Analytic Classification of Germs of Parabolic Antoholomorphic Diffeomorphisms of Codimension k

We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curve, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, commuting holomorphic and antiholomorphic parabolic germs.


Introduction
In this paper, we are interested in the local dynamics of antiholomorphic diffeomorphisms with a parabolic fixed point, i.e. a fixed point of multiplicity k + 1 (i.e. of codimension k). We study the classification under conjugacy by analytic changes of coordinate of a germ of an antiholomorphic diffeomorphism f with a parabolic fixed point. In local coordinate, it may be chosen in the form (1) f (z) = z + 1 2 z k+1 + o(z k+1 ) for some integer k ≥ 1.
The classification of parabolic fixed points in the holomorphic case for a germ (2) g(z) = z + z k+1 + k + 1 is well known. (See e.g. [5] or [6].) The dynamics of g (see Figure 1) is determined by a topological invariant, the integer k, a formal invariant, the complex number b, and an analytic invariant given by an equivalence class of 2k germs of diffeomorphisms which are the transition functions on the space of orbits of g (the Écalle horn maps). Two germs g 1 and g 2 are formally equivalent if and only if they have the same topological invariant and formal invariant. Furthermore, they are analytically equivalent if and only if they also have the same analytic invariant. The goal of this paper is to establish a local classification of antiholomorphic parabolic germs under the analytic conjugation and to describe the space of orbits of such a germ and, more generally, to explore the geometric properties of antiholomorphic parabolic germs which are invariant under analytic conjugation. This is done for fixed points of any multiplicity. It allows us to provide a solution to the following problems.

Questions.
1. (Antiholomorphic Root Extraction) The second iterate of an antiholomorphic parabolic germ f as in equation (1) is a holomorphic germ which is parabolic. When is the converse true: Given a parabolic germ of holomorphic diffeomorphism g, when is it possible to write it as g = f •f , for some antiholomorphic parabolic germ f ? We call f an antiholomorphic square root of g. More generally, when does g have an antiholomorphic root of some order? 2. Analogously, when does an antiholomorphic parabolic germ have an antiholomorphic root? When are the roots unique? 3. (Embedding) Let {v t } t , where v t : z → v t (z), be the flow of the differential equationż = v(z) = z k+1 1+bz k . Each v t (t = 0) is a holomorphic germ with a parabolic fixed point at the origin.
Then v 1 2 (·) is an antiholomorphic germ, and any antiholomorphic parabolic germ is formally conjugate to such a germ. Given an antiholomorphic parabolic germ f , when is it analytically conjugate to some v 1 2 (·)? In that case, it allows to embed f in the familly v t . 4. When does an antiholomorphic germ preserve a germ of real analytic curve? This is equivalent to say that the germ is analytically conjugate to a germ with real coefficients. 5. (Centralizer) Can we describe all the antiholomorphic parabolic germs f that commute with a holomorphic parabolic germ g? If f and g commute, then f sends the orbits of g on the orbits of g. This greatly restricts the possible f . In an analogous way, can we describe all the holomorphic and the antiholomorphic germs that commute with an antiholomorphic parabolic germ?
The above problems are questions about the equivalence classes of germs under analytic conjugacy. Therefore, the answer should be read in the modulus of classification, which will be introduced in Section 5.
The local dynamics of an antiholomorphic parabolic germ has similarities with the holomorphic case: indeed the n-th iterate f •n is holomorphic for n even. We find that the dynamics is determined by the same topological and formal invariants, but the analytic invariant is composed of k germs of diffeomorphisms, instead of 2k. This is explained by the fact that an orbit of f will usually jump between two Fatou petals of its associated holomorphic parabolic germ f •2 (see Figure 2), so that the dynamics in those petals are not independent.
We observe other differences from the holomorphic case. A holomorphic germ has 2k formal separatrices. The antiholomorphic germ has instead a privileged unique direction; a formal symmetry axis. There f •n Figure 2. An orbit of f jumping between two petals. An orbit of the second iterate f •2 will remain either in the upper petal or the lower petal.
is also a topological difference between the cases where the codimension is odd or even. When k is even, the rotation z → −z is a formal symmetry of f , whilst it is not for k odd.
Antiholomorphic dynamics has been considered before in the context of anti-polynomials, that is a polynomial function of z, p(z) = z n + · · · + a 0 . Iteration of anti-polynomials was studied by Nakane and Schleicher in [8], Hubbard and Schleicher in [4], and Mukherjee, Nakane and Schleicher in [7]. Their focus is mostly on the family of antipolynomials p c (z) = z d + c and the description of the connectedness locus M * d called the multicorn. The context is global in nature, but the local analysis contributes significantly.
An important role is played by periodic orbits of p c of odd period k, because when k is odd, p •k c is antiholomorphic. In this case, all indifferent periodic orbits are parabolic and they occur along real analytic arcs in the parameter space, as proved in [4] and [7]. However, only points of codimension 1 and 2 are observed. This is due to a choice of a special subfamily of anti-polynomials of degree d. Indeed, higher codimension is already observed in the 2-parameter family z d +c 1 z +c 0 , e.g. when c 1 = 1 and c 0 = 0.
One of the tools used for anti-polynomials is called the Écalle height, introduced by Hubbard and Schleicher in [4]. In codimension 1, on the Écalle cylinder of the attractive petal, the imaginary part of an orbit is intrinsic, and this is used to prove that landing points of curves in the parameter space where p c has a parabolic periodic orbit of odd period is are points of codimension 2. When studying the space of orbits of an antiholomorphic germ of parabolic diffeomorphism of any codimension, we see that the Écalle height has a meaning only on the Écalle cylinder of the petals containing the formal symmetry axis of f . This is seen by describing the space of orbits on a neighbourhood of a parabolic fixed point, which we do in Section 6.2.
The paper is organized as follows. In Section 2, we define the topological and formal invariants of f . We also establish a formal normal form for f .
In Section 3, we study the formal normal form.
In Section 4, we introduce the rectifying coordinate and the Fatou coordinates in order to define the transition functions (Def. 5.0.4) in Section 5, which is the analytic invariant. This leads to the modulus of classification of f .
In Section 6.1, we recall a description of the space of orbits in the holomorphic case using 2k spheres (or Écalle cylinders) glued with the horn maps (these are the expressions of the transition functions in the coordinates of the spheres). We use this space of orbits in Section 6.2 to identify the space of orbits of f to a manifold of real dimension 2 by quotienting the space of orbits of f • f by the action of f .
After describing the space of orbits, we state, in Section 6.3, the main result of the paper: the Classification Theorem 6.3.1. The idea in spirit is that two germs are equivalent if and only if their space of orbits are equivalent; the Classification Theorem is a way to rigorously express this statement.
Finally, with the Classification Theorem in hands, in Section 7 we answer Questions 1 to 5 of the introduction.

Antiholomorphic Parabolic Fixed Points
Notation. For the whole paper, we will use the following notation : • σ(z) = z is the complex conjugation; • τ (w) = 1 w is the antiholomorphic inversion; • T C (Z) = Z + C is the translation by C ∈ C; • L c (w) = cw is the linear transformation with multiplier c ∈ C; • v t is the time-t of the vector field From this definition, together with the chain rule, it follows that antiholomorphy is an intrinsic property of f under holomorphic changes of variable. Equivalently, f : is holomorphic, therefore f (z) expands in a power series in terms of z.
Note that the multiplier at a fixed point of an antiholomorphic function is not intrinsic, only its modulus is. Indeed, a scaling of λ will add a factor of λ λ to the multiplier. We will also say that f is an antiholomorphic parabolic germ.
then f is formally conjugate to a formal power series A n w n with real coefficients A n . If there exists n ≥ 2 such that A n = 0, then 0 is a parabolic fixed point of f . Let n 0 = k + 1 be the minimum such n. Then a scaling brings A k+1 to 1 2 if k is odd (resp. ± 1 2 if k is even). Proof. The proof is a mere computation. Let w = h(z) = n≥1 b n z n be a formal change of coordinate and suppose f † (w) = w + n≥2 A n w n . If we compare h • f (z) = f † • h(z) degree by degree, we find an expression for the coefficients of the form b 1 a 1 = b 1 A n = b n − b n + a n + P n (A 1 , . . . , A n−1 , a 1 , . . . , a n−1 , b 1 , . . . , b n−1 ), where P n is some polynomial. Hence, we have arg b 1 = − 1 2 arg a 1 + π, with ∈ Z. With a recursive argument, if A 1 , . . . , A n−1 are real, for A n to be real, we may choose b n = 1 2 (a n + P n ), since P n depends only on terms that were fixed in the previous steps.
Remark 2.0.3. The formal change of coordinate h is not unique. Indeed, only the imaginary part of the coefficients are determined, leaving their real part free. However, the order of the first non linear term is well defined. This leads to the following definition.
We say that f is parabolic of codimension k if the first non linear term of f † is of order k + 1.
1. The formal series with real coefficients preserves the real axis. This indicates that f has a privileged unique direction which we will call a formal symmetry axis. Hence a conjugacy between two antiholomorphic parabolic germs must preserve the formal symmetry axis. We can of course suppose that this formal axis is the real axis. Note however that in the case of even codimension, there is no canonical orientation of the formal symmetry axis. 2. The dynamics nearby the formal symmetry axis is a topological invariant. When k is odd, a rotation of angle π will flip the attractive semi-axis with the repulsive one. When k is even, both semi-axes are either attractive (when A k+1 < 0) or repulsive (when A k+1 > 0) (see Figures 3 and 4). In this paper, we will only consider the case A k+1 > 0. Indeed, when A k+1 < 0, i.e. f is of negative type, then f −1 will be of positive type and classifying f −1 is equivalent to classifying f . Definition 2.0.6. When the codimension k is even, we say f is of positive type (resp. negative type) if A k+1 > 0 (resp. A k+1 < 0), where A k+1 is the first non zero coefficient in (4).  The composition of two antiholomorphic germs is holomorphic. Therefore, we will look at g := f • f , which is a holomorphic parabolic germ. Recall that in the holomorphic case, the codimension of g is the order of the first non zero term of g(z) − z. It is linked to the multiplicity of the fixed point: g is of codimension k if and only if the fixed point has multiplicity k + 1.
The case when f • f = id is seen as a degenerate case where f is of "codimension infinity". Indeed, it only happens if f is analytically conjugate to the complex conjugation, as is shown below. This case was excluded from our definition of parabolic point, since the fixed point of σ at the origin is not isolated. Proposition 2.0.8. Let f (z) = a 1 z + a 2 z 2 + a 3 z 3 + · · · be an antiholomorphic germ at the origin. The following statements are equivalent: 1. f is formally conjugate to σ; 2. f is analytically conjugate to σ; Let us suppose that f • f = id. In particular, |a 1 | = 1. We can of course suppose f is already in a coordinate such that a 1 = 1.
We are interested in the fixed points of F , that is the zeros of F − id.
We complexify t to obtain a change of coordinate t = γ −1 (z) = u+iv that rectifies the curve γ on the real line.
The equation for fixed points F = id is equivalent to v = 0 and r(u, 0) = 0. If r(u, 0) = au s +o(u s ), a = 0, then this would contradict the fact that we must have F • F (u, 0) = u 0 . Therefore r(u, 0) ≡ 0, in other words r(u, v) = vp(u, v).
We see that the real axis is a line of fixed points forf near the origin. By the Identity Theorem, becausef • σ − id = 0 on the real axis near the origin, we havef ≡ σ. 2. ⇒ 1. This is immediate.
The formal power series with real coefficients is used to determine a formal normal form for f . Recall that a formal normal form for g := f • f may be taken as the time-1 map of the flow of (see [6]) (5)ż = z k+1 1 + bz k for some constant b ∈ C. We will call this constant b the formal invariant. It is also sometimes called the "résidu itératif" and, as mentioned in [4], it is determined by the holomorphic fixed point index, that is, the residue of 1 z−g(z) at the origin. When g = f • f is of codimension k, it is possible to get rid of the terms of degree k + 1 < j < 2k + 1 by an analytic change of coordinate. In this coordinate, g is written where b ∈ C is the formal invariant of g. When g is in this form, we will say that it is prenormalized.
As the name suggests, b is invariant under formal changes of coordinate. Since g † := f † • f † and g have the same formal invariant, where f † is as in (4), it follows that b is real because all the coefficients of g † are real.
An important consequence of this, is that the time-t map v t of (5) for t ∈ R has a power series at 0 with real coefficients, that is the complex conjugation σ and v t commute. • σ is formally conjugate to f . The formal change of coordinate conjugating f to its formal normal form can always be truncated at the (2k + 2)-th term, which yields a holomorphic change of coordinate taking f to the form that is f and σ • v 1 2 have the same first three terms.
Definition 2.0.11. When f is in the form (7), we will say that it is prenormalized.
Remark 2.0.12. In even codimension, f may only be prenormalized as in (7) when A k+1 > 0. In odd codimension, f may always be prenormalized as in (7).
The formal normal form is a model to which the germs can be compared. Now that this form has been established, we describe its properties.

Properties of the Formal Normal Form
Let us start with the following observations. The holomorphic and antiholomorphic formal normal forms are respectively where v t is the time-t of v.
We see that the real axis is a symmetry axis. We introduce a notation for the other symmetry axes.  1. v is invariant under σ for = 0, . . . , k − 1 when b is real; 2. v 1 commutes with any rotation of order k, and when b is real, it commutes with σ for = 0, . . . , k − 1; 3. When k is even, σ • v 1 2 commutes with z → −z.
We will only be interested in real values of b. We ask the following questions, which will be answered in Section 7.2.
Question 3.0.5. For a holomorphic parabolic germ g, how many distinct antiholomorphic n-th roots (n even) does it have? Question 3.0.6. For an antiholomorphic parabolic germ f and n odd, when is the formal n-th root convergent?

Fatou Coordinates
For the whole section, when the codimension of f is even, we will suppose f is of positive type (see Definition 2.0.6). The formal normal form σ • v 1 2 is a model to which it is natural to compare the antiholomorphic germ f . In the holomorphic study of parabolic germs, we use holomorphic diffeomorphisms called Fatou coordinates defined on sectors covering the origin on which the germ is conjugated to its normal form, i.e. changes of coordinates to the normal form. We then compare Fatou coordinates on the intersection of the sectors, thus yielding a conformal invariant describing the space of orbits of the germ. See [5] or [6] for the details.
The same approach can be adapted to the antiholomorphic case. It will be necessary to find a sectorial normalization (Fatou coordinates) of the antiholomorphic germ f . However, instead of adapting the construction of the holomorphic case, we will prove that it is possible to choose Fatou coordinates of f • f , which is holomorphic, that are also Fatou coordinates of f .

Rectifying Coordinates and sectors.
Suppose that an antiholomorphic parabolic germ f is of codimension k for k ≥ 1 with a formal invariant b (see Def. 2.0.9). The Fatou coordinates ϕ j are often constructed in the rectifying coordinate given by the time of the vector field (5). Since v 1 2 and v 1 are the time maps of the vector field (5), we define the time coordinate by which is multi-valued. See Figure 5 for its Riemann surface. It is the inverse of the flow of (5) with starting point z 0 . We will single out the following 2k + 1 charts of Z: where log z is determined by arg z ∈ (−π, π) for −k < j < k, and for Z k (resp. Z −k ), arg(·) will be the continuation in (0, 2π) (resp. in (−2π, 0)). In particular, we see that Z k = Z −k , and that both Z 0 and Z k commute with the complex conjugation. Now we define the sectors in the z-space (see Figure 7). On the Riemann surface of Z j , we write G j for the expression of g : We consider a vertical line j passing through Z * j and its image G j ( j ). Let B j be the domain bounded by j and G j ( j ) and containing j and G j ( j ). The sector in the Z j -coordinate is then obtained by Figure 6). We see that Figure 7). These sectors are sometimes called Fatou petals. They are described in great details in [2], although the authors only consider attractive petals. Note that there are 2k petals, with half of them being repulsive (see Figure 8). Also, S k and S −k are the same petal.
where γ j is an arc of the circle ∂D(0, δ) in S j , with endpoints z j+1 and z j , where z j = δe . The Z j defined as in (12) satisfy this condition.
The particular case ofż = z 4 . On the right, the sector U 0 in the Z 0 -coordinate, obtained from a strip (in dark gray). On the left, S 0 = Z −1 (U 0 ) the sector in z, with the preimage of the strip (in dark gray).
The sectors are ordered as in Figure 9. Note in particular that S 0 intersects the positive real axis, and S k = S −k , the negative real axis.
The time coordinate is the Riemann surface obtained from the disjoint union of the U j , glued together by the transition functions: the charts are the U j → C, with the diffeomorphism Z j : S j → U j given by  The time coordinate is conformally equivalent to a punctured disk of the origin. Now we define the complex conjugation on the time coordinate. Note that on a subdomain S 0 ⊆ S 0 such that σ(S 0 ) = S 0 , we have Z 0 (z) = Z 0 (z). The complex conjugation on the time coordinate is then obtained by analytic continuation on the other charts U j . Proposition 4.1.3 (Complex conjugation). For z ∈ S j , let be such that σ(z) = z ∈ S . We define the complex conjugation Σ on the time coordinate in the charts by Then Σ is well-defined and Σ • Σ = id.
Proof. The proof consists of showing that Σ is compatible on both charts when Z j ∈ U j ∩ U j+1 or when Σ(Z j ) ∈ U ∩ U ±1 . It is a simple computation. Note that for a subdomain S j ⊂ S j such that σ(S j ) ⊂ S −j , then in the charts, we have Σ −j,j (Z j ) = Z j .
This allows us to talk about the normal form σ • v This means that the dynamics of those two petals is no longer independent, unlike the holomorphic case. See Figure 10.
In its prenormalized form f is close to σ In the following lemma, we prove that it is also true that F and Σ • T1 2 are close in the time coordinate. Lemma 4.2.
1. Let f be in its prenormalized form (7) and let F (resp.
The proof is similar to that found in [6]. Let . We now present the existence of the Fatou coordinates. Note that Hubbard and Schleicher proved their existence in [4] (Lemma 2.3) in the codimension 1 case for a map with a parabolic periodic orbit of odd period n. We recover their case by considering f •n . This corresponds for us to a germ of antiholomorphic parabolic diffeomorphism of codimension 1. The proof in higher codimension is in the same spirit with an adaptation, since we need to work with pairs of sectors (U j , U −j ).
Proof. The proof makes use of the rigidity of the conformal structure of the doubly punctured sphere S 2 \ {0, ∞}, as in the proof of the uniqueness in the holomorphic case.
(We know that it exists since f • f is holomorphic and that it is unique up to left-composition with a translation.) In the space of the Fatou [6]). We first note that each Φ j (U j ) contains a vertical strip B j of width 1, by construction of the time coordinate U j and of the Fatou coordinate. We define Then we see that Indeed, Q j represents F in the Fatou coordinates. It is therefore natural that Q j commutes with T 1 , which represents F • F in the Fatou coordinates.
Because Q j is the composition of an antiholomorphic germ by a holomorphic diffeomorphism, Q j is antiholomorphic. In particular, Σ • Q j is holomorphic, and Σ • Q j − id is 1-periodic and holomorphic, so it has a Fourier expansion We then adjust all the c j,0 to 0 by choosing the appropriate Fatou coordinates (i.e. composing them with a translation).
The uniqueness comes from a combination of the uniqueness of the Fatou coordinates for the holomorphic f • f and having to preserve the constants c j,0 = 1 2 .

Modulus of Analytic Classification
If two antiholomorphic parabolic germs are analytically conjugate, then they have the same space of orbits. The space of orbits of an antiholomorphic parabolic germ f is a quotient of the set of orbits of the associated holomorphic parabolic germ g = f • f . Hence we start by describing the space of orbits of g; on a Fatou coordinate, it is the quotient by T 1 , which is a bi-infinite cylinder. We also need to identify some orbits represented in two different Fatou coordinates. This is done by means of the transition maps (the horn maps of Écalle).
We will describe the space of orbits of f in Section 6.1 and classify the antiholomorphic germs in Section 6.3. To do both of these, we will need the transition functions, which will form an analytic invariant.
The transition functions we describe here are the same as for the holomorphic case. We will introduce what we need here; all the details are found in [6] or [5].
In the time coordinate, if U j is a repelling (resp. attractive) petal, then U j and U j+1 intersect on a domain containing an upper half-plane (resp. a lower half-plane), see Figure 11. We can compare the Fatou coordinates Φ j and Φ j+1 by looking at where V j = Φ j (U j ) for all j. This yields a diffeomorphism defined on a domain of V j (resp. V j+1 ) containing an upper-half plane (resp. lower half-plane) with its image in V j+1 (resp. V j ) also containing some upper-half plane (resp. lower-half plane). Figure 11. Charts U j−1 and U j on the time coordinate. They intersect on a region containing (in this case) an upper half-plane.
Notice the order of the composition for Ψ j : we choose the convention that these functions will go from a repulsive petal to an attractive petal. Figure 12 is an illustration of the direction of the arrows in the z-coordinate for k = 3, where ξ j is the expression of Ψ j in the z-coordinate.
for j odd; Φ j−sgn(j) • Φ −1 j , for j even; where the composition is defined. Here, sgn(j) is the sign of j. By the uniqueness of Proposition 4.2.2, we may change Φ ±j by T C j • Φ j and T C j • Φ −j for some C j ∈ C for j = 1, . . . , k − 1, or Φ j by T R j • Φ j for some R j ∈ R for j = 0, k. This will yield another set of 2k transition functions. We will identify together these different possible choices of transition functions at the end of this section.
The following proposition is the first step towards the geometric invariant. The transition functions allow to describe the space of orbits of F and F • F .
In particular, they are transition functions of f • f and satisfy Proof. The proof is identical to the holomorphic case; it follows from the definition of Ψ j and (13).
Equation (15) says that the orbits of Σ • T1 2 in one Fatou coordinate are sent by the Ψ j on the orbits of Σ • T1 2 in another Fatou coordinate. In those coordinates, the orbits of Σ • T 1 2 correspond to those of f . Therefore, the transition functions allow to identify the same orbits of f in different coordinates.
We can rewrite Equation (15) as Thus we only need half of the transition functions of f to determine all of them. For the rest of the paper, we will work with Ψ 1 , . . . , Ψ k , knowing that Ψ −1 , . . . , Ψ −k are obtained from Equation (17).
The transition functions in the holomorphic case are well known and their properties are described in [5] by Ilyashenko. Because the transition functions of f are also those of f •f , they share the properties which we describe now.
Each Ψ j satisfies Equation (16); it follows that Ψ j − id is 1-periodic and has a Fourier expansion c n,j e 2iπnW for j > 0 odd; c n,j e 2iπnW for j > 0 even.
In particular, we see that |Ψ j −id−c j | is exponentially decreasing when W → ∞ and j is odd (resp. W → −∞ and j is even). Since the Fatou coordinates are not unique, we may change them and obtain new transition functions. This will change the constants c j and c n,j , but the following alternating sum will always be preserved We successively change Even when normalized, the transition functions are not uniquely determined. There is still a remaining degree of freedom: we may change the source and target space of each Ψ j by the same translation T C , with C ∈ R.
In the case of even codimension, if f is prenormalized, then f 1 (z) = −f (−z) is also prenormalized, and f and f 1 are conjugate. Hence, we will need to identify their moduli.  1. If k is odd, then [Ψ 1 , . . . , Ψ k ] is the equivalence class under the Relation (21); 2. If k is even, then [Ψ 1 , . . . , Ψ k ] is the equivalence under the Relation (21) and the additional relation This equivalence class is called the analytic invariant.

Remarks on the Écalle-Voronin Modulus.
In the holomorphic case, the modulus of classification is known as the Écalle-Voronin modulus (see [5] and [6]). For any holomorphic parabolic germ g (not necessarily of the form g = f • f ), we can obtain its Écalle-Voronin modulus the same way as described above, but without equation (15), so that the 2k transition functions are needed. There are 2k degrees of freedom, so we normalize the transition functions by choosing the constant terms as in (20) and with c −j = −c j ; the remaining degree of freedom is a translation in every Fatou coordinate by a constant C ∈ C. Therefore, we quotient by the equivalence relation (23) We also quotient by the action of the rotations of order k. If we note the indices Ψ −j = Ψ 2k−j+1 for j = 1, . . . , k, then we have the identification where indices are mod 2k. We will note the equivalence class of both of these identifications by [Ψ −k , . . . , We describe the link between the Écalle-Voronin modulus of g and the modulus of classification of an antiholomorphic parabolic germ f of positive type such that g = f •f , when such a f exists. For k odd (resp. k even), the symmetry axis of f appears along one of the k (resp. k 2 ) "symmetry axes" of g of the form e 2i π k R, = 0, . . . , k − 1. Therefore, we associate to the analytic invariant of g the regular k-gon with the k symmetry axes e i π k R (see Figure 13) in the following way. Divide the k-gon by its k symmetry axes to produce 2k sectors in the k-gon, then starting with the sector above the horizontal line on the right side, we identify this sector with Ψ 1 and going anti-clockwise, we associate Ψ 2 , . . . , Ψ k , Ψ −k , . . . , Ψ −1 to the subsequent sectors, as in Figure 13. The dihedral group D 2k acts on the sectors of the k-gon. The action is defined for an element u ∈ D 2k by mapping a sector to its image by the linear application represented by u (a rotation or a reflection).
Definition 5.1.1. Let u ∈ D 2k and let ∆ j be the sector of the k-gon associated with Ψ j . The action of u on the sectors of the regular k-gon defines a permutation on {−k, . . . , −1, 1, . . . , k} also noted u by abuse of notation, defined so that u −1 (∆ j ) = ∆ u(j) (see Figure 13).
Lastly, we will talk about the modulus of the inverse of a holomorphic parabolic germ g. The following proposition is probably well-known, but we could not find it in the literature, so the proof is included. Proof. Suppose g is prenormalized. The dynamics of g −1 is reversed, therefore the dynamics of g −1 in the first sector S 0 is attractive, but the dynamics in S −1 is repulsive. We apply the change of coordinate At the formal level, we apply (z, t) → (y, −t) to the vector field (5) to obtain The formal normal form of g −1 is the time-1 map w 1 of (27). In particular, g −1 has formal invariant −b. We will denote the sectors of g −1 in the y-coordinate by S j . Note that S j = L −1 λ (S j ) = S j−1 . The time coordinate on the sector S j is defined as We have the relation where the Z j 's are the time coordinates of g. Indeed, we see that It follows that the transition functions Ψ j of g −1 are given by, for j = 1, k odd, The other values of j are done similarly. Note however that for j = 1, the equation becomes

Space of Orbits and Classification Under Analytic Conjugacy
When two antiholomorphic parabolic germs are analytically conjugate, it is clear that their space of orbits are essentially the same. Our guiding principle is that the space of orbits completely describes the dynamics of the germs; when two germs have the same space of orbits, they should be analytically conjugate. A formal statement will be given in the form of the Classification Theorem 6.3.1 in Section 6.3. 6.1. Description of the Space of Orbits. The space of orbits in the holomorphic case is well known. It is briefly described in [5]. We will introduce the objects from the holomorphic case needed for the antiholomorphic description.
In each Fatou coordinate of f (and f • f ), we may choose a fundamental domain of g = f • f by taking any vertical strip B j of width 1. We quotient by the action of T 1 , obtaining the bi-infinite Écalle cylinders. Some orbits of g will appear in two consecutive cylinders. Since Ψ j • T 1 = T 1 • Ψ j , we identify together those orbits by identifying W j with Ψ j (W j ) (or Ψ −1 j (W j ) depending on j). The universal covering E : C → C * given by w = E(W ) = exp(−2iπW ) is a biholomorphism of each cylinder onto C * . This allows us to see the Écalle cylinders as Riemann spheres punctured at 0 and infinity S 2 j \ {0, ∞} (see Figure 14). We will define the horns maps using this universal covering. For positive j odd (resp. j even), the horn maps are defined on a punctured neighbourhood of the origin (resp. of infinity) with their To retrieve the ψ j for j < 0, we use Equation (17) in the coordinate where L −1 (w) = −w and τ (w) = 1 w . The space of orbits of g = f • f is described by the 2k spheres with identifications at the origin or at infinity, as seen in Figure 15. 6.2. The space of orbits of f . The complex conjugation Σ becomes τ (w) = 1 w on the spheres, where w ∈ S 2 j and τ (w) ∈ S 2 −j . The transla- To obtain the space of orbits of f , we identify w and L −1 • τ (w) in the space of orbits of f • f , that is on the 2k spheres above.
Let us first consider the case of codimension 1, so that we have two sectors S 0 and S 1 = S −1 in the z-coordinate and two spheres S 2 0 and S 2 1 . Recall that for z ∈ S 0 , f −1 (z) is still in S 0 . This means that L −1 • τ acts on S 2 0 . It sends 0 to ∞, the northern hemisphere to the southern hemisphere, and the equator on itself. On the equator |w| = 1, we identify w to −w. The resulting surface is the real projective space RP 2 . This is also true for S 1 and S 2 1 . The equators of both spheres play a special role; they each represent orbits along an invariant half-curve that each forms a "semi-axis" of reflection for f . Therefore, in codimension 1, the space of orbits is two real projective spaces with one germ of holomorphic diffeomorphism where [ψ 1 ] is the equivalence class of ψ 1 under the quotient of S 2 to RP 2 and [0] is equivalence class of the points {0, ∞} identified together. The In codimension k > 1, the spheres S 0 and S k both quotient to a real projective space, but the other spheres are identified in pairs (S j , S −j ), so that the quotient of the union of the two spheres is diffeomorphic to a sphere. The space of orbits is then described by two real projective spaces together with k − 1 spheres and k equivalence classes of horn maps [ψ j ] = {ψ j , ψ −j }. The class [ψ j ] defines a germ at [0] = [∞] on the quotiented spheres (S j , S −j ), since the representatives satisfy On the two extreme projective spaces we have a distinguished curve given by the equator. The only changes of coordinates on D preserving the equator and sending opposite points to opposite points are the linear maps L c with |c| = 1. Hence the lines W = y are invariant. This y is the generalization of the Écalle height introduced in [4]. However, on the other spheres, there does not appear to be a quantity preserved by changes of coordinates.
The two projective spaces correspond to the orbits of two Fatou petals containing the formal symmetry axis of f . The equator of each projective space corresponds to a semi-axis of symmetry in each respective petal. These axes meet at the origin, but generally they cannot be extended into a real analytic curve. We will see in Theorem 7.3.1 exactly when they extend into a real analytic curve. This formal symmetry axis explains the existence of the Écalle height in the two projective spaces and why the Écalle height does not exist in the other spheres. Proof. The proof is analogous to that in the holomorphic case, which can be found in [5] and [6].
Suppose first that f 2 (u) = h • f 1 • h −1 (u) for some germ of analytic diffeomorphism u = h(z). The germs f 1 and f 2 must have the same codimension and formal invariant, since they are topological and formal invariants. For the analytic invariant, first let F 1 , F 2 and H denote the expressions of f 1 , f 2 and h in the time coordinate. If Φ j is a Fatou coordinate of F 2 on U j , then Φ j • H is a Fatou coordinate of F 1 . It follows that they have the same transition functions.
Conversely, suppose f 1 and f 2 have the same modulus. We can choose a common normalized representative (Ψ 1 , . . . , Ψ k ) for both classes and Fatou coordinates Φ j, for f ( = 1, 2), such that the Ψ j are the transition maps for these Fatou coordinates. Let ϕ j, = Z −1 j • Φ j, • Z j , for = 1, 2, be the expression of Fatou coordinates in the z-coordinate. Then we define h j (z) = ϕ −1 j,1 • ϕ j,2 (z) on S j . It sends the orbits of f 2 to those of f 1 on S j . On the intersection of two consecutive sectors S j ∩ S j+1 , we see that so the h j 's agree on the intersection. We may define h on j S j by It sends a punctured neighbourhood of the origin into another punctured neighbourhood of the origin, so by the Riemann Bounded Extension Theorem, h extends to a holomorphic diffeomorphism of a neighbourhood of the origin. Finally, we see that h • f 2 = f 1 • h since it sends the orbits of f 2 to the orbits of f 1 on a whole neighbourhood of the origin.
2. The proof is in two steps. First we construct an abstract Riemann surface S on which Σ • T1 2 is well defined and we prove that this surface has the conformal type of a punctured disk. Then we prove that Σ • T 1 2 is the germ we are looking for on the disk.
Let (Ψ 1 , . . . , Ψ k ) be a representative of the analytic invariant and let Ψ −1 , . . . , Ψ −k be the other transition functions obtained from (17). Let σ • v 1 2 be the normal form of codimension k with formal invariant b. Let U j be the charts in the time coordinate of v.
We consider the transition functions defined on those charts. More precisely, Ψ 1 is defined on a domain of U 0 containing an upper-half plane with its image in U 1 ; Ψ 2 is defined on a domain of U 2 containing a lower half-plane with its image in U 1 , and so on. We define the Riemann surface S by where ∼ identifies W j ∈ U j with its image by Ψ j or Ψ j−1 (depending on j). As it is done in [6], we can build a smooth quasi-conformal mapping P : S → C * , and from the Ahlfors-Bers Theorem (see [1]), find a diffeomorphism Q : D → (C, 0), where D = P (S) ∪ {0}, so that the composition H = Q • P : S → (C, 0) is holomorphic. In fact, H is a biholomorphism of S onto some punctured disk of the origin.
By (15), we know that Σ • T1 2 is well defined on S. The map f = H • Σ • T1 2 • H −1 extends to the origin by f (0) = 0, since it is bounded around 0 (we can apply the Riemann Removable Singularity Theorem to f • σ to see this). Lastly, T 1 is also well-defined on S, so we set , it follows that g = f • f , and by the chain rule, ∂f ∂z 2 = g (0). By the holomorphic case, we know g is a holomorphic parabolic germ of codimension k, so it follows that f is parabolic and of codimension k. The formal invariant of f is b, since it is determined by (19).
It remains only to prove that f is of positive type. The formal symmetry axis of f is the real line, since the Ψ 1 and Ψ −1 are defined We conclude f is a germ with modulus (k, b, [Ψ 1 , . . . , Ψ k ]).

Applications of the Classification Theorem
Here we will solve Questions 1 to 5 of the introduction and other related questions. 7.1. Embedding in a Flow or the Complex Conjugate of a Flow. If a holomorphic germ g is conjugate to the normal form, that Similarly, we will say that f is embeddable if it is embedded in the fam- When is an antiholomorphic parabolic germ embeddable? This corresponds to Question 3. The answer is read in the modulus of classification. Proof. The transition functions of the normal form are translations since the time charts Z j are Fatou coordinates. Therefore, it follows from the Classification Theorem 6.3.1.
In Proposition 2.0.2, we proved that f is always formally conjugate to the sum of a formal germ with real coefficients. We ask the related question.
Question 7.1.2. When is a parabolic antiholomorphic germ analytically conjugate to a series with real coefficients?
Of course, this is the case for any embeddable germ. But we will show in Section 7.3 that the embeddable germs form a subset of infinite codimension in the set of antiholomorphic parabolic germs conjugate to a germ with real coefficients. We will answer this question in Section 7.3. We first tackle Question 1.
where s is the reflection of indices with respect to e iπ k R (see Definition 5.1.1); 3. If g is not analytically conjugate to its normal form, then i) Each family has at most one convergent root, so g has at most k distinct antiholomorphic n-th roots f j ; ii) If g has m distinct antiholomorphic roots f 1 ,. . ., f m with distinct linear parts e 2iπ j k z, j = 1, . . . , m, then the modulus of g has gcd( m − 1 , . . . , m − m−1 , k) independent transition functions.
Proof. 1. That g has a 1-parameter family of formal antiholomorphic n-th root on each symmetry axis is a consequence of the fact that g is formally conjugate to the normal form v 1 , and Proposition 3.0.4.

2.
We first prove this for the case k odd. Let be a representative of the analytic invariant of g.
First, we suppose that the representative satisfies (32). To realize an antiholomorphic root tangent to the symmetry axis e 2iπ k R, we rotate the coordinate by R (z) = e − 2iπ k z so that this axis is on the real line and the dynamics of R • g • R −1 on the side of the positive real axis is repulsive. Let g = R •g •R −1 and let r correspond to the permutation of indices defined by R (see Definition 5.1.1). The representative is permuted into (Ψ r(−k) , . . . , Ψ r(−1) , Ψ r(1) , . . . , Ψ r(k) ). Also, we have that r −1 s r = s 0 , where s 0 (j) = −j and s is the reflection of indices induced by σ . Equation (32) becomes It follows that (Ψ r −1 (−k) , . . . , Ψ r −1 (−1) , Ψ r −1 (1) , . . . , Ψ r −1 (k) ) is a representative equivalent to Ψ under Relation (23) which satisfies (32), using the fact that (r −1 s r)(j) = −j.
In the case k even, we may have antiholomorphic roots of positive and negative type. Those of positive type are tangent to an axis e iπ k R with even and are done as before. Those of negative type are tangent to an axis e iπ k R with odd. For such a root f , f −1 will be a root of positive type of g −1 . By Proposition 5.1.2, the modulus of g −1 is By simplifying the L −1 and taking the inverse on both sides, we obtain (32).
3. i) Suppose g has two roots f 1 and f 2 from the same family. In particular, they have the same linear term, so we may suppose they are tangent to σ, modulo conjugating g by a rotation of order k. In the Fatou coordinates, they take the form Σ • T 1 2 +iy j , for j = 1, 2, by Proposition 3.0.4. Since Σ • T1 2 +iy j satisfies (15) for j = 1, 2, and by combining with (18), we see that either y 1 = y 2 or the Ψ j 's are translations.
ii) Lastly, consider the dihedral group D 2k with its action on the regular k-gon. Recall that we can divide a regular k-gon by its k symmetry axes to form 2k sectors, which we associate to the transition functions (see Section 5.1 and Figure 13).
Let H = s 1 , . . . , s m be the subgroup of D 2k generated by the reflections along the symmetry axes of f 1 , . . . , f m . Each element of H acts on the modulus by introducing relations of the type , thus reducing the number of independent Ψ j (see Figure 17). The orbit by H of a sector represents the transition functions tied together, therefore the number of independent transition functions corresponds to |D 2k : H|. To compute this, we first observe that H must itself be a dihedral group, so that H = D 2j for some j (see [3]). In fact, j is given by since any rotation s m s p has order k/ gcd( m − p , k). It follows that |D 2k : H| = gcd( m − 1 , . . . , m − m−1 , k).
Proof. That f has a unique formal n-th root follows from Proposition 3.0.4. • Ψ j . To conclude, we note that gcd(1 − n, 2n) = 2 because n is odd, so there exists p, q ∈ Z such that (1 − n)p − 2nq = 2. In other words, we have 1 n = (1−n) 2n p − q. Since Ψ j commutes with T(1−n) 2n p and with T q (because q is an integer), it follows that Ψ j commutes with T 1 n . Corollary 7.2.3. f has an antiholomorphic n-th root for n odd if and only if f is the square root of g and g has a holomorphic n-th root, with n odd.
Proof. Equation (33) is independent of the representative and it is equivalent to a holomorphic parabolic germ having a holomorphic n-th root (see [5]).
Suppose g = f • f and g has a holomorphic n-th root. Then the modulus of g and f satisfies (33), so that f has an antiholomorphic n-th root.
The converse is direct.

Germs with an Invariant Real Analytic
Curve. An antiholomorphic germ with real coefficients preserves the real axis. Any germ f analytically conjugate to the latter will preserve a real analytic curve; it is a property of the equivalence class of f . Therefore, Question 7.1.2 is equivalent to asking when does an antiholomorphic parabolic germ f preserve a real analytic curve. Like the embedding problem, to preserve a germ of real axis is a condition of codimension infinity, but it is a "smaller" infinity, i.e. not every transition functions needs to be a translation, but we will see that there are infinitely many conditions of the form c n,j = c −n,−j for the Fourier coefficients in (18).
curve, as H can be extended to a C 1 diffeomorphism H : S → (C, 0), where S is obtained from S with the point ∞ = H −1 (0) added.
This curve divides a small disk D(0, δ) in two connected components A and B. By the Riemann Mapping Theorem, there exists a biholomorphism ϕ of A to the upper half-plane that sends continuously the boundary of A on the real line, see Figure 18. The image of γ corresponds to an interval [a, b]. We can extend ϕ to A ∪ γ ∪ B by This is holomorphic on B, since it is the composition of a holomorphic map with two antiholomorphic maps, and it is continuous on A ∪ γ ∪ B. The argument to prove that ϕ is holomorphic is similar to that of the Schwarz Reflection Principle. The idea is to show that the integral of ϕ along any triangle in A ∪ γ ∪ B vanishes, and it will follow from Morera's theorem. If a simple closed curve is in A ∪ γ or B ∪ γ, then it follows from Cauchy's theorem (we may take a limit of closed curves in A or B converging to the initial one). Then, we can divide any triangle in A ∪ γ ∪ B along γ to obtain a finite number of closed curves in A ∪ γ and in B ∪ γ, as in Figure 19. Thus, ϕ  Figure 19. Triangle divided in two closed curves When f is conjugate to a germ with real coefficients, then so is the holomorphic germ g = f • f . However, it is not true that every holomorphic germ analytically conjugate to a germ with real coefficients must have a germ f such that g = f • f , as the next theorem will show (see also Proposition 7.3.3). Even though this is not a property directly linked to the antiholomorphic parabolic germs, we will still prove the following necessary and sufficient condition for a holomorphic parabolic germ to preserve a real analytic curve. Theorem 7.3.2. Let g : (C, 0) → (C, 0) be a parabolic holomorphic germ and let (k, b, [Ψ −k , . . . , Ψ −1 , Ψ 1 , . . . Ψ k ]) be its Écalle-Voronin modulus (see Section 5.1). The following statements are equivalent 1. g preserves a germ of real analytic curve at the origin; 2. g is analytically conjugate to a germ with real coefficients; 3. For each representative (Ψ −k , . . . , Ψ −1 , Ψ 1 , . . . , Ψ k ), there exists y ∈ R such that the transition functions satisfy Σ • T iy • Ψ j = Ψ −j • Σ • T iy ; 4. There exists a representative (Ψ −k , . . . , Ψ −1 , Ψ 1 , . . . , Ψ k ) such that Ψ j • Σ = Σ • Ψ −j .
Proof. 1. ⇒ 2. and 4. ⇒ 1. are the same as in the previous theorem. 2. ⇒ 3. Suppose g is in a coordinate such that g = σ • g • σ. Let Φ j be a Fatou coordinate on U j for −k ≤ j ≤ k, where Φ −k = Φ k . Then Σ • Φ −j • Σ is also a Fatou coordinate of g on U j . By the uniqueness of the Fatou coordinate (for the holomorphic case), there exists a constant C ∈ C such that Σ • Φ −j • Σ • Φ −1 j = T C for all j. In particular, for j = 0, by taking the inverse and conjugating both sides by Σ of the previous equation, we obtain T C = T −C , so C must be pure imaginary, say iy with y ∈ R. For j > 0 and odd, we conclude with The other values of j are done similarly. 3. ⇔ 4. It follows from the fact that for any two representatives (Ψ −k , . . . , Ψ −1 , Ψ 1 , . . . , Ψ k ) and (Ψ −k , . . . , Ψ −1 , Ψ 1 , . . . , Ψ k ), there exists C ∈ C such that Ψ j • T C = T C • Ψ j . We choose C = − iy 2 to get a representative satisfying Ψ j • Σ = Σ • Ψ −j . Proposition 7.3.3. There exists g with real coefficients that has no antiholomorphic square root (see section 7.2).
It forms a group with Diff 1 (0, C) as a subgroup. Next, let A k,b ⊂ Diff 1 (0, C) (resp. A k,b ⊂ Diff 1 (0, C) when b is real) be the set of germs of holomorphic (resp. antiholomorphic) diffeomorphisms with a parabolic fixed point of codimension k at the origin and with formal invariant b.
We will study the centralizer of g ∈ A k,b and of f ∈ A k,b in Diff 1 (0, C, C). Let us note the following: if g ∈ A k,b commutes with f ∈ Diff 1 (0, C), then the formal invariant b of g is automatically real. In fact, we have that g and σ • g • σ are analytically conjugate by means of (σ • f ). So we are interested only in b real, since if b is not real, the centralizer of g in Diff 1 (0, C, C) is the same as the centralizer of g in Diff(0, C), which is already known (see [5]). Lemma 7.4.1. Let g ∈ A k,b and let m ∈ Diff 1 (0, C, C) be a germ that commutes with g ∈ A k,b . Then either m is the identity, or analytically conjugate to σ, or m ∈ A k,b ∪ A k,b . Moreover, in the Fatou coordinates of g, m will be of the form Proof. If m is not the identity or analytically conjugate to σ, then m is parabolic. It must have codimension k, since it maps the orbits of g on the orbits of g and the Fatou petals of g on the Fatou petals of g. To see that m has the same formal invariant, we compare degree by degree the power series on both sides of the equation m • g(z) = g • m(z).
To see m has one of the forms of (34), the first part of the proof of Proposition 4.2.2 applies almost identically to m.
The obvious germs in the centralizer of f or g are the iterates, the roots, and the iterates of the roots, which we define below as the fractional iterates. We prove in Theorem 7.4.3 and 7.4.4 that these are all the possible elements of the centralizer, provided that f or g are not conjugate to their respective normal form.
Definition 7.4.2. Let f ∈ A k,b (resp. g ∈ A k,b ). We say that m ∈ Diff 1 (0, C, C) is a fractional iterate of order p of f (resp. g) if there exists q ∈ Z such that gcd(p, q) = 1 and m •p = f •q (resp. m •p = g •q ).
Note that a fractional iterate of order one is just an iterate of f or of g.
Theorem 7.4.3. Let g ∈ A k,b with b ∈ R. Then we have one of the following cases: Then the centralizer of g is 2. g is not embeddable, then Z g contains only holomorphic and antiholomorphic fractional iterates of g and Schwarz reflections. More precisely, there exists p ∈ N \ {0} such that the centralizer of g is one of the following: • Z g = The centralizer includes a Schwarz reflection tangent to σ if and only if g is analytically conjugate to a germ with real coefficients (see Theorem 7.3.2).
Proof. 1. Since g is embeddable, each of its transition functions is a translation, so any T t will commute with them. Each T t represents a germ analytically conjugate to g t for some t ∈ C. In the case m is antiholomorphic, Σ • T t is compatible with the transition functions with no restriction on t. It corresponds to h • σ • v t • h −1 for t ∈ C.
2. Since g is not embeddable, one of the transition functions is not a translation. Suppose g 1 ∈ Diff 1 (0, C, C) is holomorphic and commutes with g. By Lemma 7.4.1, g 1 becomes T t in the Fatou coordinates for some t ∈ C. Then T t must commute with the transition functions.