pr

₃

({z

1

} × R

z₁,I

) is

₅₀₀¹

-square shaped (recall that the sets R

z₁,I

were defined in Proposition 4.1.1).

2. If S

z₁,I

and S

_z₁_,I∪{i_p+1_}

are admissible squares for R

z₁,I

and R

_z₁_,I∪{i_p+1_}

respectively, their length sides satisfy :

`

_z₁_,I∪{i_p+1_}

∈ 895

1000 `

_z₁_,I

, 905 1000 `

_z₁_,I

!

3. Let S

_z₁_,I

, S

_z₁_,I∪{11}

, S

_z₁_,I∪{12}

, S

_z₁_,I∪{21}

and S

_z₁_,I∪{22}

be admissible squares for R

_z₁_,I

, R

_z₁_,I∪{11}

, R

_z₁_,I∪{12}

, R

_z₁_,I∪{21}

and R

_z₁_,I∪{22}

respectively.

Then, S

_z₁_,I

is well subdivided by these 4 subsquares.

So pr

₃

(R

z₁,I

) contains a square which contains two subsquares included in

pr

₃

(R

z₁,I∩{1}

), pr

₃

(R

z₁,I∩{2}

) whose lengths are the length of the great square

times >

₁₀₀₀⁸⁹⁵

and the two squares are near two opposite corners of the great

square, with a change of direction at each step.

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4.2 Proof of Proposition 4.1.8

If we take ν

1

(f

1

) ⊂ ν

0

(f

1

) sufficiently small, the properties are automatically verified for (U, U

1

, U

2

). Our strategy is to consider the sets U

I⁰

as images of pieces of the sets U

I

by f which looks like locally more and more precisely to a linear function when we map the sets U

I

at a smaller scale. Then, by an infinite series argument we will get a result which looks like a bounded distorsion property.

Notation. We will call pr

_w

3,w₂

the orthogonal projection on a vector w

₃

∈ C

^cs

parallel to w

₂

∈ C

^ss

.

Definition 4.2.1. We say that a family of sets {W

I⁰

, I

⁰

∈ {I, I ∪{1}, I ∪{2}, I ∪ {11}, I ∪ {12}, I ∪ {21}, I ∪ {22}} is a family of u-oriented-subdivided tridisks if we have that :

- each W

I⁰

is an union of u-curves

- W

_I∪{1}

, W

_I∪{2}

⊂ W

I

, W

_I∪{11}

, W

_I∪{12}

⊂ W

_I∪{1}

and W

_I∪{21}

, W

_I∪{22}

⊂ W

_I∪{2}

In particular, for each finite sequence I, we easily get this new property for the sets U

_I

: (U

_I

, U

_I∪{1}

, U

_I∪{2}

, U

_I∪{11}

, U

_I∪{12}

, U

_I∪{21}

, U

_I∪{22}

) is a family of u-oriented-subdivided tridisks

Definition 4.2.2. Let I be a finite sequence and η

1

, η

2

<

₁₀¹

. Let {W

I⁰

, I

⁰

∈ {I, I ∪ {1}, I ∪ {2}, I ∪ {11}, I ∪ {12}, I ∪ {21}, I ∪ {22}} be a family of u- oriented-subdivided tridisks. We say that {W

I⁰

, I

⁰

∈ {I, I ∪ {1}, I ∪ {2}, I ∪ {11}, I ∪ {12}, I ∪ {21}, I ∪ {22}} is of (η

1

, η

2

)-type if for every s-plane P and for every w

₃

∈ P ~ ∩ C

^cs

, w

₂

∈ P ~ ∩ C

^ss

, denoting by Π

_I⁰

= pr

_w

3,w₂

(P ∩ W

_I⁰

), we have :

- For every I

⁰

, Π

_I⁰

is η

₁

-square shaped.

- If S

_I

and S

_I∪{i_p+1_}

are admissible squares for Π

_I

and Π

_I∪{i_p+1_}

respectively, their length sides satisfy :

`

_I∪{i_p+1_}

∈ 9

10 − η

2

`

I

, 9

10 + η

2

`

I

- Let S

I

, S

_I∪{11}

, S

_I∪{12}

, S

_I∪{21}

and S

_I∪{22}

be admissible squares for Π

I

, Π

I∪{11}

, Π

_I∪{12}

, Π

_I∪{21}

and Π

_I∪{22}

. Then, S

I

is well subdivided by these 4 subsquares.

Definition 4.2.3. We say that an invertible linear function h of C

³

is well adapted to the cones (C

^u

, C

^cs

, C

^ss

) if C

^u

is invariant by h, C

^ss

and C

^cs

are invariant by h

⁻¹

and for every u ∈ C

^u

, v ∈ C

^ss

, w ∈ C

^cs

, we have that :

||h(u)|| > 10||u||, ||h(v)|| <

₁₀¹

||v|| and ||h(w)|| ∈ (

₁₀⁹

||w||;

¹¹₁₀

||w||).

We easily get the following :

Proposition 4.2.4. For each f ∈ ν

₁

(f

₁

), for each point P = (P

₁

, P

₂

, P

₃

) in (D

₁⁰⁰

× D × Q) ∪ (D

⁰⁰₂

× D × Q), the differential Df

_P

is well adapted to the cones (C

^u

, C

^ss

, C

^cs

).

Proof. Df

P

is diagonalizable : there is an obvious eigenvalue of f

1

which is

µ of eigenvector e

3

such that |µ| =

₁₀⁹

which persists for f ∈ ν

1

(f

1

) with its

Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of low degree in C^{3}