Zhihang An
email: zhihang.an@insa-rennes.fr

Tony Makdissy

María Garcia Vigueras

Raphaël Gillard

Metal-Only Reflectarray Generating Two Independent Beams with Orthogonal Linear Polarizations

This paper presents a Metal-Only reflectarray antenna generating two independent beams with orthogonal linear polarizations. Such reflecctarray is based on a dual polarized cell that allows to control the reflection phase of incident waves independently in both polarizations. The simulated gains of the two beams at 25 GHz are 30.77 dBi and 30.14 dBi respectively.

INTRODUCTION

Reflectarray antennas (RAs) can be seen as a combination of planar arrays and parabolic reflectors with the advantages of high gain, low profile, low cost and easy manufacturing. For some applications, RAs are required to generate multiple independent beams. In [START_REF] Yin | Broadband Dual-Polarized Single-Layer Reflectarray Antenna With Independently Controllable 1-Bit Dual Beams[END_REF], a single layer dual-polarized 1-bit cell is used to design an independently controllable dual-beam reflectarray antenna. So far, several structures [START_REF] Encinar | Dual-Polarization Dual-Coverage Reflectarray for Space Applications[END_REF][START_REF] Chen | A Reflectarray for Generating Wideband Circularly Polarized Orbital Angular Momentum Vortex Wave[END_REF][START_REF] Encinar | Dual-polarization reflectarray in Ku-band based on two layers of dipole arrays for a transmit-receive satellite antenna with South American coverage[END_REF] have been designed with the ability to independently control the reflection phases of different polarizations. All these solutions are based on the use of dielectric substrates, which may bring certain limitations (such as the losses, and other undesired phenomena when working in harsh environments).

This contribution proposes a Metal-Only reflectarray antenna (MORA) allowing to generate two pencil beams with orthogonal polarizations. To the best knowledge of these authors, such an antenna has not yet been proposed in the open literature. The paper is organized as follows. Section II presents the MORA unit-cells, and Section III explains the design process and shows its performance based on full-wave simulations. Finally, conclusions are drawn in Section IV.

II. METAL ONLY REFLECTARRAY UNIT-CELL

A. Description of metal-only unit-Cells

Recently, a 3D Phoenix Cell (PC) was proposed by some of the authors and it was fabricated using Selective Laser Melting technology [START_REF] Makdissy | Dual linearly polarised 3D printed Phoenix cell for wide band metal only reflectarrays[END_REF]. It consists of two concentric square metallic waveguides and a square metallic block at the center. Inspired by the 3D PC, a novel metal-only cell is used here to design two beams MORA. As shown in Fig. 1, the difference between the initial 3D PC and the metal-only cell considered here is that four walls are added to split the two waveguides into two parts. This modification allows the proposed cell to independently control the incident waves with orthogonal linear polarizations. In addition, the cross-section of the cell is fixed, and the reflection phase is controlled by tuning independently the heights of both waveguides (hin and hex) as it will be explained next.

The cross-section of the cell is first designed so that the inner or outer waveguides are both propagative at the considered operating frequency, f0 = 25 GHz. The outer waveguide supports a vertically polarized mode (along x) while the inner one supports a horizontally polarized one (along y). Note that the cutoff frequency of the inner waveguide is reduced below f0 thanks to the square metallic block (Lp), which size is optimized to make propagation possible. Then, the reflection phase at f0 for both modes is controlled by adjusting only the heights of the waveguides: hin controls the phase of y polarization while hex deals with x one. In order to eliminate the coupling between both waveguides and to control the reflection phase independently in both polarizations, four walls are added to the cell (they are highlighted in red in Fig. 1). These four walls impose that the modes in the inner and outer waveguides are orthogonally polarized. Thus, the coupling between both waveguides can be minimized as much as possible by selecting appropriate wall thicknesses (w1, w2). As a result, the proposed metal-only cell has two degrees of freedom to control the reflection phase independently and it can be used to design a MORA providing two independent beams with orthogonal polarizations.

B. Performance of unit-cell

Before the design of the dual-beam MORA, the performance of the proposed metal-only cell is verified using simulations with HFSS ® . A conductivity of 1.33×10 6 S/m is used to represent the real metal material. Typical Floquet boundary conditions are employed for this simulation to mimic an infinite array of identical elements. The detailed parameters of the cell are tabulated in Table I. Figure 2 shows the simulated reflection phase of the metal-only cell at normal incidence and at 25GHz. Note that the electric field direction is parallel to the y-axis for Fig. 2 (left) and to the x-axis for Fig. 2 (right). As it can be seen, the complete 360º phase range can be achieved by increasing the height of the inner waveguide up to 15mm, and up to 9mm for the external one. Furthermore, the coupling between both waveguides is negligible: the maximum phase difference is 4° when varying hex for hin=7.5mm. This demonstrates that hin and hex can almost independently control the reflection phase in both polarizations. The reflection phases at different incident angles are shown in Fig. 3. The parameters hex and hin are set to 5mm (Fig. 3(left)) and 8mm (Fig. 3(right)), respectively. A maximum phase difference of 20° is obtained when varying the incident angle (theta, phi) for hex=3.5mm, which is completely acceptable for the design of a RA. Finally, a medium incident angle (15°, 90°) is adopted to simulate the database used to design an array. Figure 4 shows the reflection phase and loss at the incident angle (15°, 90°). It can be seen that the loss of the inner waveguide is significantly higher than that of the outer waveguide, which may deteriorate the gain. In this section, a dual-beam MORA is designed using the database in Fig. 4. The radiating aperture is 15λ0× 15λ0 while the inter-element spacing is 0.5λ0. The RA is illuminated by a linearly-polarized horn antenna with an offset angle of (15°, 90°) with respect to the normal of the panel. The f/d ratio is 1. The directions of the beams are (15°, 225°) and (15°, 315°) respectively. The hin and hex heights of each cell are selected one by one to meet the required phase distributions for both beams (hin for beam (15°, 225°) and hex for (15°, 315°)). The designed MORA is simulated in HFSS ® together with its illuminating horn antenna. Figure 5 shows the simulated normalized radiation patterns at f0 for a x-and y-polarized incident waves respectively. The horn antenna is rotated by 90° around the direction of propagation to change the polarization of the incident wave. It can be seen that the side lobe level and cross polarization level for both beams is below -29dB and -37dB respectively. The simulated gains are 30.14 dBi and 30.77 dBi respectively. The gain difference is mainly due to the loss of inner waveguide shown in Fig. 4. IV. CONCLUSION

In this work, a novel metal-only cell is proposed and analyzed. The proposed cell has the ability to independently control the reflection phase of incident waves with orthogonal linear polarizations. A two beams MORA is designed using the proposed cell. The simulated gains of two pencil beams are 30.14dBi and 30.77 dBi respectively.

Figure 1 .

 1 Figure 1. The proposed unit-cell : (left) top and (right) side views.

Figure 2 .Figure 3 .Figure 4 .

 234 Figure 2. Phase response of the cell at 25GHz at normal incidence: inner waveguide (left); outer waveguide (right)

Figure 5 .

 5 Figure 5. The simulated normalized radiation pattern: the beam controlled by inner waveguide (phi=225°) (left); the beam controlled by outer waveguide (phi=315°) (right)

TABLE I .

 I DETAILED PARAMETERS OF METAL-ONLY CELL

	Parameter	Lc	Lr	Lp	hin	hex	win	wex	w1	w2
	Value (mm) 6	4.7 3	0-15	0-9	0.3	0.2 1.6 0.4

Scientific Research Lebanon (CNRS-L) and China Scholarship Council (CSC).

ACKNOWLEDGMENT

The work was jointly funded by the Agence Universitaire de la Francophonie (Moyen-Orient), the National Council for


Jasmin Raissy

INTRODUCTION TO FATOU COMPONENTS IN HOLOMORPHIC DYNAMICS

This survey is an introduction to the classification of Fatou components in holomorphic dynamics. We start with the description of the Fatou and Julia sets for rational maps of the Riemann sphere, and finish with an updated account of the recent results on Fatou components for polynomial skew-products in complex dimension two, where we focus on the key steps in the construction giving the existence of a wandering domain for a polynomial endomorphism of C 2 .

Exercise 1.5. Assume f : C → C is a Tchebychev polynomial of degree d, i.e., such that cos(dθ

Those examples are really atypical examples, since most cases exhibit fractal and chaotic behaviour, the analysis of which requires tools from complex analysis, dynamical systems, topology, combinatorics, . . .

Introduction

Discrete dynamics concerns maps f : X → X from a space to itself. One of the main goals is to understand sequences of the form

x 0 ∈ X, x 1 := f (x 0 ), x 2 := f (x 1 ) = f •2 (x 0 ), x 3 := f (x 2 ) = f •3 (x 0 ), . . .
obtained by iterating f . The forward orbit of x 0 under f is

O + (x 0 ) := n≥0 f •n (x 0 ) .
The main problem is to understand the long term behaviour of the sequence (x n ) n≥0 . The answer will depend on the assumptions we make on the space X and the map f : X → X.

If X is a topological space, we may wonder whether the sequence converges. If this is the case, and in addition f : X → X is a continuous map, then the limit is a fixed point of f , i.e., f ( ) = .

If X is compact, then thanks to the Bolzano-Weierstrass Theorem, we may extract converging subsequences. We shall denote by ω(x 0 ) the ω-limit set of x 0 , that is the set of possible limit values:

ω(x 0 ) := n≥0 O + (x n ).
If f : X → X is continuous, then for any x ∈ X, the ω-limit set ω(x) is f -invariant.

Exercise 1.1. Prove it.

In the rest of these notes, we will study the case where X is a complex manifold and f : X → X is a holomorphic map. We shall first consider the case where X = C = C ∪ {∞} is the Riemann sphere and f : C → C is a polynomial or a rational map. This subject has a fairly long history, with contributions by Koenigs [START_REF] Koenigs | Recherches sur les intégrales de certaines équations fonctionnelles[END_REF], Schröder [START_REF] Schröder | Ueber iterirte Functionen[END_REF], Böttcher [START_REF] Böttcher | The principle laws of convergence of iterates and their applications to analysis (Russian)[END_REF] in the late 19th century, and the great memoirs of Fatou [START_REF] Fatou | Sur les solutions uniformes de certaines équations fonctionnelles[END_REF][START_REF] Fatou | Sur les équations fonctionnelles (Deuxième mémoire)[END_REF][START_REF] Fatou | Sur les équations fonctionnelles (Troisième mémoire)[END_REF] and Julia [START_REF] Julia | Mémoire sur l'itération des fonctions rationelles[END_REF] around 1920. For the history of this early period we refer the reader to the book of Alexander [START_REF] Alexander | A history of complex dynamics[END_REF]. Followed a dormant period, with notable contributions by Cremer [START_REF] Cremer | Über die Schrödersche Funktionalgleichung und das Schwartsche Eckenabbildungsproblem[END_REF][START_REF] Cremer | Über die Häufigkeit der Nichtzentren[END_REF] (1936) and Siegel [START_REF] Siegel | Iteration of analytic functions[END_REF] (1942), and a rebirth in the 1960's (Brolin, Guckenheimer, Jakobson). Since the early 1980's, partly under the impetus of computer graphics, the subject has grown vigorously, with major contributions by Douady, Hubbard, Sullivan, Thurston, and more recently Lyubich, McMullen, Milnor, Shishikura, Yoccoz . . . Although the subject still presents many open problems, there is now a substantial body of knowledge and several books have appeared: Beardon [START_REF] Beardon | Iteration of rational functions. Complex analytic dynamical systems[END_REF], Carleson and Gamelin [START_REF] Carleson | Complex dynamics. Universitext: Tracts in Mathematics[END_REF], Milnor [START_REF] Milnor | Dynamics in one complex variable[END_REF] and Steinmetz [START_REF] Steinmetz | Rational iteration. Complex analytic dynamical systems[END_REF] and the more specialized books by McMullen [START_REF] Mcmullen | Complex Dynamics and Renormalization[END_REF] [START_REF] Mcmullen | Renormalization and 3-manifolds which fibers over the circle[END_REF]. Surveys by Blanchard [START_REF] Blanchard | Complex analtic dynamics on the Riemann sphere[END_REF], Devaney [START_REF] Devaney | An introduction to Chaotic Dynamical Systems[END_REF], Douady [START_REF] Douady | Disques de Siegel et anneaux de Herman[END_REF], Keen [START_REF] Keen | Julia sets, in "Chaos and Fractals, The Mathematics behind the Computer Graphics[END_REF] and Lyubich [START_REF] Lyubich | Dynamics of rational transformations: topological picture[END_REF] are also highly recommended.

There is a small collection of polynomials, for instance

f (z) = az + b, f (z) = z d and f (z) = z 2 -2,
whose dynamics can be fairly easily understood. (1) Show that when |a| = 1 or when a = 1 and b = 0, the map

C z → ω(z) ∈ C
is constant. [START_REF] Arnold | Geometrical Methods in the Theory of Ordinary Differential Equations[END_REF] Show that in all other cases, ω(z) depends on z and is either a finite set, or a euclidean circle.

Exercise 1.3. Assume f (z) = z d with d ≥ 2.

(1) Show that ω(z) = {0} if z belongs to the unit disk D and ω(z) = {∞} if z ∈ C \ D. (2) Show that (a) ω(z) ⊂ S 1 if z belongs to the unit cirle S 1 , (b) ω(z) can be equal to S 1 and (c) ω(z) can be a finite set of arbitrary cardinality.

Exercise 1.4. Assume f (z) = z 2 -2.

(1) Show that every z ∈ C \ {0} may be written z = w + 1/w for some w

∈ C with |w| = 1 if z ∈ [-2, 2] and |w| > 1 if z ∈ C \ [-2, 2]. (2) Show that (a) ω(z) = {∞} if z ∈ C \ [-2, 2] and (b) ω(z) ⊂ [-2, 2] if z ∈ [-2, 2].

Fatou sets and Julia sets

Both Fatou's and Julia's work have as a major theme that the Riemann sphere C breaks up sharply into two complementary subsets: the open Fatou set F f with orderly dynamics, and the Julia set J f on which the dynamical behaviour of f is chaotic. In practice, understanding the dynamics of a rational map has come to mean understanding the topology of the Julia set, its geometry if possible, and classifying the components of the Fatou set.

Definition.

There are several possible definitions of the Fatou and Julia sets. We will adopt the one based on normal families, which seems very natural to us: a normal family of analytic maps is "well-behaved". But the very fact that it has become standard reflects the origins of holomorphic dynamics in complex analysis, and may explain why the subject has never quite entered the main stream of dynamical systems. Definition 2.1 (Normal families). A family (f α : U → C) α∈A of holomorphic maps on an open subset U ⊂ C is normal if from any sequence (f αn ) n≥0 one can extract a subsequence converging uniformly (for the spherical metric in the range) on compact subsets of U .

A criterion that will enable us to deal with normality is due to Montel. In particular, a bounded sequence of holomorphic maps (f n : U → C) n∈N is normal. Indeed, the family of iterates (f •n : C \ S 1 → C) n∈N takes its values in C \ S 1 , hence it omits more than three values in C. By Montel's theorem, it is normal and so, J f ⊆ S 1 . To prove that S 1 ⊆ J f , one may argue that the sequence f •(2n) n∈N converges uniformly on any compact subset of D to the constant map 0 and on any compact subset of C \ D to the constant map ∞. As any neighborhood of a point z ∈ S 1 contains points in D and points in C \ D, there is no neighborhood of z on which the sequence f •n n∈N is normal.

2.2. The polynomial case. In the case of a polynomial, there is a slightly different way of understanding things. In this case, ∞ is a fixed point and this fixed point has no preimage other than itself. It is an exceptional point.

Definition 2.5. Let f : C → C be a polynomial of degree d ≥ 2. The filled-in Julia set of f is the set K f of points with bounded orbits: This set is not only very natural for a dynamical system, but fits in very nicely with the general theory.

K f = z ∈ C f •n (z) n∈N is bounded .
Proposition 2.6. The filled-in Julia set K f is a non-empty compact set. The Julia set is the topological boundary of K f . Proof. Since f is of degree d ≥ 2, f (z)/z → ∞ as z → ∞, and so, there exists a constant R > 0 such that f (z) > 2|z| whenever |z| > R. Set D R := {z ∈ C | |z| > R}. Then, the nested intersection of compact sets

K f = n∈N f -n D R
is a non-empty compact set.

Furthermore, define U R to be the open set U R := C \ D R . Then, the sequence (f •n ) n≥0 converges uniformly to infinity on U R . Clearly, the same property holds on the open set

Ω ∞ = n∈N f -n (U R ) Observe that Ω ∞ = C \ K f and ∂Ω ∞ = ∂K f .
We will first show that ∂K f ⊆ J f . Choose a point z 0 ∈ ∂Ω ∞ = ∂K f and assume that there is a neighborhood U of z 0 on which the family of iterates of f is normal. Then, U intersects Ω ∞ and the sequence (f •n ) n≥0 must converge to infinity on the whole set U . This contradicts the fact that the sequence (f •n (z 0 )) n≥0 is bounded.

We will now show that J f ⊆ ∂K f . As mentioned previously, the sequence (f •n ) converges locally uniformly to a map which is constant and equal to ∞ outside K f . Thus, the complement of K f is contained in the Fatou set F f and J f ⊆ K f . Similarly, on the interior of K f , the sequence (f •n ) is bounded, thus normal by Montel's theorem. As a consequence, the interior of K f is contained in F f and J f ⊆ ∂K f . The reason why K f is called a filled-in Julia set is the following.

Definition 2.7. A compact subset K ⊂ C is called full if it is connected and its complement is connected. Proposition 2.8. Let f : C → C be a polynomial. Each connected component of K f is full.
Proof. Of course, any component K 0 is compact and connected. Any component U of the complement must satisfy ∂U ⊂ K 0 , and if the complement is not connected at least one such component must be bounded. Then, by the maximum principle

sup U f •n (z) = sup ∂U f •n (z) ≤ sup K0 f •n (z) .
Hence, it is uniformly bounded, so U ⊂ K 0 .

2.3.

Invariance of Fatou sets and Julia sets. Proposition 2.9. Both the Julia set and the Fatou set are completely invariant:

f (J f ) = J f , f -1 (J f ) = J f , f (F f ) = F f , f -1 (F f ) = F f .
Proof. As the Julia set is the complement of the Fatou set, it is enough to prove the statements about the Fatou set. This is an immediate consequence of the following lemma.

Lemma 2.10. Let U be an open subset of C, and let φ : U → C be holomorphic and non-constant. Then the family (g α : Proof. (⇒) This is obvious. If the sequence g αn converges to g, then f αn := g αn • φ converges to g • φ.

φ(U ) → C) α∈A is normal if and only if (f α := g α • φ : U → C) α∈A is normal.
(⇐) Assume that the sequence f α k := g α k • φ is uniformly convergent on every compact subset of U . Let K ⊂ φ(U ) be compact, and choose a cover of U by open subsets U i with compact closure. Since φ is open, φ(U i ) is an open cover of K, and there exists a finite subcover φ(U ij ) . The set

K := φ -1 (K) ∩ j U ij is a compact subset of U such that φ(K ) = K. Then, sup z∈K dist C (g αn (z), g αm (z)) = sup w∈K dist C (f αn (w), f αm (w)) → 0 as n, m → ∞. The lemma follows.
To prove F f = f -1 (F f ), apply this lemma to the families

f •(m+1) : f -1 (F f ) → C and f •m : F f → C
and to the holomorphic map f :

f -1 (F f ) → F f . Applying f to both sides of the equality F f = f -1 (F f ), we deduce that f (F f ) = F f .
Proposition 2.12. The Julia and Fatou sets of the k-th iterate f •k are equal to the Julia and Fatou sets of f :

J f •k = J f and F f •k = F f Proof.
It is enough to prove this proposition for the Fatou sets. If the family of iterates f •n is normal on an open set U , then the subfamily f •(kn) is also normal. Hence,

F f ⊆ F f •k . Now, assume that z ∈ F f •k
and let U be a neighborhood of z on which the family of iterates f •(kn) is normal. For any sequence (n i ) i∈N , there exists an integer N such that n i = N mod (k) for infinitely many i's. Let us extract this subsequence which, with an abuse of notations, we still denote by n i . By assumption, the sequence f •(ni-N ) : U → C is normal and we can extract a subsequence converging to a map g :

U → C. As f •ni = f •N • f •(ni-N ) , we can extract a subsequence of the sequence f •ni : U → C converging to f •N • g : U → C. This shows that the family f •n : U → C n∈N is normal and that F f •k ⊆ F f .

Conjugacy

In every part of mathematics, the morphisms are the maps that preserve the structure. Thus the morphisms of a dynamical system f : X → X to a dynamical system g : Y → Y are the maps sending sequences of iterates of f to sequences of iterates of g. This is just what maps φ : X → Y such that φ • f = g • φ do; such maps are called semi-conjugacies between f and g. Indeed, we see by induction that then φ

• f •n = g •n • φ for all n ≥ 1: by hypothesis it is true for n = 1, and if it is true for n -1, then φ • f •n = φ • f • f •(n-1) = g • φ • f •(n-1) = g • g •(n-1) • φ = g •n • φ. When φ is invertible, φ • f = g • φ can be rewritten f = φ -1 • g • φ,
so f and g are conjugate: conjugacies are the isomorphisms of dynamical systems. When studying dynamical systems, we constantly try to construct conjugacies or semi-conjugacies to simpler dynamical systems, linear ones in particular. Definition 3.1 (Conjugate rational maps). Two rational maps f and g are topologically (respectively analytically) conjugate if there exists a homeomorphism φ :

C → C (respectively an analytic isomorphism) such that φ • f = g • φ.
One should think of φ as a change of coordinates on C. The only analytic isomorphisms of C are the Möbius transformations z → (az + b)/(cz + d) and the only isomorphisms of C are the affine maps z → az + b. Being a zero or a pole of a rational map depends on the coordinates, whereas being a fixed point or a critical point of a rational map does not depend on the choice of coordinates. Proposition 3.2. If f and g are conjugate (topologically or analytically) by φ :

C → C, then φ(F f ) = F g and φ(J f ) = J g . Proof. The family (f •n : U → C) n∈N is normal if and only if the family (φ • f •n • φ -1 : φ(U ) → C) n∈N is normal.
Example 3.3. Example We will see below that the Newton's methods

N P : z → z - P (z) P (z) and N Q : z → z - Q(z) Q (z)
are analytically conjugate as soon as Q(z) = λP (αz + β).

Exercise 3.4. Let a = b be complex numbers and set P (z) := (z -a)(z -b).

(1) Show that the Newton's method N P is conjugate to f : w → w 2 via the isomorphism z → w := (z -a)/(z -b).

(2) Determine the Julia set of N P and the set ω(z) for z ∈ C \ J N P .

Periodic point and critical points

Periodic points and critical points are important objects in holomorphic dynamics as we shall see in this section.

T α C → T α C is called the multiplier of f at α. The fixed point is called • superattracting if λ = 0; • attracting if 0 < |λ| < 1; • repelling if |λ| > 1; • indifferent if |λ| = 1, and more precisely -parabolic if λ is a root of 1, -elliptic otherwise, that is if λ = e 2πiθ with θ ∈ R \ Q. Exercise 4.2.
Let α be a fixed point of a holomorphic map f .

(1) Show that if α = ∞, then the multiplier of f at α is λ = f (α).

(2) Show that if α = ∞ and if f (z) ∼ az as z → ∞, then the multiplier is λ = 1/a, whereas if f (z) ∼ az k with k ≥ 2, the multiplier is λ = 0. (3) Show that if f is a polynomial of degree d ≥ 2, then ∞ is a superattracting fixed point of f .

The multiplier at a fixed point is invariant under analytic conjugacy. In other words, if φ : ( C, α) → ( C, β) is a local isomorphism conjugating f : ( C, α) → ( C, α) to g : ( C, β) → ( C, β), then the multiplier of f at α is equal to the multiplier of g at β. Indeed, the derivative of φ

• f = g • φ at α is D α φ • D α f = D β g • D α φ. In other words, D α φ conjugates D α f : T α C → T α C to D β g : T β C → T β C
and the two endomorphisms have the same eigenvalue.

Definition 4.3 (Periodic points).

A periodic point of f is a fixed point of f •p for some p ≥ 1. The smallest such integer p ≥ 1 is called the period of α. In this case, we say that α, f (α), . . . , f •(p-1) (α) is a cycle of f . Proposition 4.4. If α 1 , α 2 , . . . , α p is a cycle, then the multiplier of f •p at all points of the cycle is the same.

Proof. According to the Chain Rule, the derivative of f •p at any point of the cycle is the composition of the derivatives of f , as linear transformations, along the cycle. If the derivative of f vanishes at a point of the cycle, then the multiplier is 0 at all points of the cycle. If the derivative never vanishes, then f is locally invertible at each point of the cycle and f conjugates f •p at α i to f •p at α i+1 . Thus, the multiplier of f •p at α i is equal to the multiplier of f •p at α i+1 . Definition 4.5 (Multiplier along a cycle). The multiplier of f along a cycle of period p is the multiplier of f •p at any point of the cycle. The cycle is superattracting (respectively attracting, indifferent or repelling) for f if the points of the cycle are superattracting (respectively attracting, indifferent or repelling) fixed points of f •p .

If α 1 , α 2 , . . . , α p is a cycle of period p avoiding ∞, then its multiplier is given by the product

λ = (f •p ) (α 1 ) = p i=1 f (α i ).
Proposition 4.6. Let f : C → C be a rational map, and let α be a periodic point of f . If α is superattracting or attracting, then α ∈ F f . If α is repelling, then α ∈ J f . Proof. Replacing f by f •p if necessary, we may assume that α is a fixed point of f . Conjugating by a Möbius transformation if necessary, we may assume that α = 0. If α is (super)attracting, i.e., |f (α)| < 1, then there exists a bounded neighborhood U of α which is mapped into itself. In particular, for all n ∈ N, f •n (U ) ⊂ U . Hence, the family (f

•n : U → C) n∈N is normal, which implies that α ∈ F f . If α is repelling, then (f •n ) (α) = f (α) n -→ n→+∞ ∞.
So, the family (f •n ) n∈N cannot be normal in a neighborhood of α.

A cycle is superattracting if and only if it contains a point at which the derivative of f vanishes, i.e., a critical point of f . 4.2. Critical points and critical values. We will see that the dynamical properties of a rational map f : C → C are strongly related to the dynamical behaviour of the critical points of f . For example, the Julia set of a polynomial f is connected if and only if the orbit of every critical point of f is bounded. Definition 4.7 (Critical points and critical values). Let U and V be two Riemann surfaces and f : (U, x) → (V, y) be an analytic map which is not locally constant at x. The point x is a critical point if the derivative D x f : T x U → T y V is zero. In that case, y is a critical value. More precisely, in local coordinates z vanishing at x and w vanishing at y = f (x), the expression of f is of the form z → aw k + O(w k+1 ) for some integer k ≥ 1 and some complex number a = 0.

The integer k does not depend on the choice of coordinates and is called the local degree of f at x. The point x is a critical point if and only if k ≥ 2. In that case, x is called a critical point of f of multiplicity k -1.

Example 4.8. Example The map z → z 3 has a critical point at 0. The local degree of the map at 0 is 3, and the multiplicity of this critical point is 2.

We shall use the following result which is a consequence of the Riemann-Hurwitz Formula (which will not be proved in those notes).

:= f -1 (V ). Then, χ(U ) = d • χ(V ) -N
where N is the number of critical points of f in U , counting multiplicities. Exercise 4.11. Let f : C → C be a rational map of degree d.

(1) Prove that f has 2d -2 critical points counting multiplicities.

(2) Prove that f has at least 2 distinct critical values.

(3) Prove that if f is a polynomial, then ∞ is a critical point of multiplicity d -1 and that there are d -1 critical points in C counting multiplicities.

5. Description of the Julia set 5.1. Topology of the Julia set.

Proposition 5.1. The Julia set J f is non empty and compact.

Proof. If the Julia set were empty, there would be a subsequence of f •m converging uniformly on C; let f 0 be the limit. The rational map f 0 must have some finite degree. Since the degree is an invariant of homotopy, the approximating iterates f •mi must have the same degree. But the degree of f •mi is d mi , which leads to a contradiction. Evidently the Fatou set F f is open, hence the Julia set J f is closed. Since C is compact, the Julia set J f is also compact. Proposition 5.2. Either the Julia set J f has empty interior, or it is the entire Riemann sphere.

Proof. Suppose U ⊂ J f is an open subset of C. The family {f •m : U → C} must avoid F f , which is open and will contain more than two points if it is not empty. But this would make {f •m : U → C} normal, a contradiction. Thus if the interior of J f is non empty, J f is the entire Riemann sphere.

We have seen that the Julia set J f is completely invariant. We will now determine whether there are any other completely invariant closed sets than J f . For this purpose, we will use the Riemann Hurwitz formula. Proposition 5.3. Let E be a completely invariant closed set. If E is finite, then E contains at most two points.

Proof. If E is finite, then its complement U is connected and completely invariant. Its Euler characteristic χ(U ) = 2-card(E) is finite. Let us denote by N the number of critical points of f in U counted with multiplicity. The Riemann-Hurwitz formula applied to f : U → U gives

χ(U ) = d • χ(U ) -N. Thus χ(U ) = N d -1 ≥ 0, which leads to card(E) ≤ 2.
A union of completely invariant sets is still completely invariant, therefore the following definition makes sense.

Definition 5.4. The exceptional set E f is the largest finite completely invariant set.

Remark 5.5. The grand-orbit of a point is a totally invariant set. If it is a finite set, then, by definition of the exceptional set E f , we have z ∈ E f .

Observe that when two rational maps f : C → C and g : C → C are conjugate by a homeomorphism φ, then the homeomorphism φ sends the forward (resp. backward) orbit of a point z ∈ C under the action of f to the forward (resp. backward) orbit of φ(z) under the action of g. Hence, f -invariant sets are mapped by φ to g-invariant sets. Thus, the largest finite completely f -invariant set is mapped by φ to the largest finite completely g-invariant set. In other words, when it is not empty, the exceptional set E f is mapped by φ to the exceptional set E g .

The exceptional set E f is a minor irritant, which will come back to annoy us on several occasions. The following description should remove any doubts about its lack of significance.

Proposition 5.6. Up to conjugacy, the exceptional set E f is not empty in precisely two cases:

• when f (z) = z d with |d| ≥ 2, E f = {0, ∞} and • when f is a polynomial but is not conjugate to z → z d , E f = {∞}.
Proof. It is clear from the construction that E f is completely invariant; so if it has only one point, this point must be fixed, and if it consists of two points, these must be fixed or exchanged by f . We will examine these three cases one at a time.

If E f has just one point a, move it to ∞, for example by making the change of coordinates w = 1/(z -a). Then f = P/Q is a rational map such that infinity is its only inverse image. Hence, the polynomial Q has no roots, it is constant, and f is a polynomial.

If E f has two points a and b, both fixed, then we can put one at ∞ and the other at 0 (make the change of coordinates w = (z -b)/(z -a)). We see that f is a polynomial and that 0 is the only root of the polynomial equation f (w) = 0, so f (w) = λw d for some λ. Choose a number ν such that ν d-1 = λ, and make the change of coordinates z = νw. Then, it is easy to see that the expression of f becomes z → z d .

If E f has two points exchanged by f , put one at ∞ and the other at 0, and write f = P/Q. Since ∞ is the only inverse image of 0, we see that P has no roots, so it is a constant, which we may take to be 1. Since 0 is the only inverse image of ∞, we see that Q(w) = λw d for some λ = 0, so that f (w) = 1/(λw d ). We can do a change of coordinate z = νw, where ν d+1 = λ. Again it is easy to check that the expression of f becomes z → z -d . Proposition 5.7. The exceptional set E f is always contained in the Fatou set F f . Proof. In all three cases, E f is a union of superattracting cycles. Proposition 4.6 shows that such cycles are always contained in the Fatou set F f .

We can now give a new characterization of the Julia set.

Proposition 5.8. The Julia set of a rational map is the smallest completely invariant closed set containing at least three points.

Proof. The Julia set is a completely invariant closed set. It contains at least three points, since otherwise it would be contained in the exceptional set E f ⊂ F f . Furthermore, if E is a completely invariant closed set containing at least 3 points, the complement Ω = C \ E of E is a completely invariant open set omitting more than three points. Hence, by Montel's theorem, the family of iterates f •n is normal on Ω. Thus Ω ⊂ F f and J f ⊂ E.

The following proposition asserts that if z does not belong to the exceptional set E f , then the closure of the backward orbit of z,

O -(z) := m≥0 f -m {z},
contains the Julia set J f . Proposition 5.9. For any z / ∈ E f , we have

J f ⊆ O -(z).
Proof. Take a point z 0 ∈ J f and let U be an arbitrary neighborhood of z 0 . We must show that the set Ω = n∈N f •n (U ) contains z. We will show that its complement E = C \ Ω is contained in the exceptional set E f . Indeed, Ω is a forward invariant open set which omits at most two points, since otherwise the family f •n would be normal on U . Thus, E = C \ Ω is a backward invariant set. Since E is finite, it is permuted by f , so it is forward invariant. Hence, by definition of the exceptional set, it is contained in E f .

Proposition 5.10. The Julia set J f is perfect.

Proof. Since the Julias set is closed, it suffices to prove that it has no isolated points. The Julia set J f is not finite. Thus, it contains a non-isolated point z 0 .

Since the backward orbit of z 0 is dense in J f , any point in J f is non-isolated. This proves that J f is perfect, hence uncountable.

5.2. Julia sets and periodic points. Fatou and Julia both proved that the Julia set is the closure of the set of repelling periodic points. This "dynamical" definition of the Julia set is probably more natural to specialists of dynamical systems, and relates holomorphic dynamics to "Axiom A attractors" and hyperbolic dynamics. Fatou's and Julia's proofs are different but both involve results which will be proved later. Fatou's proof is based on the fact that there are only finitely many nonrepelling cycles. Julia's proof is based on the existence of a repelling cycle.

In these notes, we will only prove the following weaker result which follows directly from Montel's Theorem. This result is the first step in Fatou's proof. The density of repelling cycles in the Julia set then follows from the finiteness of non-repelling cycles.

Proposition 5.11. The closure of the set of periodic points contains the Julia set.

Proof. Let z 0 ∈ J f be arbitrary. Since J f is perfect, we may assume that z 0 is not a critical value of f •2 , so there exist a neighborhood U of z 0 and three branches g 1 , g 2 , g 3 of f -2 with mutually disjoint images. We are considering the second iterate because if d = 2, there would only be two branches of f -1 .

Now consider the family of maps

h m (z) := f •m (z) -g 1 (z) f •m (z) -g 2 (z) g 3 (z) -g 1 (z) g 3 (z) -g 2 (z) . If f •m (z) = g i (z), i = 1, 2,
3 for all z in some neighborhood U of z, then the maps h m (z) are never 0, 1 or ∞ in U , hence they form a normal family. But then the family f •m is also normal, since we can express f •m in terms of h m :

f •m = g 1 (g 3 -g 2 ) -g 2 (g 3 -g 1 )h m (g 3 -g 2 ) -(g 3 -g 1 )h m .
This is a contradiction, so in every neighborhood of z there are roots of f •m (z) = g i (z) for some i and some m, i.e., roots of f •(m+2) (z) = z.

Connectivity of Julia sets of polynomials.

Here is a first relation between the behaviour of critical orbits and the topology of the Julia set.

Theorem 5.12 (Fatou). The filled-in Julia set K f of a polynomial f is connected if and only if all critical points of f belong to K f . Remark 5.13. Note that since the Julia set

J f is the boundary of K f , J f is connected if and only if K f is connected.
Proof. Since f is a polynomial of degree d ≥ 2, we have that f (z)/z tends to infinity as z tends to infinity, hence there exists R > 0 so that

f (z) ≥ 2|z| for |z| > R. Set U 0 := C \ D R and define inductively U n+1 := f -1 (U n ).
By the Riemann-Hurwitz formula, we have

χ(U n+1 ) = dχ(U n ) -(d -1)
, where the number d -1 is the multiplicity of the critical point at ∞. Since U 0 is simply connected, we have χ(U 0 ) = 1, and by induction, we see that when all the critical points of f belong to K f , then χ(U n ) = 1 for all n ≥ 0. It follows that

Ω ∞ := n≥0 U n is simply connected and its complement K f is connected. Conversely, let Ω ∞ ⊂ C be the connected component of the Fatou set F f which contains ∞. Since the only inverse image of ∞ is ∞ itself, f -1 (Ω ∞ ) = Ω ∞ . Applying the Riemann-Hurwitz formula to f : Ω ∞ → Ω ∞ , we get χ(Ω ∞ ) = dχ(Ω ∞ ) -(d -1) -k,
where the number d-1 is contributed by the critical point at ∞, and the number k is the number of other critical points in Ω ∞ (i.e., not in

K f ) counted with multiplicity. If χ(Ω ∞ ) is finite, then χ(Ω ∞ ) = 1 + k/(d -1). But the Euler characteristic of a connected plane domain is at most 1. So χ(Ω ∞ ) = 1 if no critical point is attracted to ∞; otherwise χ(Ω ∞ ) is infinite.

Description of the Fatou set

We now return to the case where f is a rational map, not necessarily a polynomial. Our goal is to give a classification of periodic Fatou components.

Definition 6.1. A map f : U → V is a proper map if for any compact set K ⊆ V , the preimage f -1 (K) is compact. Proposition 6.2. If U is a connected component of F f , then f (U ) is also a con- nected component of F f and f : U → f (U ) is a proper map.
Proof. The rational map f : C → C is a proper map. Hence, for any open set V ⊂ C, f : f -1 (V ) → V is a proper map. In particular, this holds for U = F f . But since f -1 (F f ) = F f by Proposition 2.9, we get that f : F f → F f is a proper map. Then the proposition follows, since the restriction of a proper map to a connected component of the domain is still proper.

Definition 6.3. A connected component U of F f is called a Fatou component of f . Definition 6.4. A Fatou component is periodic if there exists an integer k ≥ 1 such that f •k (U ) = U. Note that if U is periodic of period k for f , then it is invariant for f •k . Definition 6.5. An invariant Fatou component U of f is called • a (super)attracting domain if there is a (super)attracting fixed point α ∈ U (f (α) = α and 0 ≤ |f (α)| < 1)
and the sequence f •n converges uniformly to α on every compact subset of U ; • a parabolic domain if there is a fixed point α ∈ ∂U with f (α) = 1, and the sequence f •n converges uniformly to α on every compact subset of U ; • a Siegel disk if it is simply connected, and if there exists an isomorphism h :

U → D such that h • f • h -1 (z) = e 2iπθ z
with θ ∈ R \ Q; • a Herman ring if it is doubly connected, and if there exists a radius r and an isomorphism

h : U → A r := {z ∈ C ; r < |z| < 1}, such that h • f • h -1 (z) = e 2iπθ z with θ ∈ R \ Q.
Remark 6.6. Clearly, the possibilities are mutually exclusive. Theorem 6.9 below asserts that these are the only possibilities and it is known that they all occur. The existence of Siegel disks was proved in 1942 by Siegel, and the existence of Herman rings was proved by Herman in 1981.

Figure 2 shows the Julia set of a polynomial having a Siegel disk and the Julia set of a rational map having a Herman ring. Proposition 6.7. If U is an invariant Fatou component of a rational map f of degree d ≥ 2, then it is either simply connected, or doubly connected or its complement has infinitely many connected components. It is doubly connected if and only if it is a Herman ring.

Proof. We can apply the Riemann-Hurwitz formula to f : U → U . It gives

χ(U ) = dχ(U ) -n 0 J P S 1 J f Figure 2.
Left: the polynomial P : z → e iπ( √ 5-1) z + z 2 has a Siegel disk (colored grey). We have drawn the orbits of some points in the Siegel disk. Each orbit accumulates on a R-analytic circle. Right: if t ∈ R/Z is chosen carefully enough, the rational map f (z) = e 2iπt z 2 (z -4)/(1 -4z) has a Herman ring. It leaves the circle S 1 invariant and is conjugate to an irrational rotation on S 1 . The picture is drawn for t = 0.61517321588 . . . where d is the degree of f : U → U and n is the number of critical points of f in U , counting multiplicities. If χ(U ) is finite, it follows that χ(U ) = n/(d -1) is non negative, and since U is not the entire Riemann sphere, we have χ(U ) = 0 or 1. Moreover, if χ(U ) = 0, i.e., if U is doubly connected, then n = 0 and f : U → U is a covering map. In addition, the degree of f : U → U is 1 since it preserves the modulus of U . Thus, f : U → U is an isomorphism. This isomorphism cannot be of finite order, since otherwise an iterate of f would have infinitely many fixed points. Therefore U is a Herman ring. Remark 6.8. If f is a polynomial and U is a bounded Fatou component, then thanks to the maximum principle U is simply connected. In particular, there are no Herman rings.

The following result is due to Fatou. We will not give the detailed proof. Case 1. There is a non constant limit value. Case 1.1. There is a sequence (n k ) such that f •n k → Id. This is a first place where we do not provide the details. One proves that

• either f has finite order (which is not the case since if f •p = Id on U for some p ≥ 1, then f •p = Id on C by analytic continuation, which is not possible since f •p is a rational map of degree d p > 1); • or U is simply connected, in which case it is a Siegel disk; • or U is doubly connected, in which case it is a Herman ring. Case 1.2. There is a sequence (n k ) such that f •n k → φ with φ non constant. By continuity, φ takes its values in U . According to the Hurwitz Theorem, φ takes its values in U . Extracting a subsequence, we may assume that n k+1 -n k → +∞ and that f •(n k+1 -n k ) → ψ. Then, ψ • φ = φ, so that ψ is the identity on the image of φ, thus on the whole component U by analytic continuation. In other words, we are in the previous situation : U is a Siegel disk or a Herman ring. Case 2. Every limit value of the sequence (f

•n ) is a constant. Note that if the sequence (f •n k ) converges to α, then α is a fixed point of f . Indeed, since f (z) ∈ U for all z ∈ U , we have that α ←- k→+∞ f •n k f (z) = f f •n k (z) -→ k→+∞ f (α).
Note also that any point z ∈ U can be joined to its image f (z) ∈ U by a path compactly contained in U . It follows that the set of limit values of the sequence f •n (z) is a continuum. Since f has finitely many fixed points, the whole sequence (f •n ) converges to a fixed point α of f . Case 2.1. If α ∈ U , the derivative of f •n at α tends to 0 as n → ∞. As a consequence, α is a (super)attracting fixed point and U is a (super)attracting domain. Case 2.2. If α ∈ ∂U , then α cannot be (super)attracting since it belongs to the Julia set J f . It cannot be repelling since the sequence (f •n ) converges to α. So, it is indifferent. We claim that D α f = 1 and so, U is a parabolic domain.

Fix z 0 ∈ U and set z n = f •n (z 0 ). Let ρ : [0, 1] → U be a continuous path with ρ(0) = z 0 and ρ(1) = z 1 . Then we can extend ρ to a continuous path ρ : [0, +∞) → U by setting ρ(n + t) = f •n • ρ(t) for all n ∈ N and all t ∈ [0, 1). Then lim t→+∞ ρ(t) = α and ρ satisfies ρ(t + 1) = f • ρ(t) for all t ∈ [0, +∞) and we conclude using the following lemma. Lemma 6.10 (Snail lemma). Let V be a neighborhood of the origin in C and let f :

V → f (V ) be a holomorphic function such that f (α) = α and f (α) = 0. If there is a continuous path ρ : [0, +∞) → V \ {α} such that lim t→+∞ ρ(t) = α and f • ρ(t) = ρ(t + 1) for all t ∈ [0, +∞) then either |f (α)| < 1 or f (α) = 1.
This is a second place where we do not provide the details. For a proof of this result see for example [START_REF] Milnor | Dynamics in one complex variable[END_REF]Lemma 16.2].

The following theorem, due to Fatou (1905), is probably the result which started the entire field of holomorphic dynamics. Theorem 6.11. A (super)attracting domain always contains at least one critical point.

Proof. If U is a superattracting domain, then it contains a fixed critical point, and the result is obvious. If U is an attracting domain, it contains an attracting fixed point α with multiplier λ. We can show that there is a map φ : (C, 0) → ( C, α) which conjugates the multiplication by λ to f . If U does not contain any critical point, then φ extends to an entire map φ : C → U via the relation φ(λz) = f • φ(z). This contradicts the Liouville Theorem. Remark 6.12. The same is true for parabolic domains, but the proof is more difficult and requires a detailed study of the local theory near a fixed point with multiplier 1. A proof can be found for example in [START_REF] Milnor | Dynamics in one complex variable[END_REF]Theorem 10.15].

Fatou's classification of invariant Fatou components clearly provides also a classification of periodic Fatou components since they are invariant for an iterate of the rational map, and even pre-periodic Fatou components, that is components U so that there exist m ≥ 0 and p ≥ 1 such that f •(m+p) (U ) = f •m (U ). The following fundamental result due to Sullivan completes the description for rational maps of degree d ≥ 2. Theorem 6.13 (Sullivan [START_REF] Sullivan | Quasiconformal Homeomorphisms and Dynamics I. Solution of the Fatou-Julia Problem on Wandering Domains[END_REF]). Let f : C → C be a rational map of degree d ≥ 2. Then every Fatou component of f is pre-periodic.

The proof of Sullivan's non-wandering Theorem 6.13 strongly relies on the Ahlfors-Bers mesurable mapping Theorem for quasi-conformal functions and we refer to the original paper of Sullivan [START_REF] Sullivan | Quasiconformal Homeomorphisms and Dynamics I. Solution of the Fatou-Julia Problem on Wandering Domains[END_REF] for it. The recent notes [START_REF] Buff | Wandering Fatou component for polynomials[END_REF] present a proof due to Adam Epstein and based on a density result of Bers for quadratic differentials. Such results are strongly one-dimensional and do not have an analogue in higher dimension, making impossible to mimic Sullivan's proof there. Besides this observation, little was known about this problem until recently. In Section 8 we will present some recent results on Fatou components in dimension 2.

Parabolic implosion in dimension 1

In this section, we recall the main ingredients in the proof that the Julia set J f does not depend continuously on f for the Hausdorff topology on the space of compact subsets of C. Lavaurs proved the following result (see also [START_REF] Douady | Does a Julia set depend continuously on the polynomial? Complex dynamical systems[END_REF]).

Theorem 7.1 (Lavaurs [START_REF] Lavaurs | Systèemes dynamiques holomorphes, explosion de points péeriodiques paraboliques[END_REF]). Assume f : C → C is a polynomial fixing 0 with f (z) = z + z 2 + O(z 3 ). Then,

J f lim sup δ→0 J f +δ .
The proof is based on the description of limits of iterates of f + δ in terms of maps called Fatou coordinates that will be introduced in subsection 7.2. Those limits are called Lavaurs maps.

7.1. Parabolic basin. In the rest of this section, we assume that f : C → C is a polynomial whose expansion near 0 is of the form

f (z) = z + z 2 + az 3 + O(z 4 ) with a ∈ C.
To understand the local dynamics of f near 0, it is convenient to consider the change of coordinates Z := -1/z. In the Z-coordinate, the expression of f becomes

F (Z) = Z + 1 + b Z + O 1 Z 2 with b := 1 -a.
In particular, if R > 0 is large enough, F maps the right half-plane { (Z) > R} into itself, so that if r > 0 is close enough to 0, f maps the disk D(-r, r) into itself.

In addition, the orbit under f of any point z ∈ D(-r, r) converges to 0 tangentially to the real axis. Similarly, if r > 0 is sufficiently close to 0, there is a branch of f -1 which maps the disk D(r, r) into itself and the orbit under that branch of f -1 of any point z ∈ D(r, r) converges to 0 tangentially to the real axis.

Definition 7.2. The basin B f is the open set of points whose orbit under iteration of f intersects the disks D(-r, r) for all r > 0.

7.2. Fatou coordinates. In order to understand further the local dynamics of f near 0, it is customary to use local attracting and repelling Fatou coordinates. In the case of a polynomial, those Fatou coordinates have global properties.

Proposition 7.3. There exists a (unique) attracting Fatou coordinate Φ f :

B f → C which semi-conjugates f : B f → B f to the translation T 1 : C Z → Z + 1 ∈ C: Φ f • f = T 1 • Φ f .
and satisfies the normalization:

Φ f (z) = - 1 z -b log - 1 z + o(1) as - 1 z → +∞.
Proof. For z ∈ B f , set

Z := - 1 z , Z n := - 1 f •n (z)
and

Φ n (z) := Z n -n -b log Z n .
We have that

(Z n ) → +∞ and Z n+1 = Z n + 1 + b Z n + O 1 Z 2 n , so that 1 Z n = O 1 Z + n as n → +∞.
As a consequence,

Φ n+1 (z) -Φ n (z) = Z n+1 -Z n -1 -b log Z n+1 Z n = O 1 Z 2 n = O 1 (Z + n) 2 .
So, the sequence (Φ n ) converges to a limit Φ f :

B f → C which satisfies Φ f (z) = Φ 0 (z) + O 1 Z = - 1 z -b log - 1 z + o(1) as - 1 z → +∞.
Passing to the limit on the relation Φ n •f = T 1 •Φ n+1 yields the required result.

Figure ?? gives a rough idea of the behaviour of the Fatou coordinate Φ f for the cubic polynomial f (z) := z + z 2 + 0.95z 3 . The basin B f contains the two critical points of f . Those points and their iterated preimages form the critical points of Φ f . Denote by c + the critical point with positive imaginary part and by c -its complex conjugate. Set v ± := Φ f (c ± ). Points z ∈ B f are colored according to the location Φ f (z): dark grey when Φ f (z) < (v -), light grey when (v -) < Φ f (z) < (v + ) and medium grey when (v + ) < Φ f (z) .

Proposition 7.4. There exists a (unique) repelling Fatou parametrization Ψ

f : C → C which semi-conjugates T 1 : C → C to f : C → C: Ψ f • T 1 = f • Ψ f .
and satisfies the normalization:

Ψ f (Z) = - 1 Z + b log(-Z) + o(1) as (Z) → -∞.
Proof. Choose r > 0 sufficiently close to 0 so that there is a branch g of f -1 which maps the disk D(r, r) into itself. For z ∈ D(r, r), set

Z := - 1 z , Z n := - 1 g •n (z) and Φ n (z) := Z n + n -b log(-Z n ).
Note that

Z n = Z n+1 + 1 + b Z n+1 + O 1 Z 2 n+1 .
So, as in the previous proof

Φ n+1 (z) -Φ n (z) = Z n+1 -Z n + 1 + b log Z n+1 Z n = O 1 Z 2 n+1 = O 1 (Z -n) 2 .
The sequence Φ n converges in the left half-plane (Z) < -1/(2r) to a limit Φ g which satisfies

Φ g (z) = Φ 0 (z) + O 1 Z = Z -b log(-Z) + o(1) as - 1 z → -∞.
Passing to the limit on the equation

Φ n+1 • f = T 1 • Φ n shows that Φ g conjugates f to T 1 . The inverse Ψ f of Φ g conjugates T 1 to f and Z = Φ g •Ψ f (Z) = - 1 Ψ f (Z) -b log - 1 Ψ f (Z) +o(1) = - 1 Ψ f (Z) -b log(-Z)+o(1)
as (Z) → -∞.

L f := Ψ f • Φ f : B f → C.
Note that the Lavaurs map L f commutes with f . Indeed,

Φ f • f = T 1 • Φ f and Ψ f • T 1 = f • Ψ f , so that L f • f = Ψ f • Φ f • f = Ψ f • T 1 • Φ f = f • Ψ f • Φ f = f • L f .
Figure 3 gives a rough idea of the behaviour of the Lavaurs map L f for the cubic polynomial f (z) := z + z 2 + 0.95z 3 . Points in the basin B f are colored according to the location of their image by L f : grey when L f (z) ∈ B f , white otherwise. The restriction of L f to each bounded white domain is a covering of C \ B f . 7.4. Discontinuity of the Julia set. Set

K(L f ) := n≥0 L -n f (K f ) and J (L f ) := ∂K(L f ).
Figure 4 shows K(L f ) for the cubic polynomial f (z) = z + z 2 + 0.95z 3 . The Lavaurs map L f has two complex conjugate sets of attracting fixed points. The fixed points of L f are indicated and their basins of attraction are colored (dark grey for one of the fixed points, and light grey for the others). Those basins form the interior of K(L f ). The black set J (L f ) is the topological boundary of K(L f ).

The following result may be considered as the main reason why Lavaurs maps are studied in holomorphic dynamics. Proposition 7.6 (Lavaurs [START_REF] Lavaurs | Systèemes dynamiques holomorphes, explosion de points péeriodiques paraboliques[END_REF]). Let f : C → C be a polynomial whose expansion at 0 is f (z) = z + z 2 + O(z 3 ). Let (N n ) be a sequence of integers tending to +∞ and (ε n ) be a sequence of complex numbers tending to 0, such that

N n - π ε n → 0.
Then,

(f + ε 2 n ) •Nn → L f locally uniformly on B f .
In addition,

J f J (L f ) ⊆ lim inf J f +ε 2 n and lim sup K f +ε 2 n ⊆ K(L f ) K f .
We do not present the proof of this result which is rather technical, and can be found in [START_REF] Douady | Does a Julia set depend continuously on the polynomial? Complex dynamical systems[END_REF] or [START_REF] Shishikura | Bifurcation of parabolic fixed points[END_REF] for example. Instead, we will show that the result holds for the Möbius transformation

g(z) = z 1 -z = z + z 2 + O(z 3 ).
In that case, all maps involved are Möbius transformations and the computations are explicit. If we perform the change of coordinates Z = -1/z, the Möbius transformation g gets conjugated to T 1 . Thus,

Φ g (z) = - 1 z and Ψ g (Z) = - 1 Z , so that L g = Id.
Note that g + ε 2 is also a Möbius transformation. It has two fixed points

α ± = ±iε + O(ε 2 ) with multipliers λ ± = exp ±2iε + O(ε 3 ) . So, if N → +∞ and N - π ε → 0, so that ε = π N + o(1) = π N + o 1 N 2 , then (g + ε 2 )
•N is a Möbius transformation fixing α ± with multipliers

µ ± := (λ ± ) N = exp (±2πi + o(1/N )) = 1 + o 1 N .
This Möbius transformation is

z → α + + µ + • (z -α + ) 1 - µ + -1 α + -α -(z -α + ) . Since µ + -1 = o 1 N = o(α + -α -), we see that indeed, (g + ε 2 ) •N -→ N →+∞ L g .
Recently, in [START_REF] Vivas | Non-autonomous parabolic bifurcation[END_REF], Vivas studied a non-autonomous parabolic implosion in dimension 1 for the map f (z) = z 1-z . In higher dimension, (semi-)parabolic implosion was recently studied for dissipative polynomial automorphisms of C 2 by Bedford, Smillie and Ueda in [START_REF] Bedford | Semi-parabolic bifurcations in complex dimension two[END_REF] (see also [START_REF] Dujardin | tability and bifurcations for dissipative polynomial automorphisms of C 2[END_REF]) and their strategy was recently adapted by Bianchi in [START_REF] Bianchi | Parabolic implosion for endomorphisms of C 2[END_REF] for a perturbation of a class of holomorphic endomorphisms tangent to identity, establishing a two-dimensional Lavaurs theorem for such a class.

Fatou components for polynomial maps in dimension 2

We end these notes by giving an updated account of the recent results on Fatou components for polynomial skew-products in complex dimension two in a neighbourhood of a periodic fiber. We divide our discussion according to the different possible kinds of periodic fibers. 8.1. Preliminaries. Let F : C 2 → C 2 be a holomorphic endomorphism of C 2 , and consider the discrete holomorphic dynamical system given by the iteration of F . In the investigation of the global behaviour of such a system it is natural to give the generalize the definition of the Fatou set of F as the largest open set F(F ) where the family of iterates (F •n ) n∈N of F is normal. A connected component of the Fatou set is called a Fatou component.

We have seen in the previous sections that in complex dimension one, Fatou components of rational maps of degree at least 2 on the Riemann sphere are well understood, thanks to Theorem 6.9 and Theorem 6. [START_REF] Bracci | Dynamics of one-resonant biholomorphisms[END_REF].

In complex dimension two, the understanding of Fatou components is far less complete. A considerable progress in the classification of periodic Fatou components has been achieved thanks to Bedford and Smillie [START_REF] Bedford | Polynomial diffeomorphisms of C 2 : currents, equilibrium measure and hyperbolicity[END_REF] [7] [8], Fornaess and Sibony [START_REF] Fornaess | Classification of recurrent domains for some holomorphic maps[END_REF], Lyubich and Peters [START_REF] Lyubich | Classification of invariant Fatou components for dissipative Hénon maps[END_REF] and Ueda [START_REF] Ueda | Fatou sets in complex dynamics on projective spaces[END_REF].

The question of the existence of wandering (i.e., not pre-periodic) Fatou components in higher dimension was put forward by several authors since the 1990's (see e.g., [START_REF] Fornaess | Some open problems in higher dimensional complex analysis and complex dynamics[END_REF]). Higher dimensional transcendental (i.e., non polynomial) maps with wandering domains can be constructed from one-dimensional examples by taking direct products. An example of a transcendental biholomorphism of C 2 with a wandering Fatou component oscillating to infinity was constructed by Fornaess and Sibony in [START_REF] Fornaess | Fatou and Julia sets for entire mappings in C k[END_REF]. Nonetheless, until recently very little was known about the existence of wandering Fatou components for holomorphic endomorphisms of P 2 (C) or for polynomial endomorphisms of C 2 .

A first natural class of polynomial endomorphisms of C 2 to consider are direct product, that is maps

F : C 2 → C 2 of the form F (z, w) = (f (z), g(w)),
where f and g are complex polynomials in one variable. This allows us to recover the generalizations of one-dimensional dynamical behaviours in dimension two. However, direct products are a very particular class and they do not give us a complete understanding of all possible behaviours of polynomial endomorphisms in C 2 .

A more interesting class to consider is given by polynomial skew-products in C 2 , namely polynomial maps F :

C 2 → C 2 of the form (1) F (z, w) = (f (z, w), g(w)),
where g is a complex polynomial in one variable and f is a complex polynomial in two variables. Since they leave invariant the fibration {w = const.}, skew-products allow us to build on one-dimensional dynamics and to get a first flavour of the richness of the higher dimension setting we are working in. This idea has been used by several authors to construct maps with particular dynamical properties. Dujardin, for example, used in [START_REF] Dujardin | A non-laminar dynamical Green current[END_REF] specific skew-products to construct a non-laminar Green current. Boc-Thaler, Fornaess and Peters constructed in [START_REF] Boc-Thaler | Fatou components with punctured limit sets[END_REF] a map having a Fatou component with a punctured limit set. Last but not least, as we will explain in subsection 8.2, skew-products are one of the key ingredients in the construction we recently obtained in [START_REF] Astorg | A two-dimensional polynomial mapping with a wandering Fatou component[END_REF] with Astorg, Dujardin, and Peters of holomorphic endomorphisms of P 2 (C) having a wandering Fatou component. The investigation of the holomorphic dynamics of polynomial skew-products was started by Heinemann [START_REF] Heinemann | Julia sets of skew-products in C 2[END_REF] and then continued by Jonsson [START_REF] Jonsson | Dynamics of polynomial skew-products on C 2[END_REF]. The topology of Fatou components of skew-products has been studied by Roeder in [START_REF] Roeder | A dichotomy for Fatou components of polynomial skew-products[END_REF].

Given a Fatou component Ω of a polynomial skew-product F in C 2 , its projection on the second coordinate Ω 2 = π 2 (Ω) is a Fatou components for g and hence thanks to Sullivan's non-wandering Theorem 6.13, up to considering an iterate of F , it has to fall into one of the three cases given by Theorem 6.9, and moreover, since we are considering polynomials, Herman rings cannot occur. Therefore, since (pre-)periodic points for g correspond to (pre-)periodic fibers for F , up to considering an iterate of F , we can restrict ourselves to study what happens in neighbourhoods of invariant fibers of the form {w = w 0 }. One-dimensional theory also describes the dynamics on the invariant fiber, which is given by the action of the one-dimensional polynomial f (z, w 0 ) := f w0 (z), and hence the Fatou components of f c will be again all pre-periodic and, up to consider an iterate, we can assume that they are either attracting basins, or parabolic basins or Siegel disks. This structure leads us to two immediate questions.

(1) Do all Fatou components of f c bulge to two-dimensional Fatou components of F ? (2) Is it possible to have wandering Fatou components for F in a neighbourhood of an invariant fiber? In the following we shall call an invariant fiber {w = w 0 } attracting, parabolic or elliptic according to whether w 0 is an attracting, parabolic or elliptic fixed point for g. A bulging Fatou component will be a Fatou component Ω of F such that Ω ∩ {w = w 0 } is a one-dimensional Fatou component of f w0 on the invariant fiber {w = w 0 }. We shall say that a Fatou component Ω w0 of f w0 on the invariant fiber {w = w 0 } is bulging if there exists a bulging Fatou component Ω of F so that Ω w0 = Ω ∩ {w = w 0 }.

Attracting invariant fiber.

Let us consider a polynomial skew-product F :

C 2 → C 2 of degree d ≥ 2 F (z, w) = (f (z, w), g(w)),
with an attracting invariant fiber. We can assume without loss of generality that the invariant fiber is {w = 0}. Therefore we have g(0) = 0 and |g (0)| < 1. In this case we know that there exists an attracting basin, containing the origin, of points whose iterates converge to the origin. The rate of convergence to the fixed point depends on whether the fixed point is superattracting, i.e, g (0) = 0, or attracting or geometrically attracting, i.e., g (0) = 0. 8.2.1. Superattracting case. This setting was studied by Lilov in [START_REF] Lilov | Fatou Theory in Two Dimensions[END_REF] who was able to answer both questions stated in the introduction. He first proved the following result giving a positive answer to Question 1. Theorem 8.1 (Lilov [45]). Let F : C 2 → C 2 be a polynomial skew-product of the form (1) of degree d ≥ 2. Let {w = w 0 } be a superattracting invariant fiber for F . Then all one-dimensional Fatou components of f w0 bulge to Fatou components of F .

Idea of the proof. We can assume w 0 = 0 without loss of generality. Thanks to Theorem 6.9 and Theorem 6.13, all Fatou components of the restriction f 0 (z) = f (z, 0) of f (z, w) to the invariant fiber are (pre-)periodic and are either attracting basins, or parabolic basins or Siegel domains. The strategy of the proof is to prove separately for each of these cases that the corresponding component is contained in a two-dimensional Fatou component of F . The bulging of one-dimensional Fatou components of attracting periodic points of f 0 (z) is well-known and follows for instance from the results of Rosay and Rudin [START_REF] Rosay | Holomorphic maps from C n to C n[END_REF]. For the remaining cases, by [START_REF] Lilov | Fatou Theory in Two Dimensions[END_REF]Theorem 3.17] there exists a strong stable manifold through all point in the one-dimensional Fatou components of parabolic or elliptic periodic points of f 0 (z), and so the corresponding bulging Fatou components simply consist of the union of such manifolds.

Then Lilov proved the following result implying the non-existence of wandering Fatou components in a neighbourhood of a superattracting invariant fiber. Theorem 8.2 (Lilov [START_REF] Lilov | Fatou Theory in Two Dimensions[END_REF]). Let F : C 2 → C 2 be a polynomial skew-product of the form (1) of degree d ≥ 2. Let {w = w 0 } be a superattracting invariant fiber for F and let B be the immediate basin of the superattracting fixed point w 0 . Take w ∈ B and let D w be a one-dimensional open disk lying in the fiber over w. Then the forward orbit of D w must intersect one of the bulging Fatou components of f w0 .

The proof relies on the repeated use of [START_REF] Lilov | Fatou Theory in Two Dimensions[END_REF]Lemma 3.2.4] applied to the orbit of a disk lying in a fiber over a point in the attracting basin, in order to obtain estimates from below for the radii of the images. Thanks to [START_REF] Lilov | Fatou Theory in Two Dimensions[END_REF]Proposition 3.2.8], by studying the geometry of the bulging Fatou components, it is also possible to obtain an upper bound on the largest possible disk lying in a fiber over a point in the attracting basin that can lie in the complement of a bulging Fatou component, depending on the distance to the invariant fiber. The conclusion then follows combining these two estimates.

All bulging Fatou components are (pre-)periodic, therefore all Fatou components for F in a neighbourhood of a superattracting invariant fiber are (pre-)periodic, and then the non-existence of wandering Fatou components in a neighbourhood of a superattracting invariant fiber follows immediately.

Corollary 8.3 (Lilov [45]). Let F : C 2 → C 2 be a polynomial skew-product of the form (1) of degree d ≥ 2. Let {w = w 0 } be a superattracting invariant fiber for F and let B be the immediate basin of the superattracting fixed point w 0 . Then there are no wandering Fatou components in B × C. 8.2.2. Geometrically attracting case. The geometrically attracting case was first partially addressed by Lilov in [START_REF] Lilov | Fatou Theory in Two Dimensions[END_REF] even if not stated explicitly. In fact, the proof of Theorem 8.1 can be readily adjusted to this case obtaining the following statement answering Question 1.

Theorem 8.4 (Lilov [START_REF] Lilov | Fatou Theory in Two Dimensions[END_REF]). Let F : C 2 → C 2 be a polynomial skew-product of the form (1) of degree d ≥ 2. Let {w = w 0 } be an attracting invariant fiber for F . Then all one-dimensional Fatou components of f w0 bulge to Fatou components of F .

On the other hand, the proof of Theorem 8.2 cannot be generalized to this setting, which is indeed more complicated than the superattracting case. In fact, Theorem 8.2 does not hold in general, as showed by Peters and Vivas with the following result. Theorem 8.5 (Peters-Vivas [START_REF] Peters | Polynomial skew-products with wandering Fatou-disks[END_REF]). Let F : C 2 → C 2 be a polynomial skew-product of the form

(2)
F (z, w) = (p(z) + q(w), λw), with 0 < |λ| < 1 and p and q complex polynomials. Then there exists a triple (λ, p, q) and a holomorphic disk D ⊂ {w = w 0 } whose forward orbit accumulates at a point (z 0 , 0), where z 0 is a repelling fixed point in the Julia set of f 0 .

As a consequence, the forward orbits of D cannot intersect the bulging Fatou components of f 0 .

The family (F | •n D ) n∈N is normal on the disks D, and so these are Fatou disks. However such disks are completely contained in the Julia set of F , which is the complement in C 2 of the Fatou set (see [START_REF] Peters | Polynomial skew-products with wandering Fatou-disks[END_REF]Theorem 6.1]).

The geometrically attracting case has been further investigated by Peters and Smit in [START_REF] Peters | Fatou components of attracting skew products[END_REF]. They focused their investigation on polynomial skew-products such that the action on the invariant attracting fiber is subhyperbolic, that is the polynomial does not have parabolic periodic points and all critical points lying on the Julia set are pre-periodic. They proved the following result. Proposition 8.6 (Peters-Smit [START_REF] Peters | Fatou components of attracting skew products[END_REF]). Let F : C 2 → C 2 be a polynomial skew-product of the form (1). Assume that the origin is an attracting, not superattracting, fixed point for g with corresponding basin B g , and the polynomial f 0 (z) := f (z, 0) is subhyperbolic. Then there exists a set E ⊂ C of full mesure, such that for every w 0 ∈ E the forward orbit of every disk in the fiber {w = w 0 } must intersect a bulging Fatou component of f 0 .

Idea of the proof. Notice that it suffices to prove the proposition in a neighbourhood of the attracting fiber {w = 0}. Therefore, up to considering a smaller neighbourhood, we can assume without loss of generality that g(w) = λw, and

f (z, w) = a 0 (w) + a 1 (w)z + • • • + a d (w)z d
where a 0 (w), . . . , a d (w) are holomorphic functions in w. The subhyperbolicity of the polynomial f 0 implies that its Fatou set is the union of finitely many attracting basins, and the orbits of the critical points contained in the Fatou set converge to one of these attracting cycles. The proof can be divided into 5 main steps.

Step 1. Fix R > 0 large enough so that for all z such that |z| > R we have |f 0 (z)| > 2|z| and set

W 0 = {|z| > R} ∪ y∈Att(f0) W y
where Att(f 0 ) is the set of all attracting periodic points of f 0 , and for each y ∈ Att(f 0 ) the set W y is an open neighbourhood of the orbit of y such that f 0 (W y ) ⊂ W y . Fix a neighbourhood U of the post-critical set of f 0 . Then by [START_REF] Peters | Fatou components of attracting skew products[END_REF]Proposition 15], there exists a set E ⊂ C of full mesure in a neighbourhood of the origin such that for all w 0 ∈ E there exists a constant C = C( w 0 , U ) such that for all n ∈ N we have [START_REF] Astorg | Wandering domains arising from Lavaurs maps with Siegel disks[END_REF] Card z :

∂F •n 1 ∂z (z, w 0 ) = 0 and F •n 1 (z, w 0 ) ∈ W 0 × U ≤ C √ n,
where F •n 1 is the first component of the n-th iterate of F . Step 2. Assume by contradiction that a fiber {w = w 0 }, with w 0 ∈ E, contains a disk D whose forward orbit avoids the bulging Fatou components of f 0 . Then the restriction of F •n to D is bounded and hence a normal family. Therefore, up to shrinking D there exists a subsequence F •nj such that F •nj | D converges, uniformly on compact subsets of D, to a point ζ in the Julia set of f 0 . Moreover, there exists ε > 0 so that F •n (D) ∩ (W 0 × D(0, ε)) is empty for all n ∈ N.

Step 3. Each critical point x contained in the Julia set is eventually mapped into a repelling periodic point, and up to considering an iterate of F we may assume that it is eventually mapped into a repelling fixed point with multiplier µ, with |µ| > 1.

The main tool to control the orbits of the critical points of F is obtained using a linearization map of the unstable manifold of the repelling fixed point, given by a map Φ :

C → C satisfying Φ(µt) = f •k 0 • Φ(t) for some k ∈ N.
Thanks to [52, Proposition 10], there exist C > 1 and 0 < γ < 1 so that ( 4)

Area(F •n (D)) ≤ Cγ n .
Step 4. We may assume that ζ does not lie in the post-critical set, and we may choose U and r > 0 such that

D(ζ, r) ∩ (U ∪ W 0 ) = ∅. Let j 1 ∈ N be such that F •nj 1 (D) ⊆ D(ζ, r
2 ) for all j ≥ j 1 , and consider O j the connected component of (F √ n j critical points.

•nj ) -1 (D(ζ, r) × {λ nj w 0 }) containing D. Then D ⊆ O j ⊆ D(0, R) × { w 0 },
Step 5. It is possible (see [START_REF] Peters | Fatou components of attracting skew products[END_REF]Proposition 28]) to find a uniform constant

C 1 > 0 so that if f : D → D is a proper holomorphic function of degree d, the set R ⊂ D has Poincaré area equal to A, and d • A 1/2d < 8, then the Poincaré area of f -1 (R) is at most C 1 d 3 A 1/d . Then, setting R j = F •nj 1
(D) and denoting by A j its Poincaré area Area D(ζ,r) (R j ) with respect to D(ζ, r), for j ≥ j 1 , we have R j ⊆ D(ζ, r), and we can estimate A j applying (4). Therefore there exists j 2 ≥ j 1 such that d j A 1/2dj j < 1/8 for all j ≥ j 2 . This implies

Area D(0,R) (D) ≤ Area Oj (D) ≤ C 2 d 3 j A 1/dj j ≤ M n 3/2 j γ n 3/2 j
where M > 0. The contradiction follows from the fact that the last expression will converge to zero as j increases towards infinity.

Thanks to the fact that in particular E is dense, Peters and Smit are able to give a negative answer to Question 2 when the action on the invariant fiber is subhyperbolic. They also obtain as a corollary that the only Fatou components of F are the bulging ones, since the topological degree of F equals the one of f 0 , implying that the only Fatou components that can be mapped onto the bulging Fatou components of f 0 are exactly those bulging Fatou components. Theorem 8.7 (Peters-Smit [START_REF] Peters | Fatou components of attracting skew products[END_REF]). Let F : C 2 → C 2 be a polynomial skew-product of the form [START_REF] Alexander | A history of complex dynamics[END_REF]. Assume that the origin is an attracting fixed point for g with corresponding basin B g , and the polynomial f 0 (z) := f (z, 0) is subhyperbolic. Then F has no wandering Fatou component over B g .

Very recently, Ji was able in [START_REF] Ji | Non-wandering Fatou components for strongly attracting polynomial skew products[END_REF] to generalize Lilov's Theorem 8.2 to polynomial skew-products with an invariant geometrically attracting fiber under the hypothesis that the multiplier of the invariant fiber is sufficiently small. More precisely he proved the following result. Theorem 8.8 (Ji [START_REF] Ji | Non-wandering Fatou components for strongly attracting polynomial skew products[END_REF]). Let F : C 2 → C 2 be a polynomial skew-product of the form (1) of degree d ≥ 2. Let {w = w 0 } be an attracting invariant fiber for F and let B w0 be the basin of the attracting fixed point c. Then there exists λ 0 (w 0 , f ) > 0 depending only on f and c such that if |g (w 0 )| < λ 0 , then there are no wandering Fatou components in B w0 × C.

The proof of this result follows Lilov's strategy. The main difficulty is due to the breaking down of Lilov's argument in the geometrically attracting case as we pointed out at the beginning of this section. Ji is able to overcome such difficulty by adapting a one-dimensional result due to Denker, Przytycki and Urbanski in [START_REF] Denker | On the transfer operator for rational functions on the Riemann sphere Ergodic Theory and Dynamical Systems[END_REF] in this case. Such result is used to obtain estimates of the size of bulging Fatou components and of the size of forward images of wandering Fatou disks.

Ji also proved in [START_REF] Ji | Non-uniform hyperbolicity in polynomial skew products, to appear in IMRN[END_REF] that if F : C 2 → C 2 is a polynomial skew-product with an attracting invariant fiber L such that the restriction of F to L is non-uniformly hyperbolic, then the Fatou set in the basin of L coincides with the union of the basins of attracting cycles, and the Julia set in the basin of L has Lebesgue measure zero. Therefore there are no wandering Fatou components in the basin of the invariant attracting fiber. 8.3. Parabolic invariant fiber and wandering domains. A first contribution to the investigation of this case is due to Vivas, who proved a parametrization result [START_REF] Vivas | Parametrization of unstable manifolds and Fatou disks for parabolic skewproducts[END_REF]Theorem 3.1] for the unstable manifolds for special parabolic skew-product of C 2 . Vivas used this parametrization as the main tool to prove the analogue of Theorem 8.5 for special parabolic skew-product. However, this construction does not allow to construct a wandering Fatou component in a neighbourhood of the parabolic invariant fiber.

In [START_REF] Astorg | A two-dimensional polynomial mapping with a wandering Fatou component[END_REF], together with Astorg, Dujardin and Peters, we proved the existence of polynomial skew-products of C 2 , extending to holomorphic endomorphisms of P 2 (C), having a wandering Fatou component. The key tool consists in using parabolic implosion techniques on polynomial skew-products, and this idea was initially suggested by Lyubich. The main strategy is to combine slow convergence to an invariant parabolic fiber and parabolic transition in the fiber direction, to produce orbits shadowing those of the so-called Lavaurs map, that we defined in the previous section.

Theorem 8.9 ([4]

). There exists a holomorphic endomorphism F : P 2 (C) → P 2 (C), induced by a polynomial skew-product mapping F : C 2 → C 2 , having a wandering Fatou component. More precisely, let f : C → C and g : C → C be polynomials of the form [START_REF] Beardon | Iteration of rational functions. Complex analytic dynamical systems[END_REF] f (z) = z + z 2 + O(z 3 ) and g(w) = w -w 2 + O(w 3 ).

If the Lavaurs map L f : B f → C has an attracting fixed point, then the skew-product

F : C 2 → C 2 defined by (6) F (z, w) := f (z) + π 2 4 w, g(w)
has a wandering Fatou component.

The orbits in these wandering Fatou components are bounded and the approach used in the proof is essentially local. Notice that if f and g have the same degree, F extends to a holomorphic endomorphism of P 2 (C). Moreover we can obtain examples in arbitrary dimension k ≥ 2 by simply considering products mappings of the form (F, Q), where Q has a fixed Fatou component.

A first step in the proof of Theorem 8.9 is to find a parabolic polynomial f whose Lavaurs map L f admits an attracting fixed point. Idea of the proof. We consider

U f := ψ -1 f (B f ) and E f := ϕ f • ψ f : U f → C.
The open set U f contains an upper half-plane and a lower half-plane, and it is invariant under T 1 . Like the Lavaurs map, the map E f commutes with T 1 , therefore E f -Id is periodic of period 1 and admits a Fourier expansion in a upper half-plane:

E f (Z) = Z + k≥0 c k e 2πikZ .
Using the expansion of ϕ f and ψ f near infinity, we obtain with an elementary computation:

E f (Z) = ϕ f • ψ f (Z) = Z + (1 -a) log(-Z) + o(1) -(1 -a) log Z + (1 -a) log(-Z) + o(1) + o(1) = Z + (1 -a) log(Z) -πi(1 -a) -(1 -a) log(Z) + o(1) = Z -πi(1 -a) + o(1),
and so c 0 = -πi(1 -a).

Thanks to a more elaborate argument, based on the notion of finite type analytic map introduced by Adam Epstein, it is possible to prove that:

E f (Z) = Z -πi(1 -a) + c 1 e 2πiZ + o(e 2πiZ ) with c 1 = 0.
It then follows that for a = 1 close to 1, E f has a fixed point Z f with multiplier ρ f satisfying

c 1 e 2πiZ f ∼ πi(1 -a) and ρ f -1 ∼ 2πic 1 e 2πiZ f ∼ -2π 2 (1 -a) as a → 1.
Therefore, for r > 0 sufficiently close to 0 and a ∈ D(1 -r, 1), the multiplier ρ f belongs to the unit disk and Z f is an attracting fixed point of E f . This concludes the proof since ψ f : U f → B f semi-conjugates E f to L f , and so, the fact that Z f is an attracting fixed point of E f implies that the point ψ f (Z f ) is an attracting fixed point of L f .

The key result in the proof of Theorem 8.9 relies on a non-autonomous analogue of Lavaurs estimates in the setting of skew-products.

Let B f and B g be the parabolic basins of 0 under iteration of respectively f and g. One of the key points is to choose ( z 0 , w 0 ) ∈ B f × B g so that the first coordinate of F •m ( z 0 , w 0 ) returns infinitely many times close to the attracting fixed point of L f . The proof is designed so that the return times are the integers n 2 for n ≥ n 0 . Therefore, we need to analyze the orbit segment between n 2 and (n + 1) 2 , which is of length 2n + 1.

Proposition 8.11 ([4]).

As n → +∞, the sequence of maps

C 2 (z, w) → F 2n+1 z, g n 2 (w) ∈ C 2 converges locally uniformly in B f × B g to the map B f × B g (z, w) → L f (z), 0 ∈ C × {0}.
Idea of the proof. Let B g the parabolic basin of 0 under iteration of g. For all w ∈ B g , the orbit g •m (w) converges to 0 like 1/m. We want to analyze the behaviour of F starting at z, g •n 2 (w) during 2n + 1 iterates. For large n, the first coordinate of F along this orbit segment is approximately

f (z) + ε 2 with π ε 2n.
As we saw in the previous section, a rough statement of Lavaurs Theorem from parabolic implosion gives us that if π ε = 2n, then for large n, the (2n

) th iterate of f (z) + ε 2 is approximately equal to L f (z) on B f .
Our setting is different since in our case ε keeps decreasing along the orbit. Indeed on the first coordinate we are taking the composition of 2n + 1 transformations of the form

f (z) + ε 2 k with π ε k 2n + k n and 1 ≤ k ≤ 2n + 1.
The key step in the proof of the statement consists in a detailed analysis of this nonautonomous situation, proving that the decay of ε k is counterbalanced by taking exactly one additional iterate of F .

With this proposition in hand, the proof of Theorem 8.9 is easily completed.

Proof of Theorem 8.9. Let ξ be an attracting fixed point of L f and let V ⊂ B f be a disk centered at ξ and such that L f (V ) is compactly contained in V . Therefore L •k f (V ) converges to ξ as k → +∞. Let W B g be an arbitrary disk. Thanks to Proposition 8.11, there exists n 0 ∈ N such that for every n ≥ n 0 ,

π 1 • F •(2n+1) (V × g •n 2 (W )) V,
where π 1 : C 2 → C denotes the projection on the first coordinate, π 1 (z, w) := z.

Let U be a connected component of the open set F -n 2 0 V × g •n 2 0 (W ) . Then for every integer n ≥ n 0 , we have

(7) F •n 2 (U ) ⊆ V × g •n 2 (W ).
In fact, this holds by assumption for n = n 0 . Now if the inclusion is true for some n ≥ n 0 , then

π 1 • F •(n+1) 2 (U ) = π 1 • F •(2n+1) F •n 2 (U ) ⊂ π 1 • F •(2n+1) V × g •n 2 (W ) ⊂ V,
from which [START_REF] Bedford | Polynomial diffeomorphisms of C 2 : II. Stable manifolds and recurrence[END_REF] follows. This yields that the sequence (F •n 2 ) n≥0 is uniformly bounded, and hence normal, on U . Moreover, any cluster value of this sequence of maps is constant and of the form (z, 0) for some z ∈ V , and (z, 0) is a limit value (associated to a subsequence (n k ) k∈N ) if and only if L f (z), 0 is a limit value (associated to the subsequence (1 + n k ) k∈N ). Therefore the set of cluster limits is totally invariant under L f : V → V , and so it must coincide with the attracting fixed point ξ of L f . Therefore the sequence (F •n 2 ) n≥0 converges locally uniformly to (ξ, 0) on U . The sequence (F •m ) m∈N is locally bounded on U if and only if there exists a subsequence (m k ) k∈N such that (F •m k | U ) k∈N has the same property. In fact, since W is compact, there exists R > 0 such that if |z| > R, then for every w ∈ W , (z, w) escapes locally uniformly to infinity under iteration. The domain U is therefore contained in the Fatou set of F .

Let Ω be the component of the Fatou set F F containing U . For any integer j ≥ 0, the sequence of maps (F •n 2 +j ) n∈N converges locally uniformly to F •j (ξ, 0) = (f •j (ξ), 0) on U and hence on Ω. Therefore the sequence (F •n 2 ) n∈N converges locally uniformly to (f •j (ξ), 0) on F •j (Ω). If i, j are nonnegative integers such that F

•i (Ω) = F •j (Ω), then f •i (ξ) = f •j (ξ)
, and so i = j because ξ cannot be pre-periodic under iteration of f , since it belongs to the parabolic basin B f . This proves that Ω is not (pre-)periodic under iteration of F , and so it is a wandering Fatou component for F .

We end this subsection with some explicit examples satisfying the assumption of Theorem 8.9.

Example 8.12. Example As a consequence of Proposition 8.10 we obtain that if f : C → C is the cubic polynomial f (z) = z + z 2 + az 3 , and g is as in [START_REF] Beardon | Iteration of rational functions. Complex analytic dynamical systems[END_REF], then the polynomial skew-product F defined in (6) admits a wandering Fatou component for r > 0 sufficiently small and a ∈ D(1 -r, r).

It is also interesting to search for real polynomial mappings with wandering Fatou domains intersecting R 2 . We also have such examples. There exist parameters b ∈ (-8/27, 0) such that for g as in [START_REF] Beardon | Iteration of rational functions. Complex analytic dynamical systems[END_REF], the polynomial skew-product F defined in (6) has a wandering Fatou component intersecting R 2 .

Astorg, Boc-Thaler and Peters constructed in [START_REF] Astorg | Wandering domains arising from Lavaurs maps with Siegel disks[END_REF] a second example of polynomial skew-products, arising from similar techniques, but with distinctly different dynamical behaviour. Instead of wandering domains arising from a Lavaurs map with an attracting fixed point, they constructed a domain arising from a Lavaurs map with a fixed point of Siegel type.

In [START_REF] Hahn | A polynomial automorphism with a wandering Fatou component[END_REF], Hahn and Peters constructed polynomial automorphisms of C 4 with wandering Fatou components. Their four-dimensional automorphisms lie in a oneparameter family, depending on a parameter δ ∈ C \ {0}, and degenerating to the two-dimensional polynomial map in equation ( 5) of Theorem 8.9 as δ converges to 0.

Berger and Biebler recently proved in [START_REF] Berger | Emergence of wandering stable components[END_REF] the existence of a locally dense set of polynomial automorphisms of C 2 with real coefficients, having a wandering Fatou component. These Fatou components have nonempty real trace and their statistical behaviour is historic with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. 8.4. Elliptic invariant fiber. With Peters we investigated in [START_REF] Peters | Fatou components of polynomial elliptic skew products[END_REF] the case of invariant fibers at the center of a Siegel disk. More precisely, we considered a polynomial skew-product of the form (1) having an elliptic invariant fiber. As before, we can assume without loss of generality that the invariant fiber is {w = 0} and so we have g(0) = 0 and g (0) = e 2πiθ with θ ∈ R\Q. We assume that the origin belongs to a Siegel disk for g and hence g is locally holomorphically linearizable near w = 0. Therefore, up to a local change of coordinates we may assume that F is of the form

F (z, w) = (f (z, w), λ • w),
where f w (z) := f (z, w) is a polynomial in z with coefficients depending holomorphically on w. We assume that the degree of the polynomial f w is constant near {w = 0}, and at least 2.

In this case we only have a partial answer to Question 1. In fact, while we already know that the attracting Fatou components of f 0 always bulge, the general situation appears to be more complicated as there might be resonance phenomena. In [START_REF] Peters | Fatou components of polynomial elliptic skew products[END_REF] we proved the following local result, implying that all parabolic Fatou components of a polynomial skew-product with an elliptic invariant fiber bulge if the rotation number satisfies the Brjuno condition. where, for j ≥ h, α k+j+2 (w) is a holomorphic function in w such that α k+j+2 (0) = f 0,k+j+2 .

We recall that a complex number is called Brjuno (see [START_REF] Brjuno | Analytic form of differential equations[END_REF] for more details) if ( 10)

+∞ k=0 1 2 k log 1 ω(2 k+1 ) < +∞ ,
where ω(m) = min 2≤k≤m |λ k -λ| for any m ≥ 2, and a quadratic polynomial λw + w 2 is linearizable near 0 if and only if λ is Brjuno (see [START_REF] Brjuno | Analytic form of differential equations[END_REF][START_REF] Brjuno | Analytic form of differential equations[END_REF][START_REF] Yoccoz | Théorème de Siegel, nombres de Brjuno et polynômes quadratiques[END_REF]). For higher degree polynomials the Brjuno condition (10) is sufficient, but necessity is still an open question. The proof of the previous proposition makes use of a procedure inspired by the Poincaré-Dulac normalization process [START_REF] Arnold | Geometrical Methods in the Theory of Ordinary Differential Equations[END_REF]Chapter 4] aiming to conjugate the given polynomial skew-product to a skew-product in a simpler form. At each step of the usual Poincaré-Dulac normalization procedure we can use a polynomial change of coordinates to eliminate all non-resonant monomials of a given degree. A similar idea is used here, and thanks to the skew-product structure of the germ and the Brjuno assumption we are able to eliminate all non-resonant terms of a given degree in the powers of z by means of changes of coordinates that are polynomial in z with holomorphic coefficients in w. It turns out that there are some differences in between the first degrees as it is pointed out in [51, Section 2].

The arithmetic Brjuno condition is strongly needed in the proof of Proposition 8.14 and it is natural to ask whether parabolic Fatou components of f 0 always bulge, or it is possible to characterize when they bulge. We do not have a complete answer to such question. However we can prove the following result. Proposition 8.15 ([51, Proposition 10]). Let F (z, w) = (f z (w), g(w)) be a holomorphic skew-product with an elliptic linearizable invariant fiber. If F does not admit a holomorphic invariant curve on the invariant fiber, then the parabolic Fatou components of f 0 do not bulge. with λ a Cremer number (see [START_REF] Peters | Fatou components of polynomial elliptic skew products[END_REF]Example 8] for details). More generally, arguing as in Cremer's example [START_REF] Cremer | Über die Häufigkeit der Nichtzentren[END_REF] we can show (see [START_REF] Peters | Fatou components of polynomial elliptic skew products[END_REF]Proposition 9]) the existence of holomorphic skew-products without invariant holomorphic curves of the form {w = ϕ(z)}, having an elliptic invariant fiber that is not point-wise fixed.

Describing the general situation can also be complicated by resonance phenomena. For example, an invariant fiber at the center of a Siegel disk was used in [START_REF] Boc-Thaler | Fatou components with punctured limit sets[END_REF] to construct a non-recurrent Fatou component with limit set isomorphic to a punctured disk, and in that construction the invariant fiber also contains a Siegel disk, but with opposite rotation number. Moreover, it might happen that Fatou components on the invariant fiber do not bulge. For example, we can consider the skew-product F (z, w) = (λz(1 + azw), λ -1 w), with a ∈ C * , λ = e 2πiθ and θ ∈ R \ Q. We have F (z, 0) = (λz, 0), but the Siegel disk around the origin in {w = 0} is not bulging, and in fact it follows from [START_REF] Bracci | Dynamics of one-resonant biholomorphisms[END_REF] and [START_REF] Bracci | Dynamics of multi-resonant biholomorphisms[END_REF] that there exists a Fatou component of parabolic type having on its boundary the origin of C 2 , which is fixed by F . We also have an answer to Question 2, under the assumption that the multiplier at the elliptic invariant fiber is Brjuno and all critical points of the polynomial acting on the invariant fiber lie in basins of attracting or parabolic cycles. and let {w = w 0 } be an elliptic invariant fiber with multiplier λ. If λ is Brjuno and all critical points of the polynomial f w0 lie in basins of attracting or parabolic cycles, then all Fatou components of f w0 bulge, and there is a neighbourhood of the invariant fiber {w = w 0 } in which the only Fatou components of F are the bulging Fatou components of f w0 . In particular there are no wandering Fatou components in this neighbourhood.

Idea of the proof. The proof relies on the fact that, in dimension 1, if every critical point of the polynomial either (1) lies in the basin of an attracting periodic cycle, (2) lies in the basin of a parabolic periodic cycle, or (3) lies in the Julia set and after finitely many iterates is mapped to a periodic point, then it is possible to construct a conformal metric µ, defined in a backward invariant neighbourhood of the Julia set minus the parabolic periodic orbits, so that f c is expansive with respect to this metric. It then follows that there can be no wandering Fatou components.

The key step in the proof is that for fibers {w = w 0 } sufficiently close to the invariant fiber {w = w 0 } we can define conformal metrics µ w0 that depend continuously on w 0 , and so that F acts expansively with respect to this family of metrics (see [START_REF] Peters | Fatou components of polynomial elliptic skew products[END_REF]Section 3] for details). That is, for a point ( z 0 , w 0 ) ∈ C 2 lying in the region where the metrics are defined, and for a non-zero tangent vector ξ ∈ T z0 (C w0 ) we have that µ w1 ( z 1 , df w0 ξ) > µ w0 ( z 0 , ξ), where ( z 1 , w 1 ) = F ( z 0 , w 0 ). Therefore, if ( z 0 , w 0 ) does not lie in one of the bulging Fatou components and its orbit ( z n , w n ) remains in the neighbourhood where the expanding metrics are defined, for any tangent vector ξ ∈ T z0 (C w0 ) we have that

µ wn df zn-1 • • • df z0 ξ → ∞,
from which it follows that the family (F •n ) n∈N cannot be normal on any neighbourhood of ( z 0 , w 0 ). This proves that there are no Fatou components but the bulging components near the invariant fiber {w = w 0 }.

It is unclear whether the same techniques could be used to deal with pre-periodic critical points lying on the Julia set. The difficulty is that the property that critical points are eventually mapped onto periodic cycles is not preserved in nearby fibers.

Exercise 1 . 2 .

 12 Assume f (z) = az + b with a ∈ C and b ∈ C.

Theorem 2 . 2 (

 22 Montel). A family (f α : U → C) α∈A of holomorphic maps on an open subset U ⊂ C which omits at least three values in C is normal.

Definition 2 . 3 (

 23 Fatou sets and Julia sets). The Fatou set F f of a rational map f : C → C is the largest open subset of C on which the family of iterates (f •n ) n∈N is normal. The Julia set J f is the complement of the Fatou set. Example 2.4. Example For f (z) = z d , with d ≥ 2 or d ≤ -2, the Julia set J f is the unit circle S 1 := {z ∈ C ; |z| = 1}.

Figure 1 .

 1 Figure 1. The Julia set of the quadratic polynomial f : C → C defined by f (z) = z 2 -1.

Remark 2 . 11 .

 211 Since φ is a non-constant analytic map, it is an open map. Thus φ(U ) is open and the family g α is defined on an open set.

4. 1 .

 1 Periodic points. Definition 4.1 (Multiplier at a fixed point). A fixed point of f is a point α such that f (α) = α. The derivative of f at α is an endomorphism of the tangent line D α f : T α C → T α C, thus a multiplication by a number λ ∈ C. The eigenvalue λ of the endomorphism D α f :

Definition 4 . 9 (

 49 Euler characteristic). If U is an open subset of C with m boundary components, the Euler characteristic of U is χ(U ) := 2 -m.

Theorem 4 . 10 .

 410 Let f : C → C be a rational map of degree d, V ⊆ C be an open set and U

Theorem 6 . 9 (

 69 Classification of invariant Fatou components). Let f : C → C be a rational function of degree d ≥ 2. An invariant Fatou component U of f is a (super)attracting domain, a parabolic domain, a Siegel disk or a Herman ring. Sketch of the proof. The classification starts by studying what are the possible limit values of the sequence of iterates (f •n : U → U ).

7. 3 .Definition 7 . 5 .

 375 Lavaurs maps. The Lavaurs map L f : B f → C is the map

Figure 3 .

 3 Figure 3. Behavior of L f for f (z) := z + z 2 + 0.95z 3 .

Figure 4 .

 4 Figure 4. The set K(L f ) for f (z) = z + z 2 + 0.95z 3 .

1 :

 1 and we can study the proper holomorphic function F •nj O j → D(ζ, r). Thanks to (3), such a map has at most d j = C

Proposition 8 .

 8 10 ([4, Proposition B]). Let f : C → C be the cubic polynomial defined byf (z) = z + z 2 + az 3 with a ∈ C.If r > 0 is sufficiently close to 0 and a belongs to the disk D(1 -r, r), then the Lavaurs map L f : B f → C admits an attracting fixed point.

Example 8 . 13 .

 813 Example [4, Proposition C] Let f : C → C be the degree 4 polynomial defined by f (z) := z + z 2 + bz 4 with b ∈ R.

Proposition 8 .

 8 14 ([51, Proposition 2]). Let F be a holomorphic skew-product of the form(8) F (z, w) = (f w (z), g(w))with g(w) = λw + O(w 2 ), f 0 (0) = 0, and f 0 (0) = 1. Assume λ is a Brjuno number. If f 0 (w) ≡ w, then F is holomorphically linearizable at the origin. If f 0 (z) = z+f 0,k+1 z k+1 +O(z k+2 ) with f 0,k+1 = 0 for some k ≥ 1, then for any h ≥ 0 there exists a local holomorphic change of coordinates near the origin conjugating F to a map of the form F (z, w) = ( f (z, w), λw) satisfying(9) f (z, w) = z + f 0,k+1 z k+1 + • • • + f 0,k+h+1 z k+h+1 + j≥h z k+j+2 α k+j+2 (w),

  Moreover, we can construct examples showing that the Brjuno condition cannot be completely omitted. In fact by taking Cremer numbers, that is λ ∈ C with |λ| = 1 and such that lim sup we can construct examples of polynomial skew-product not admitting a holomorphic invariant curve on the invariant fiber. An elementary example is the following polynomial skew-product F (z, w) = (z + w + zw, λw)

Theorem 8 .

 8 16 ([51, Theorem 1]). Let F be a polynomial skew-product of the form

( 11 )

 11 F (z, w) = (f w (z), g(w)),

This work is part of the C.I.M.E. Lecture Notes "Modern Aspects of Dynamical Systems" to be published in the Springer Lecture Notes in Mathematics -C.I.M.E. subseries, corresponding to the course taught by the second author in Cetraro in August 2021.1

Acknowledgements

We thank John H. Hubbard for letting us use the book in preparation [20] to write this survey. The first seven sections are adapted from the first chapter of this book. We also thank Marco Abate and Fabrizio Bianchi for useful comments on the first draft of these notes.