Analytic moduli for parabolic Dulac germs

First return maps of planar polynomial vector fields are classically used for studying limit cycles, that is, isolated periodic orbits. They are considered when dealing with the possibility of an accumulation of limit cycles (the Dulac problem) or of their creation by deformation (the cyclicity problem related to Hilbert's 16th problem). Most interesting is the study of first return maps in a neighbourhood of polycycles, which can be hyperbolic or non-hyperbolic. The non-accumulation problem posed by Dulac [4] was solved independently by Écalle [5] and Ilyashenko [8], [9].

We call Dulac germs the type of germs that appear as first return maps of hyperbolic polycycles. In this case, Ilyashenko's approach [7] consists in extending a Dulac germ (he calls them almost regular germs) from the positive real line to a sufficiently large domain in the complex plane, a so-called standard quadratic domain, which is a biholomorphic image of $\mathbb C_+$. This extension ensures the injectivity of the map that assigns to a Dulac germ its power-logarithmic asymptotic expansion, called the Dulac expansion.

However, for non-hyperbolic polycycles, the proof is much more involved ([5] and [8], [9]). Then the first return maps expand asymptotically as transseries that also include iterated logarithms and exponentials. And moreover, determining the domain of their complex extension is not as straightforward.

In this paper we consider parabolic Dulac germs (that is, those tangent to the identity map) and give complete invariants of their analytic classification.

Our construction resembles the construction of Voronin [22] for analytic invariants of parabolic germs analytic at 0, giving the so-called Écalle–Voronin moduli, except that here we get a doubly infinite sequence of pairs of germs giving the moduli. In [15], we study the inverse realization problem for moduli. The problem is solved in a broader class of generalized Dulac germs. This is the reason that here we formulate our results on moduli of analytic classification in this broader class.

The solutions of Dulac's problem by Ilyashenko and Écalle are still not very well understood. They show the necessity of a very precise analysis of germs of first return maps and, in particular, of the relationship between their asymptotic transserial expansion and the germ itself. In this paper we develop techniques such as integral summability, its iterations, and $\log$ -Gevrey asymptotic expansions. They are related to difference equations or differential equations that the objects in question satisfy. We hope that these techniques will be of use also in more general problems of associating germs with given transserial expansions, in particular, in studying the first return map of non-hyperbolic polycycles.

On the other hand, we note that, among Dulac germs, the parabolic germs studied in this paper are the most interesting from the point of view of cyclicity (Hilbert's $16$th problem). Even in the hyperbolic case very little is known about the cyclicity, other than the result of Mourtada [20] in the generic (non-parabolic) case, and some results for polycycles with a few vertices. In the generic case the study of the cyclicity of a hyperbolic polycycle is based on Mourtada's normal form [19], which is a composition of an alternating sequence of powers and translations. The argument breaks down in the parabolic case, and a more precise knowledge of the Dulac map is required. We hope that some combinatorial arguments based on our moduli (and their unfolding) can give this extra information.

Let $\mathcal R_C$ be a subset of the Riemann surface of the logarithm, given in the logarithmic chart with coordinate $\zeta=-\log z$ by

$$\begin{equation*} \varphi\bigl(\mathbb C^+\setminus \overline{K(0,R)}\,\bigr),\qquad \varphi(\zeta)=\zeta+C(\zeta+1)^{1/2},\quad C>0,\quad R>0, \end{equation*}$$

where $\mathbb C^+=\{\zeta\in\mathbb C\colon \operatorname{Re}(\zeta)>0\}$ and $\overline{K(0,R)}=\{\xi\in\mathbb C\colon|\xi|\leqslant R\}$. Such a domain $\mathcal R_C$ (or its image in the logarithmic chart of the variable $\zeta=-\log z$) is called a standard quadratic domain [8], [9] (see Fig. 1).

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For $C=0$ we get the Riemann surface of the logarithm with bounded radii, while for $C>0$ we get a subdomain of it such that the radii become smaller on each level of the surface. Moreover, they tend to zero at an exponential rate. Indeed, if by $\varphi\in[(k-1)\pi,(k+1)\pi)$ we denote the $k$th level of the surface $\mathcal R_C$, $k\in\mathbb Z$, and we set $\varphi_k=k\pi$, $k\in\mathbb Z$, then the radii $r(\varphi_k)$ by levels $k\in\mathbb Z$ decrease at most at the rate

$$\begin{equation} C\exp\biggl(-D\sqrt{\frac{|k|\pi}{2}}\,\biggr),\quad |k|\to\infty,\quad \text{for some } C>0,\ D>0. \end{equation} \tag{ 2.1 }$$

By a germ on a standard quadratic domain [8], [9], we mean an equivalence class of functions that coincide on some standard quadratic domain (for arbitrarily large $R>0$ and $C>0$).

Definition  (adapted from [8], [9], [21]). We say that a germ $f$ is a Dulac germ if:

1) it is holomorphic and bounded on some standard quadratic domain $\mathcal R_C$ and real on the set $\{z\in\mathcal R_C\colon \arg(z)=0\}$;

2) it admits in $\mathcal R_C$ a Dulac asymptotic expansion ¹

$$\begin{equation} \widehat f(z)=\sum_{i=1}^{\infty}z^{\lambda_i} P_i(-\log z),\qquad c>0,\quad z\to 0, \end{equation} \tag{ 2.2 }$$

where $\lambda_i\geqslant 1$, $i\in\mathbb N$, are strictly positive, finitely generated, ² and strictly increasing to $+\infty$, and $P_i$ is a sequence of polynomials with real coefficients, with $P_1\equiv A$, $A>0$.

A Dulac germ tangent to the identity map, that is, with $\lambda_1=1$ and $P_1\equiv 1$ in (2.2), is called a parabolic Dulac germ.

A series of the form (2.2) is called a formal Dulac series. Note that all coefficients in the expansion are real.

The choice of the class of Dulac germs is motivated by the fact that the first return maps near hyperbolic saddle polycycles are analytic germs on the open positive real line having Dulac power-logarithmic asymptotic expansions as $x\to 0$ which extend holomorphically to a standard quadratic domain $\mathcal{R}_C$. We note that in [8], [9] Ilyashenko calls Dulac germs almost regular germs. The existence of an extension to a sufficiently large complex domain was important in proving the fact that almost regular germs are uniquely determined by their Dulac asymptotic expansion, which was the key step in proving the non-accumulation theorem for limit cycles (see [4], [8], [9]).

By [8], [9], Dulac germs form a group under the operation of composition. Here we work with parabolic Dulac germs, that is, with germs tangent to the identity map: $f(z)=z+o(z)$. They form a subgroup under the operation of composition. We expect the hyperbolic cases $f(z)=\lambda z+o(z)$, $|\lambda|\ne 1$, or $f(z)=Az^{\lambda_1}+o(z^{\lambda_1})$, $A\in\mathbb C$, $\lambda_1\ne 1$, to be analytically linearizable, as was the case for germs analytic at $0$ (see, for example, [2]). Parabolic Dulac germs are thus the most interesting from the point of view of cyclicity (see a comment in the Introduction of [17] for more details).

In [15], we deal with the realization problem for moduli of analytic classification of parabolic Dulac germs. We solve there the realization problem in a broader class of parabolic generalized Dulac germs, which we introduce here. This class contains parabolic Dulac germs. Its definition is well adapted to the solution of the realization problem considered in [15].

For convenience let

$$\begin{equation*} \boldsymbol\ell:=-\frac{1}{\log z}\,. \end{equation*}$$

Generalized Dulac expansions and germs are natural generalizations of Dulac expansions and germs: instead of polynomials in $\boldsymbol\ell^{-1}$ multiplying powers of $z$ in Dulac expansions, in generalized Dulac expansions we take series in $\boldsymbol\ell$ that are canonically summable on petals (in a certain generalized Gevrey sense) as analytic germs. Here, in order to be able to uniquely define the asymptotic expansion of such a germ, the problem arises of canonical summation at steps with numbers equal to limit ordinals, since we do not assume that the series in $\boldsymbol\ell$ multiplying the powers of $z$ reduce to polynomials. This problem is solved by introducing the notion of $\log$ -Gevrey summable series (see § 4). Then in § 9 we construct the moduli of analytic classification for parabolic generalized Dulac germs.

We recall from [16] that by $\widehat{\mathcal L}_1$ (or simply $\widehat{\mathcal L}$) we denote the class of power- logarithmic transseries of the form

$$\begin{equation*} \widehat{f}(z)=\sum_{i=1}^{\infty}\,\sum_{m=N_i}^{\infty} a_{i,m} z^{\alpha_i} \boldsymbol\ell^{m},\qquad a_{i,m}\in\mathbb C, \end{equation*}$$

where the $\alpha_i>0$ are finitely generated and strictly increasing to $+\infty$, and $N_i\in\mathbb Z$ for $i\in\mathbb N$. By $\widehat{\mathcal L}_0$, we denote the subclass of $\widehat{\mathcal L}$ consisting of power series with strictly increasing, finitely generated powers. Furthermore, by $\widehat{\mathcal L}_k$, $k\in\mathbb N$, we denote the class of power-iterated logarithmic transseries in the variables

$$\begin{equation*} z,\quad \boldsymbol\ell,\quad \boldsymbol\ell_2=-\dfrac{1}{\log\boldsymbol\ell}\,,\quad\dots,\quad\boldsymbol\ell_k= -\dfrac{1}{\log \boldsymbol\ell_{k-1}}\,, \end{equation*}$$

where the powers of $z$ are strictly increasing and finitely generated, and the powers of $\boldsymbol\ell_i$, $i=1,\dots,k$, belong to $\mathbb Z$. We put $\widehat{\mathfrak{L}}:=\bigcup_{i=0}^{\infty}\widehat{\mathcal L}_i$. If we allow the powers $\alpha_i$ of $z$ to start with negative powers, but to stay finitely generated and strictly increasing to $+\infty$, then we denote the corresponding classes by $\widehat{\mathcal L}_k^\infty$, $k\in\mathbb N_0$, and $\widehat{\mathfrak L}^{\infty}$. The subset of all formal transseries of class $\widehat{\mathcal L}_k$ (or $\widehat{\mathcal L}_k^{\infty}$), $k\in\mathbb N_0$, whose leading term does not contain a logarithm is a group under composition. Note also that if $\widehat f\in\widehat{\mathcal L}$ does not contain a logarithm in the leading term, then $\boldsymbol\ell\mapsto -\dfrac{1}{\log \widehat f(e^{-1/\boldsymbol\ell})}$ is an exponential-power transseries in the variable $\boldsymbol\ell$. By $\widehat{\mathcal L}_k^{\rm inv}\subseteq \widehat{\mathcal L}_k$ we denote the subset of power-logarithmic formal diffeomorphisms (of the form $f(z)=cz+o(z)$, $c\ne 0$), and by $\widehat{\mathcal L}_k^{\mathrm{id}}\subseteq \widehat{\mathcal L}_k$ we denote the subset of transseries tangent to the identity map (that is, of the form $\widehat f(z)=z+\dotsb$). Both are groups under composition. For more details, see [16].

To express that the coefficients of transseries are real, we will write simply $\widehat{\mathcal L}_k(\mathbb R)$ for $k\in\mathbb N_0$, and $\widehat{\mathfrak L}(\mathbb R)$. Parabolic Dulac and parabolic generalized Dulac series belong to $\widehat{\mathcal L}(\mathbb R)$. They both form groups under compositions (see Proposition 8.2).

In Theorem A of [16] we provided a formal classification of transseries in $\widehat{\mathcal L}(\mathbb R)$. The results can be directly applied to a formal classification of formal parabolic Dulac series and of formal parabolic generalized Dulac series. The formal classification is provided in the subgroup $\widehat{\mathcal L}^{\rm inv}(\mathbb R)$ of the group $\widehat{\mathcal L}(\mathbb R)$ of power-logarithmic formal diffeomorphisms (that is, of the form $f(z)=cz+o(z)$, $c\ne 0$) with real coefficients and with finitely generated support. Let

$$\begin{equation*} f(z)=z-az^\alpha\boldsymbol \ell^{m}+o(z^\alpha\boldsymbol \ell^{m}),\qquad \alpha>1,\quad m\in\mathbb Z, \end{equation*}$$

be a parabolic generalized Dulac series. Then it is formally equivalent in $\widehat{\mathcal L}^{\rm inv}(\mathbb R)$ to either of the normal forms

$$\begin{equation} \begin{aligned} \, \nonumber f_F(z)&=z-z^\alpha\boldsymbol\ell^{m}+ \rho z^{2\alpha-1}\boldsymbol\ell^{2m+1}, \\ f_1(z)&=\operatorname{Exp}(X_1).\mathrm{id},\quad X_1(z)=\frac{-z^\alpha\boldsymbol\ell^{m}} {1+(-\alpha/2)z^{\alpha-1}\boldsymbol\ell^{m}+ (m/2+\rho)z^{\alpha-1}\boldsymbol\ell^{m+1}}\,\frac{d}{dz}\,. \end{aligned} \end{equation} \tag{ 2.3 }$$

Let $f_c$ ($c\in\mathbb R$) be the time-$c$ map of the vector field $X_1$:

$$\begin{equation*} f_c(z):=\operatorname{Exp}(cX_1).\mathrm{id}. \end{equation*}$$

It is analytic on $\mathcal R_C$. Its (unique) power-logarithmic expansion will be denoted by $\widehat f_c(z)$. The formal invariants of such series are $(\alpha,m,\rho)$, $\alpha>1$, $m\in\mathbb Z$, $\rho\in\mathbb R$. If $f$ is normalized, that is, if $a=1$, then the reduction to the normal forms can be done in the smaller subgroup $\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$, consisting of transseries tangent to the identity map.

We note that the formal changes of variables needed for reducing parabolic Dulac series to formal normal forms leave the class of formal Dulac series and are transserial (belong to $\widehat{\mathcal L}(\mathbb R)$). Moreover, the blocks in $\boldsymbol\ell$ in them are in general not convergent and do not belong to the class of parabolic generalized Dulac series that we introduce below in Definition 2.3. The class of parabolic generalized Dulac germs is here introduced as the class for which we obtained in [15] a complete result on realization of analytic moduli (namely, realization of every symmetric sequence of diffeomorphisms). For the subclass of parabolic Dulac germs we expect a weaker realization result. The characterization of moduli that can be realized for true parabolic Dulac germs will be the subject of future investigations. On the other hand, the formal changes of variables for parabolic Dulac and parabolic generalized Dulac series belong to a class of so-called block iterated integrally summable series to be introduced in § 6.2. It is an even larger subclass of $\widehat{\mathcal L}(\mathbb R)$, containing the parabolic generalized Dulac series.

To be able to uniquely define generalized Dulac expansions in the definition below of parabolic generalized Dulac germs, we introduce in § 4 the definition of $log$ -Gevrey asymptotic expansions on certain cusps that are $\boldsymbol\ell$-images of sectors, and we state some of their properties. The definition is motivated by the notion of Gevrey asymptotic expansions of order $k\in\mathbb N$ on sectors. For the precise definition of an asymptotic expansion on a sector and of a Gevrey asymptotic expansion of some order on a sector, see, for example, [11], § 1, or [1]. The origin of the name $\log$ -Gevrey is related to the fact that a $\log$-Gevrey asymptotic expansion on some domain is Gevrey of every order $k$, but not necessarily convergent at $0$, therefore being somewhere between Gevrey-type and convergent.

We will prove in Theorem THA, (i), in § 8 a more general fact that a flower-like dynamics of a germ $f$ defined on a standard quadratic domain $\mathcal R_C$ follows merely from the assumption that $f$ is an analytic germ on $\mathcal R_C$ whose leading terms satisfy the uniform estimate

$$\begin{equation*} \bigl|f(z)-(z-az^{\alpha}\boldsymbol\ell^m)\bigr|\leqslant c |z^\alpha\boldsymbol\ell^{m+1}|,\qquad z\in\mathcal R_{C},\quad a>0,\quad c>0. \end{equation*}$$

More precisely, we will prove in § 8 the following proposition.

Proposition 2.2  (flower-like dynamics of a parabolic germ). The invariant sets for the dynamics of a parabolic germ $f$ on $\mathcal R_C$ satisfying the condition

$$\begin{equation} \begin{gathered} \, |f(z)-z+az^\alpha\boldsymbol\ell^{m}|\leqslant c|z^\alpha\boldsymbol\ell^{m+1}|, \\ a>0,\quad\alpha>1,\quad m\in\mathbb Z,\quad c>0,\quad z\in\mathcal R_C, \end{gathered} \end{equation} \tag{ 2.4 }$$

are open attracting petals $V_j^+$ ($j\in\mathbb Z$) on the domain $\mathcal R_C$ (or possibly $\mathcal R_{C'}\subset\mathcal R_C$) that have opening $2\pi/(\alpha-1)$ and are symmetric with respect to the tangential complex directions

$$\begin{equation*} 1^{-1/(\alpha-1)}. \end{equation*}$$

Analogously, the invariant sets for the dynamics of the inverse germ $f^{-1}(z)=z+az^\alpha\boldsymbol\ell^{m}+o(z^\alpha\boldsymbol\ell^{m})$ are open repelling petals $V_j^-$ ($j\in\mathbb Z$) on $\mathcal R_C$ (or possibly $\mathcal R_{C'}\subset\mathcal R_C$) that have opening $2\pi/(\alpha-1)$ and are symmetric with respect to the tangential complex directions

$$\begin{equation*} (-1)^{-1/(\alpha-1)}. \end{equation*}$$

At the intersections of the attracting and repelling petals the orbits for the discrete dynamics are closed.

This in particular applies to parabolic generalized Dulac germs defined below. Therefore, in Definition 2.3 below we may assume in advance the existence of attracting and repelling petals $V_j^\pm$ for the dynamics of $f$ ($j\in\mathbb Z$). These petals are described in detail in Theorem THA, (i), in the next section.

Definition 2.3  (parabolic generalized Dulac germs). A parabolic germ $f$, analytic on a standard quadratic domain $\mathcal R_C$, that maps the ray $\{\arg(z)=0\}\cap\mathcal R_C$ to itself and satisfies the estimate

$$\begin{equation} \begin{aligned} \, \begin{gathered} \, |f(z)-z+az^{\alpha}\boldsymbol\ell^m|\leqslant c|z^{\alpha}\boldsymbol\ell^{m+1}|, \\ a\ne 0,\quad \alpha>1,\quad m\in\mathbb Z, \quad c>0,\quad z\in\mathcal R_C, \end{gathered} \end{aligned} \end{equation} \tag{ 2.5 }$$

is called a parabolic generalized Dulac germ if on every invariant petal $V_j^{\pm}$ ($j\in\mathbb Z$) with opening $2\pi/(\alpha-1)$ it admits an asymptotic expansion of the form

$$\begin{equation} f(z)=z+\sum_{i=1}^{n} z^{\alpha_i} R_i^{j,\pm}(\boldsymbol\ell)+ o(z^{\alpha_n+\delta_n}), \qquad \delta_n>0, \end{equation} \tag{ 2.6 }$$

for every $n\in\mathbb N$ as $z\to 0$ on $V_j^{\pm}$. Here $\alpha_1=\alpha$, the $\alpha_i>1$ are strictly increasing to $+\infty$ and finitely generated, and the $R_i^{j,\pm}(\boldsymbol\ell)$ are analytic functions on open cusps $\boldsymbol\ell(V_j^{\pm})$ which admit common $\log$ -Gevrey asymptotic expansions ³ $\widehat R_i(\boldsymbol\ell)$ of order strictly greater than $(\alpha-1)/2$ as $\boldsymbol\ell\to 0$:

$$\begin{equation*} \widehat R_i(\boldsymbol\ell)= \sum_{k=N_i}^{\infty}a_k^i \boldsymbol\ell^k,\qquad a_k^i\in\mathbb R,\quad N_i\in\mathbb Z. \end{equation*}$$

We then say that the series $\widehat f\in\widehat{\mathcal L}$ given by

$$\begin{equation*} \widehat f(z):=z+\sum_{i=1}^{\infty}z^{\alpha_i}\widehat R_i(\boldsymbol\ell) \end{equation*}$$

is the generalized Dulac asymptotic expansion of $f$. Such an $\widehat f$ is called a parabolic generalized Dulac series.

We prove in Proposition 8.2 that the parabolic generalized Dulac germs form a group under composition.

The assumption (2.5) is automatically satisfied for parabolic Dulac germs (see Proposition 8.1), since Dulac asymptotic expansions are uniform on the whole of $\mathcal R_C$. For parabolic generalized Dulac germs, on the other hand, we assume the existence of the asymptotic expansions (2.6) only on petals, and the estimates do not necessarily hold uniformly from petal to petal. The additional uniform condition (2.5) in their definition is therefore important to ensure that the rate of decrease of the radii of petals for the dynamics of a parabolic generalized Dulac germ on levels of the Riemann surface of the logarithm is not greater than the rate dictated by the geometry of a standard quadratic domain (see § 8).

We note also that a generalized Dulac asymptotic expansion is an asymptotic expansion in the class of transseries $\widehat {\mathcal L}(\mathbb R)$. As discussed in [17], an asymptotic expansion of a germ in $\widehat{\mathcal L}(\mathbb R)$ is in general not unique and not well defined. A generalized Dulac expansion is a sectional asymptotic expansion that becomes unique after a canonical choice of section functions (summation rules) at limit ordinal steps. See [17] for the precise definition of sectional asymptotic expansions on $\mathbb R$, and for a generalization to complex sectors see Definition 10.1 in the Appendix. The existence of a canonical choice of section functions is here guaranteed by the assumption that the functions $R_i^{j,\pm}(\boldsymbol\ell)$ admit $\log$ -Gevrey power asymptotic expansions of some order on cusps which are the $\boldsymbol\ell$-images of sectors with sufficiently large opening, called $\boldsymbol\ell$ -cusps. Then by Corollary 4.4 (a variation of Watson's lemma), their $\log$ -Gevrey sum on each petal is unique. This $\log$-Gevrey condition is a stronger condition than Gevrey summability of order $m$ [11] for any $m>0$ (see § 4). It ensures unique summability on $\boldsymbol\ell$-cusps, which do not contain sectors of any positive opening. The assumption of $\log$-Gevrey summability for parabolic generalized Dulac germs generalizes the polynomiality assumption for Dulac germs.

Parabolic Dulac germs are trivially parabolic generalized Dulac germs. In that case we have a canonical choice of polynomial sections. Polynomial functions of $\boldsymbol\ell^{-1}$ are convergent Laurent series in the variable $\boldsymbol\ell$.

In [15] for the converse problem of realizing a sequence of diffeomorphisms as analytic moduli of power-logarithmic germs, we obtain parabolic generalized Dulac germs as realizing germs.

Below we suppose in addition that $f(z)=z-az^\alpha\boldsymbol\ell^m+o(z^\alpha\boldsymbol\ell^m)$, where $a>0$, $\alpha>1$, $m\in\mathbb Z$, so that $\{\arg(z)=0\}\cap \mathcal R_C$ is an invariant attracting direction. (In the case $a<0$ we switch to the inverse germ $f^{-1}$, which is by Proposition 8.2 also a parabolic generalized Dulac germ.) By a real homothety $\varphi(z)=a^{-1/(\alpha-1)}z$, $a^{-1/(\alpha-1)}\in\mathbb R$, we transform $f(z)=z-az^\alpha\boldsymbol\ell^m+o(z^\alpha\boldsymbol\ell^m)$ into a normalized germ

$$\begin{equation} f(z)=z-z^\alpha\boldsymbol\ell^m+o(z^\alpha\boldsymbol\ell^m). \end{equation} \tag{ 2.7 }$$

Note that we cannot transform the first coefficient $-1$ to $+1$ by a homothety conjugation, since we would then lose the invariance of the positive real line. Therefore, below we always suppose that we are working with a (parabolic or parabolic generalized) Dulac germ whose first coefficient $a$ is equal to $1$. This first normalization step, performed for simplicity, is not important from the viewpoint of analytic conjugacy of generalized Dulac germs (discussed later), since a homothety is an analytic germ on a standard quadratic domain.

We will work both in the $z$-chart (which we will call the original chart) and in the $(\zeta=-\log z)$-chart (which we will call the logarithmic chart), each time choosing the chart which is more convenient.

Here we are interested in complete invariants of analytic classification, analogous to the Écalle–Voronin moduli for analytic diffeomorphisms. We first define in § 5 what we mean by saying that two parabolic (generalized) Dulac germs $f$ and $g$ are analytically conjugate. In the case of analytic germs $\mathbb C\{z\}$ it was clear that $f,g\in\mathbb C\{z\}$ are analytically equivalent if there exists a germ of an analytic diffeomorphism $\varphi\in\mathbb C\{z\}$ such that

$$\begin{equation*} g=\varphi^{-1}\circ f\circ \varphi. \end{equation*}$$

Obviously, this analytic equivalence implies formal equivalence in the sense of formal Taylor series in the class $\widehat{\mathcal L}_0(\mathbb R)$. In this paper we will stay within the limits of the $\widehat{\mathcal L}(\mathbb R)$-formal classes, and we adopt the following definition of analytic equivalence of parabolic generalized Dulac germs.

Definition 2.4  (analytic conjugacy). We say that two normalized parabolic generalized Dulac germs $f$ and $g$ of the form (2.7) defined on a standard quadratic domain $\mathcal R_C$ are analytically conjugate if

1) $f$ and $g$ are $\widehat{\mathcal L}(\mathbb R)$-formally conjugate, ⁴ and

2) there exists a germ of a diffeomorphism $h(z)=z+o(z)$ of a standard quadratic domain $\mathcal R_C$ such that

$$\begin{equation*} g=h^{-1}\circ f\circ h\quad\text{on }\mathcal R_C. \end{equation*}$$

If we allow non-normalized germs $f$ and $g$, then $h$ must possibly be composed with an additional homothety, so that $h$ is assumed to be of the more general form $h(z)=cz+o(z)$, $c>0$. For simplicity we assume that $f$ and $g$ are normalized and that $h$ is tangent to the identity map. Two non-normalized germs are analytically conjugate if and only if their normalizations are analytically conjugate. Definition 2.4 is analyzed in more detail in § 5.

We prove Proposition 2.5 below in § 7.4. It explains why it is natural to define an analytic conjugacy of two parabolic generalized Dulac germs in Definition 2.4 simply as a bounded analytic germ tangent to the identity map on a standard quadratic domain, without any other asymptotic conditions. Proposition 2.5 states that it is automatically coherent with the formal conjugacy in $\widehat{\mathcal L}^{\rm inv}(\mathbb R)$.

Proposition 2.5.  Let $f$ and $g$ be two normalized parabolic generalized Dulac germs that are analytically conjugate on a standard quadratic domain $\mathcal R_C$ (in the sense of Definition 2.4). Let $h(z)=z+o(z)$, $z\to 0$, $z\in\mathcal R_C$, be their analytic conjugacy. Let $\widehat h\in\widehat {\mathcal L}^{\mathrm{id}}(\mathbb R)$ be their formal conjugacy:

$$\begin{equation*} \widehat g=\widehat h^{-1}\circ \widehat f\circ\widehat h. \end{equation*}$$

Then $h$ admits $\widehat h$ as its generalized block iterated integral asymptotic expansion, ⁵ up to the non-uniqueness of the formal conjugacy $\widehat h$ described after Proposition 6.14.

In § 7.4 we define the sense in which an analytic conjugacy of two parabolic (generalized) Dulac germs has an asymptotic expansion in $\widehat{\mathcal L}(\mathbb R)$. In fact, because we are working with transserial asymptotic expansions, which are not unique, we need to determine the section function (the summation rule) that we use at each limit ordinal step. The conjugacies are no longer parabolic generalized Dulac germs, but more complicated. We introduce integral sums of length $k\in\mathbb N_0$ in Definition 6.1, and the notion of generalized block iterated integral summability in Definitions 6.12 and 7.8. We prove in Proposition 7.9 that any analytic conjugacy of two parabolic (generalized) Dulac germs admits a formal conjugacy in $\widehat{\mathcal {L}}(\mathbb R)$ as its generalized block iterated asymptotic expansion, up to non-uniqueness of the formal conjugacy described after Proposition 6.14.

Theorem THA describes the local dynamics of a parabolic (generalized) Dulac germ on a standard quadratic domain $\mathcal R_C$ (considered on the Riemann surface of the logarithm or in the logarithmic chart of the variable $\zeta=-\log z$ as a ramified neighbourhood of the point $z=0$ or the set $\operatorname{Re}(\zeta)=+\infty$). The local dynamics resembles the well-known flower dynamics for parabolic analytic diffeomorphisms given by the Leau–Fatou flower theorem (see, for instance, [13]), but with countably many tangential attracting and repelling directions situated equidistantly on the Riemann surface of the logarithm.

Theorem THA  (Leau–Fatou theorem for parabolic generalized Dulac germs). Let

$$\begin{equation*} f(z)=z-az^\alpha\boldsymbol\ell^m+o(z^\alpha\boldsymbol\ell^m),\qquad \alpha>1,\quad a>0,\quad m\in\mathbb Z, \end{equation*}$$

be a parabolic generalized Dulac germ on a standard quadratic domain $\mathcal R_C$, $C>0$.

(i) There exist countably many overlapping attracting and repelling open petals ⁶ $V_i^+$ and $V_i^-$, $i\in\mathbb Z$, respectively, for $f$ on $\mathcal R_C$, which are maximal invariant domains for the dynamics of $f$ and $f^{-1}$, respectively. They are symmetric with respect to the complex directions $1^{1/(\alpha-1)}$ and $(-1)^{1/(\alpha-1)}$, respectively, on $\mathcal R_C$ which are tangential directions for the dynamics of $f$ and $f^{-1}$, respectively. The opening angle of each petal is equal to $2\pi/(\alpha-1)$.

(ii) On each open petal $V_i^{\pm}$ there exists a (unique up to an additive constant) holomorphic Fatou coordinate $\Psi_{i}^{\pm}$ which conjugates $f$ to a translation:

$$\begin{equation*} \Psi_i^{\pm}(f(z))-\Psi_i^{\pm}(z)=1,\qquad z\in V_i^{\pm}, \end{equation*}$$

and admits a formal Fatou coordinate $\widehat \Psi\in\widehat{\mathcal L}_2^\infty(\mathbb R)$ as its integral asymptotic expansion ⁷ as $z\to 0$ (up to a constant).

(iii) On each open petal $V_i^{\pm}$ the germ $f$ is analytically conjugate by a germ of the form $\varphi_i^{\pm}(z)=a^{-1/(\alpha-1)}z+o(z)$ to its formal normal form $f_1=\operatorname{Exp}(X_1).\mathrm{id}$ in (2.3). The conjugation on each petal is unique, up to a precomposition $\varphi_i\circ f_c$ with the time-$c$ map ($c\in\mathbb R$) along trajectories of the vector field $X_1$. Its block iterated integral asymptotic expansion ⁸ is, up to precomposition with $\widehat f_c$, equal to the formal conjugacy $\widehat{\varphi}\in\widehat{\mathcal L}^{\rm inv}(\mathbb R)$.

The formal Fatou coordinate $\widehat\Psi$ is a formal solution of the Abel equation for the parabolic generalized Dulac expansion $\widehat f$ of $f$:

$$\begin{equation*} \widehat{\Psi}(\widehat f)-\widehat\Psi=1. \end{equation*}$$

The proof of Theorem THA is given in § 8. As was done in [17] for the Fatou coordinate of a parabolic Dulac germ, we prove in § 8 that the formal Fatou coordinate of a parabolic generalized Dulac germ is unique (up to a constant) in the class $\widehat{\mathfrak L}(\mathbb R)$ and moreover belongs to the class $\widehat{\mathcal L}_2^\infty(\mathbb R)$.

In Proposition 2.2 we prove the statement of Theorem THA, (i), under weaker conditions on the germ $f$, without assuming the full generalized Dulac asymptotic expansion, but just the first part of it. However, the proof of the existence of the petalwise analytic Fatou coordinate (and then also of the sectorial conjugacy) in § 8 is based on the existence of the transasymptotic expansion of a generalized Dulac germ $f$ at least up to the residual term of the form $z^{2\alpha-1}\boldsymbol\ell^{2m+1}$. This enables us to deduce the infinite part of the Fatou coordinate in the proof of Theorem THA, (ii), and then prove the convergence and analyticity of the infinitesimal remainder using a modified Abel equation.

The local dynamics of a parabolic generalized Dulac germ $f$ is illustrated on a standard quadratic domain in the logarithmic chart in Fig. 2.

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In Theorem THB we describe the complete system of analytic invariants for a normalized parabolic generalized Dulac germ $f$, with respect to analytic conjugations $\varphi(z)=z+o(z)$, $z\to 0$, on a standard quadratic domain. This result is similar to the result of Écalle and Voronin [22] for parabolic analytic germs of diffeomorphisms, where the analytic class was given by the formal class and a finite number of analytic diffeomorphisms (analytic moduli), after appropriate identifications (see [22] or [13]).

Let $f$ be a normalized parabolic generalized Dulac germ defined on a standard quadratic domain. We suppose for simplicity that $\alpha=2$. This can always be done without loss of generality, since we can apply an analytic change of variables $h(z)=(\alpha-1)^{-m/(\alpha-1)}z^{1/(\alpha-1)}$ carrying one standard quadratic domain to another (see Proposition 9.1 for more details). It is just more convenient, since every level of the surface of the logarithm contains only one petal of opening $2\pi$. In this situation it is easier to express the decrease of the size of the domain of definition of the analytic moduli (see (3.1) below). The case $\alpha>1$, $\alpha\ne 2$, can be analyzed in the same way, but then a finite number of moduli that correspond to one level of the Riemann surface of the logarithm ($\lfloor2\pi/\alpha\rfloor$ of them) have the same radius.

Let

$$\begin{equation*} f(z)=z-z^2\boldsymbol\ell^m+o(z^2\boldsymbol\ell^m),\qquad m\in\mathbb Z,\quad z\in\mathcal{R}_C, \end{equation*}$$

be a parabolic generalized Dulac germ. Let $(\Psi_{\pm}^i)_{i\in\mathbb Z}$ be its analytic Fatou coordinates on attracting or repelling petals $V_{\pm}^i$ (of opening $2\pi$) along the domain. We prove in § 9 that there exists a symmetric (with respect to the $\{\arg(z)=0\}$-axis) sequence $(h_{0,\infty}^i)_{i\in\mathbb Z}$ of analytic germs of diffeomorphisms called horn maps for $f$ such that

$$\begin{equation*} \begin{aligned} \, h_0^i(t)&:=\exp\biggl\{-2\pi \mathrm{i}\Psi_{i-1}^+\circ(\Psi_{i}^{-})^{-1} \biggl(-\frac{\log t}{2\pi \mathrm{i}}\biggr)\biggr\}, \\ h_\infty^i(t)&:=\exp\biggl\{2\pi \mathrm{i} \Psi_i^-\circ (\Psi_i^{+})^{-1} \biggl(\frac{\log t}{2\pi \mathrm{i}}\biggr)\biggr\}, \end{aligned}\qquad t\in(\mathbb C,0),\quad i\in\mathbb Z, \end{equation*}$$

and that this sequence and the formal class $(2,m,\rho)$ are a complete system of analytic invariants of the parabolic generalized Dulac germ $f$.

Due to the standard quadratic domain of definition of $f$, the radii $R_i$ of the domains of the sequence of its horn maps are bounded from below:

$$\begin{equation} R_i\geqslant K_1\exp\bigl\{-Ke^{C\sqrt{|i|}}\bigr\},\qquad K_1,K,C>0,\quad i\in\mathbb Z. \end{equation} \tag{ 3.1 }$$

Also, the converse holds: if all the horn maps are defined at least on discs of radii given by (3.1), then all the sectorial Fatou coordinates glue together to form an analytic map at least on a standard quadratic domain.

Theorem THB  (analytic invariants of a parabolic generalized Dulac germ). Let $f$ and $g$ be two normalized parabolic generalized Dulac germs that belong to the same $\widehat{\mathcal L}(\mathbb R)$-formal class $(2,m,\rho)$, $m\in\mathbb Z$, $\rho\in\mathbb R$. Let

$$\begin{equation*} (h_{0}^i,h_{\infty}^i;R_i^f)_{i\in\mathbb Z}\quad\textit{and}\quad (k_{0}^i,k_{\infty}^i;R_i^g)_{i\in\mathbb Z} \end{equation*}$$

be their sequences of horn maps with radii of convergence $(R_i^f)_i$ and $(R_i^g)_i$, respectively, where $R_i^f$ and $R_i^g$ satisfy estimates of the type (3.1). Then $f$ and $g$ are analytically conjugate (by an analytic change of variables that is tangent to the identity) on some standard quadratic domain if and only if there exist sequences $(a_i)_i,(b_i)_i\in\mathbb C^*$ such that

$$\begin{equation} h_0^i(t)=a_{i-1}\cdot k_0^i(b_i t),\quad h_\infty^i(t)=b_i\cdot k_\infty^i(a_{i} t),\qquad t\in(\mathbb C,0),\quad i\in\mathbb Z. \end{equation} \tag{ 3.2 }$$

More details and the proof of Theorem THB are given in § 9. We also show in Proposition 9.2 that the horn maps of a parabolic generalized Dulac germ in an $\mathcal L(\mathbb R)$-formal class $(2,m,\rho)$ are symmetric with respect to the real axis, because the coefficients are real. That is,

$$\begin{equation*} (h_{0}^{-i+1})^{-1}(t)\equiv \overline{h_{\infty}^i(\,\overline t\,)},\qquad i\in\mathbb Z, \end{equation*}$$

on their domains of definition.

Note that the assumption that the germs are normalized as in (2.7) is just convenient for simplicity, but is not very important. In fact, two parabolic generalized Dulac germs are analytically conjugate by a change of variables $\varphi(z)=bz+o(z)$, $z\to 0$, $b>0$, on a standard quadratic domain if and only if their normalizations are analytically conjugate by a change of variables tangent to the identity on a standard quadratic domain. Indeed, we only need one additional homothety that is analytic on a standard quadratic domain. Therefore, it suffices to consider the moduli and the question of analytic conjugacy only for normalized generalized Dulac germs. That is, to consider conjugacies that are tangent to the identity maps.

Here we define the $\log$-Gevrey classes of germs and formal series used in Definition 2.3 of parabolic generalized Dulac germs. We also state and prove some of their properties: closedness with respect to addition, almost closedness with respect to multiplication and differentiation. That is needed for describing Fatou coordinates and sectorial analytic reductions to normal form for parabolic generalized Dulac germs in § 7, and also in [15]. The proofs are somewhat similar to the proofs in [1], [11] for classical Gevrey function classes which are differential algebras.

Below we define an $\boldsymbol\ell$ -cusp to be an open cusp that is the image of an open sector $V$ of positive opening at $0$ under the change of variables $\boldsymbol\ell=-1/\log z$, and we denote it by $S=\boldsymbol\ell(V)$ (see Fig. 3). Any open $\boldsymbol\ell$-cusp $\boldsymbol\ell(V')\subset S$, where $V'\subset V$ is a proper subsector, will be called a proper $\boldsymbol\ell$ -subcusp of $S$.

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It will be shown below in Corollary 4.8 that asymptotic series on such $\boldsymbol\ell$-cusps inherit the property of termwise differentiation from power asymptotic series on the corresponding sectors of positive opening (a property that cannot be claimed for $\mathbb R_+$ or for general cusps).

Definition 4.1  ($\log$-Gevrey asymptotic expansions on $\boldsymbol\ell$-cusps). Let $F$ be a germ analytic on an $\boldsymbol\ell$-cusp $S=\boldsymbol\ell(V)$. We say that $F$ has the series $\widehat F(\boldsymbol\ell)=\sum_{k=0}^{\infty} a_k \boldsymbol\ell^k$, $a_k\in\mathbb C$, as its $\log$ -Gevrey asymptotic expansion of order $m>0$ if, for every proper $\boldsymbol\ell$-subcusp $S'=\boldsymbol\ell(V')\subset S$, $V'\subset V$, there exists a constant $C_{S'}>0$ such that for every $n\in\mathbb N$ with $n\geqslant 2$ we have

$$\begin{equation} \biggl|F(\boldsymbol\ell)-\sum_{k=0}^{n-1}a_k \boldsymbol\ell^k\biggr| \leqslant C_{S'} m^{-n}(\log n)^n \exp\biggl\{-\frac{n}{\log n}\biggr\} |\boldsymbol\ell|^n,\qquad \boldsymbol\ell\in S'. \end{equation} \tag{ 4.1 }$$

By $\operatorname{LG}_m(S)\subseteq\mathcal O(S)$ with $S$ an $\boldsymbol\ell$-cusp and $m>0$ we denote the set of all germs analytic on $S$ that have a $\log$-Gevrey asymptotic expansion of order $m>0$ on $S$ as $\boldsymbol\ell\to 0$. By $\widehat{\operatorname{LG}}_m(S)\subseteq \mathbb C[[\boldsymbol\ell]]$ we denote the set of their $\log$-Gevrey asymptotic expansions of order $m>0$ on $S$.

For the definition of generalized Dulac germs, we will need the following generalization of Definition 4.1.

Definition 4.2  (generalization of Definition 4.1 to formal Laurent expansions). We say that a function $F$ analytic on an $\boldsymbol\ell$-cusp $S$ has a formal Laurent series $\widehat F(\boldsymbol\ell)\in\mathbb C((\boldsymbol\ell))$, $\widehat F(\boldsymbol\ell)=\sum_{k=N}^{\infty}a_k\boldsymbol\ell^k$, $N\in\mathbb Z$, as its $\log$-Gevrey expansion of order $m>0$ if $F(\boldsymbol\ell)\boldsymbol\ell^{-k}$ has $\widehat F(\boldsymbol\ell)\boldsymbol\ell^{-k}$ as its $\log$-Gevrey expansion of order $m>0$ on $S$ in the sense of the above definition. ⁹ We use the same notation $\operatorname{LG}_m(S)$ for the set of all such germs, and $\widehat{\operatorname{LG}}_m(S)\subseteq \mathbb C((\mathbb \ell))$ for the set of their $\log$-Gevrey asymptotic expansions of order $m>0$.

Recall that we say that $F(z)$ admits $\widehat F(z)$ as its Gevrey asymptotic expansion of order $k>0$ on a sector $V$ if, for every proper subsector $V'\subset V$, there exist constants $C,D>0$ such that for all $n\in\mathbb N$

$$\begin{equation*} \biggl|F(z)-\sum_{k=0}^{n-1}a_k z^k\biggr|\leqslant C D^n (n!)^{1/k} |z|^n,\qquad z\in V'. \end{equation*}$$

We can now check using (4.1) and Stirling's formula $\sqrt{n}\,(n/e)^n\sim n!$, $n\to\infty$, that a $\log$-Gevrey asymptotic expansion of some order $m>0$ is also a Gevrey asymptotic expansion of every order ¹⁰ $k>0$, as introduced in [11], [1], and [12].

In this paper we have to consider functions on $\boldsymbol\ell$-cusps that do not contain any sectors of positive opening. This is why we need a stronger, $\log$-Gevrey, growth estimate in order to apply a Watson-type lemma. Recall that, classically, if $F$ admits a zero Gevrey asymptotic expansion of order $k>0$ on $V$, then, for every proper subsector $V'\subset V$, there exist constants $C>0$ and $M>0$ such that

$$\begin{equation*} |F(z)|\leqslant C\exp\biggl\{-\frac{M}{|z|^{k}}\biggr\},\qquad z\in V'. \end{equation*}$$

Also, the classical Watson's lemma guarantees that if $F$ is an analytic function with a zero Gevrey asymptotic expansion of order $k>0$ on a sector of opening strictly greater than $\pi/k$, then $F$ is necessarily equal to zero on this sector.

Proposition 4.3.  Suppose that $F$ admits a zero $\log$-Gevrey asymptotic expansion of order $m>0$ on an $\boldsymbol\ell$-cusp $S=\boldsymbol\ell(V)$. Then for every $\boldsymbol\ell$-subcusp $S'\subset S$ and every $\delta>0$, there exist constants $C,M>0$ such that

$$\begin{equation} |F(\boldsymbol\ell)|\leqslant C\exp\{-Me^{(m-\delta)/|\boldsymbol\ell|}\},\qquad \boldsymbol\ell\in S'. \end{equation} \tag{ 4.2 }$$

Proof.  Let $S'\subset S$. Putting $|\boldsymbol\ell|=a$ and considering a zero expansion of the form (4.1), we get that

$$\begin{equation} |F(\boldsymbol\ell)|\leqslant C m^{-n}(\log n)^n \exp\biggl\{-\frac{n}{\log n}\biggr\} a^n,\qquad |\boldsymbol\ell|=a, \quad n\in\mathbb N,\quad n\geqslant 2. \end{equation} \tag{ 4.3 }$$

We maximize the right-hand side by $n=n(a)$ (take the logarithm, differentiate with respect to $n$ and find the critical point $n_a$). Then $n_a\sim e^{m/a}$ and $\log n_a\sim m/a$ as $a\to 0$. From (4.3) we get that there exists a $C>0$ such that

$$\begin{equation*} |F(\boldsymbol\ell)|\leqslant C\exp\biggl\{-\frac{1}{m}\,ae^{m/a}\biggr\},\qquad \boldsymbol\ell\in S'. \end{equation*}$$

In other words, returning to $a=|\boldsymbol\ell|$, we get that

$$\begin{equation*} |F(\boldsymbol\ell)|\leqslant C\exp\biggl\{-\frac{1}{m}|\boldsymbol\ell| e^{m/|\boldsymbol\ell|}\biggr\},\qquad \boldsymbol\ell\in S'.\qquad\square \end{equation*}$$

Corollary 4.4  (variant of Watson's lemma for $\log$-Gevrey expansions). Let $m>0$, and let $F(\boldsymbol\ell)$ be analytic on the $\boldsymbol\ell$-cusp $S_m=\boldsymbol\ell(V_m)$, where $V_m$ is a sector of opening strictly greater than $\pi/m$. Let $F$ have a zero $\log$-Gevrey asymptotic expansion of order $m$ on $S_m$. Then $F$ is equal to zero on $S_m$.

Proof.  Take any $\boldsymbol\ell$-subcusp $S'=\boldsymbol\ell(V')\subset S_m$, where $V'$ is any subsector of $V_m$ of opening strictly less than the opening of $V_m$. By (4.2), since $\widehat F(\boldsymbol\ell)=0$, we get that for any $\delta>0$ there exist constants $C,M>0$ such that

$$\begin{equation*} |F(\boldsymbol\ell)|\leqslant C\exp\{-Me^{(m-\delta)/|\boldsymbol\ell|}\},\qquad \boldsymbol\ell\in S'. \end{equation*}$$

Passing to the variable $z=e^{-1/\boldsymbol\ell}$ and letting $\widetilde F(z):=F(\boldsymbol\ell)$, we get that (since $|\log z|\geqslant -\log |z|$, $z\in\mathbb C$)

$$\begin{equation*} |\widetilde F(z)|\leqslant C\exp\biggl\{-\frac{M}{|z|^{m-\delta}}\biggr\},\qquad z\in V'\subset V_m. \end{equation*}$$

Now take $V'$ of opening strictly greater than $\pi/m$, and take $\delta>0$ small enough that $V'$ has opening strictly greater than $\pi/(m-\delta)$. By the classical Watson's lemma (see [1]), $\widetilde F\equiv 0$ on $V'$. Therefore, $\widetilde F\equiv 0$ on the whole of $V_m$ by the uniqueness of the analytic extension. $\square$

Unlike $m$-Gevrey classes (see, for example, [11] or [18] for the $1$-Gevrey class), the $\log$- Gevrey classes $\operatorname{LG}_m(S)$ with $m>0$ and $S$ an $\boldsymbol\ell$-cusp in Definition 4.1 are not differential algebras. However, we prove a weaker statement about closedness with respect to the operations $+$, $\,\cdot\,$, and $d/d\boldsymbol\ell$ in Propositions 4.5–4.7.

Proposition 4.5  (addition in $\log$-Gevrey classes). Let $m>0$ and let $S=\boldsymbol\ell(V)$ be an $\boldsymbol\ell$-cusp such that the opening of the sector $V$ is strictly greater than $\pi/m$. Then the sets $\operatorname{LG}_m(S)$ and $\widehat{\operatorname{LG}}_m(S)$, are groups under the operation $+$. The map $f\mapsto\widehat f$ associating the series $\widehat f\in \widehat{\operatorname{LG}}_m(S)$ with a function $f\in \operatorname{LG}_m(S)$ is an isomorphism of these groups with respect to the operation $+$.

Proof.  The uniqueness of the $\log$-Gevrey asymptotic expansion is obvious. The injectivity follows by Corollary 4.4. The homomorphism property is obvious by the definition of $\log$-Gevrey asymptotic expansions and the formula (4.1). $\square$

Proposition 4.6  (multiplication in $\log$-Gevrey classes). Let $m,n>0$, let $S=\boldsymbol\ell(V)$ be an $\boldsymbol\ell$-cusp, and let $f\in \operatorname{LG}_m(S)$ and $g\in \operatorname{LG}_n(S)$, with asymptotic expansions $\widehat f\in\widehat{\operatorname{LG}}_m(S)$ and $\widehat g\in\widehat{\operatorname{LG}}_n(S)$. Let $s:=\min\{m,n\}$. Then $fg\in \operatorname{LG}_r(S)$ for every $0<r<s$. Moreover, $fg$ has the formal product $\widehat f\cdot \widehat g$ as its $\log$-Gevrey asymptotic expansion of order $r$ on $S$.

The proof is in the Appendix (see § 10).

Proposition 4.7  (differentiation in $\log$-Gevrey classes). Let $f\in \operatorname{LG}_m(S)$ have the series $\widehat f(\boldsymbol\ell)$ as a $\log$-Gevrey asymptotic expansion of order $m$ on an $\boldsymbol\ell$-cusp $S=\boldsymbol\ell(V)$. Then for every $0<r<m$ all the derivatives $f^{(k)}$, $k\in\mathbb N$, belong to $\operatorname{LG}_r(S)$ and have formal derivatives $\widehat f^{(k)}(\boldsymbol\ell)$ as their $\log$-Gevrey asymptotic expansions of order $r$ on $S$.

The proof is in the Appendix (§ 10).

In the course of proving Proposition 4.7 we also prove the following corollary. The Cauchy formula in the variable $z$ is used here as in the proof of Proposition 4.7.

Corollary 4.8  (termwise differentiation of power asymptotic expansions on $\boldsymbol\ell$-cusps). The power asymptotic expansions of the form $f(\boldsymbol\ell)\sim \sum_{n=0}^{\infty} a_n \boldsymbol\ell^n$, $\boldsymbol\ell\in S$, $\boldsymbol\ell\to 0$, on $\boldsymbol\ell$-cusps $S:=\boldsymbol\ell(V)$, where $V$ is a sector of positive opening at $0$, can be differentiated term-by-term. That is,

$$\begin{equation*} \frac {d}{d\boldsymbol\ell}f(\boldsymbol\ell)\sim \sum_{n=1}^{\infty} n a_n \boldsymbol\ell^{n-1},\qquad \boldsymbol\ell\in S,\quad \boldsymbol\ell\to 0. \end{equation*}$$

More precisely, we say here that $f$ has an asymptotic expansion $\widehat f(\boldsymbol\ell)\,{=}\sum_{n=0}^{\infty} a_n \boldsymbol\ell^n$ on an $\boldsymbol\ell$-cusp $S=\boldsymbol\ell(V)$ if, for every proper $\boldsymbol\ell$-subcusp $\boldsymbol\ell(V')\subset S$, where $V'\subset V$ is a proper subsector, there exists a constant $C_n>0$ such that

$$\begin{equation*} \biggl|f(\boldsymbol\ell)-\sum_{k=0}^{n-1}a_k\boldsymbol\ell^{k}\biggr| \leqslant C_n|\boldsymbol\ell|^n,\qquad \boldsymbol\ell\in S',\quad n\in\mathbb N. \end{equation*}$$

We remark that in the classical situation, by the Cauchy formula we need a sector of positive opening to be able to differentiate asymptotic expansions term-by-term (see, for example, [23]). Here this property is inherited due to the special form of $\boldsymbol\ell$-cusps, which are $\boldsymbol\ell$-images of sectors.

Proposition 4.9  (compositions with analytic germs for $\log$-Gevrey classes). Let $S=\boldsymbol\ell(V)$ be an $\boldsymbol\ell$-cusp. ¹¹ Let $f,h\in \operatorname{LG}_m(S)$ and $g\in \operatorname{LG}_p(S)$, $m,p>0$, have $\log$-Gevrey expansions $\widehat f$, $\widehat h\in\widehat{\operatorname{LG}}_m(S)$ and $\widehat g\in\widehat{\operatorname{LG}}_p(S)$. Assume that $f(\boldsymbol\ell)=o(1)$ as $\boldsymbol\ell\to 0$ on $V$. Let $F$ be an analytic or meromorphic germ at $0$, with Taylor expansion or Laurent expansion $\widehat F$ at $0$. Then:

(i) $F\circ f\in \operatorname{LG}_r(S)$ for every $0<r<m$, with $\log$-Gevrey expansion $\widehat F\circ \widehat f\in\widehat{\operatorname{LG}}_r(S)$ of order $r$;

(ii) if $\widehat g\ne 0$, then ¹² $h/g\in \operatorname{LG}_q(S)$ for every $0<q<\min\{m,p\}$, and $\widehat h/\widehat g\in \widehat{\operatorname{LG}}_q(S)$ is its $\log$-Gevrey asymptotic expansion of order $q$.

The proof is in the Appendix (§ 10).

In the case of regular parabolic diffeomorphisms, analytic equivalence implies formal equivalence. For parabolic generalized Dulac germs $f$ defined on a standard quadratic domain $\mathcal R_C$ the situation is more complicated. We will see that for two parabolic (generalized) Dulac germs on $\mathcal R_C$ an analytic conjugacy tangent to the identity map does not necessarily imply that the asymptotic expansion of the conjugacy germ is in $\widehat{\mathcal L}(\mathbb R)$. However, this will be true in $\widehat{\mathcal L}_2(\mathbb R)$ (see Lemma 7.5). The reason is the possibility of formal reduction of $\widehat f(z)$ to a simpler normal form by a formal conjugacy belonging to a broader class $\widehat{\mathcal L}_2(\mathbb R)\supset \widehat{\mathcal L}(\mathbb R)$, which allows us also to eliminate the residual term, as in Remark 5.5 below.

However, in this paper when we define the analytic equivalence of two parabolic generalized Dulac germs, we stay within the limits of the $\widehat{\mathcal L}(\mathbb R)$-formal classes. The analytic equivalence of two parabolic generalized Dulac germs is defined in Definition 2.4 of § 2.

In the following example we show that in Definition 2.4 the condition 1) does not follow from 2). Indeed, if only 2) is satisfied, then $h$ has a generalized block iterated integral expansion in the larger class $\widehat{\mathcal L}_2^{\,\mathrm{id}}(\mathbb R)$, a subgroup of the class $\widehat{\mathcal L}_2(\mathbb R)$ of transseries tangent to the identity map.

Example 5.1.  There exist parabolic generalized Dulac germs that satisfy 2) in Definition 2.4 but are not formally equivalent in the class $\widehat {\mathcal L}(\mathbb R)$. They are in fact formally equivalent in $\widehat{\mathcal L}_2(\mathbb R)$. This broader class can be used to eliminate the residual invariant term (see Proposition 5.2 below). For example, take two parabolic generalized Dulac germs that are models for two different formal classes with respect to $\widehat{\mathcal L}(\mathbb R)$ and give the time-one maps along trajectories of the two vector fields:

$$\begin{equation*} f(z)=\operatorname{Exp}\biggl(\frac{z^2}{1+z-z\boldsymbol\ell}\, \frac{d}{dz}\biggr).\mathrm{id}\quad\text{and}\quad g(z)=\operatorname{Exp}\biggl(\frac{z^2}{1+z}\, \frac{d}{dz}\biggr).\mathrm{id}. \end{equation*}$$

Both are obviously parabolic Dulac germs, defined on a whole neighbourhood of zero of the Riemann surface $\mathcal R$ of the logarithm, and polynomial in $\boldsymbol\ell$. They both admit global Fatou coordinates $\Psi_f,\Psi_g\colon\mathcal R\to\widetilde{\mathcal R}$, where $\widetilde{\mathcal R}$ denotes a neighbourhood of the infinity on the Riemann surface of the logarithm. The Fatou coordinates are, up to a complex additive constant, given by

$$\begin{equation*} \Psi_f(z)=\int \frac{dz}{z^2}+\int\frac{dz}{z}- \int\frac{\boldsymbol\ell}{z}\,dz=-\frac{1}{z}+\log z+\log(-\log z) \end{equation*}$$

and

$$\begin{equation*} \Psi_g(z)=-\frac{1}{z}+\log z,\qquad z\in\mathcal R. \end{equation*}$$

The function $\Psi_g$ is injective on $\mathcal R$. By the Abel equation for the Fatou coordinate, one analytic conjugacy conjugating $g$ to $f$, defined and analytic on $\mathcal R$ and tangent to the identity, is

$$\begin{equation*} \varphi:=\Psi_g^{-1}\circ \Psi_f. \end{equation*}$$

On the other hand, $f$ and $g$ have different formal invariants $\rho$ in the class $\widehat{\mathcal L}(\mathbb R)$. Indeed, the asymptotic expansion $\widehat\varphi$ of $\varphi$ belongs to the class $\widehat{\mathcal L}_2(\mathbb R)$, as is easily seen, for example, from the Taylor expansion of the left-hand side of the equality $\Psi_g(x+h(x))=\Psi_f$, $\varphi=\mathrm{id}+h$, or by writing $\widetilde\Psi_f=-e^{1/\boldsymbol\ell}-1/\boldsymbol\ell- \log\boldsymbol\ell$ and $\widetilde\Psi_g=-e^{1/\boldsymbol\ell}- 1/\boldsymbol\ell$ in terms of the variable $\boldsymbol\ell$. Here it is obvious that we cannot generate the logarithmic term in $\widetilde\Psi_f$ by precomposing $\widetilde\Psi_g$ with a power-exponential transseries.

Let $\widehat{h}\in\widehat{\mathcal L}(\mathbb R)$ denote the formal conjugacy between the generalized Dulac expansions $\widehat f$ and $\widehat g$ of two analytically conjugate (in the sense of Definition 2.4) parabolic generalized Dulac germs $f$ and $g$:

$$\begin{equation*} \widehat g=\widehat h^{-1}\circ \widehat f\circ \widehat h,\qquad \widehat h\in\widehat{\mathcal L}(\mathbb R). \end{equation*}$$

The formal conjugacy $\widehat h\in \widehat{\mathcal L}(\mathbb R)$ is derived in [16]. We will show in § 7.4 that, up to some completely determined changes, the $h$ in Definition 2.4 then admits the formal conjugacy $\widehat h \in\widehat{\mathcal L}(\mathbb R)$ as its generalized block iterated integral sectional asymptotic expansion (see Definition 7.8 and Proposition 7.9). In fact, the transserial asymptotic expansions of germs in $\widehat{\mathfrak L}(\mathbb R)$ are not unique (see [17]). To make them unique we must choose a canonical summation rule at limit ordinal steps. That is, we must fix an appropriate section function (see [17], and Definition 10.1 for a complex version).

We repeat here the $\widehat{\mathcal L}(\mathbb R)$-normal form result derived in [16] for parabolic generalized Dulac germs. We give here an alternative proof working block-by-block, not term-by-term as in [16]. As opposed to termwise eliminations done previously in [16], the value of blockwise eliminations is that they determine the form of integrals in each block. This will be important in finding the appropriate integral asymptotic expansions for conjugacies in § 7.4. The proof here is inductive, by block-by-block eliminations. By a block, we mean all monomials in a transseries that have the same powers of $z$. In the proof we need Lemma 5.3 below.

Proposition 5.2  (formal normal forms of parabolic generalized Dulac transseries; see [16]). Let $\widehat f(z)=z-z^\alpha\boldsymbol\ell^{m}+\mathrm{h.o.t.}$, ¹³ $\alpha>1$, $m\in\mathbb Z$, be a normalized parabolic generalized Dulac transseries. By a formal series $\widehat\varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ tangent to the identity map it can be reduced to either of the normal forms

$$\begin{equation} \begin{aligned} \, \widehat f_F(z)&=z-z^\alpha\boldsymbol\ell^{m}+ \rho z^{2\alpha-1}\boldsymbol\ell^{2m+1}, \nonumber \\ \widehat f_1(z)&=\operatorname{Exp}(X_1).\mathrm{id},\quad X_1(z)=\frac{-z^\alpha\boldsymbol\ell^{m}} {1+(-\alpha/2)z^{\alpha-1}\boldsymbol\ell^{m}+(m/2+\rho)z^{\alpha-1} \boldsymbol\ell^{m+1}}\,\frac{d}{dz}\,. \end{aligned} \end{equation} \tag{ 5.1 }$$

The $\widehat{\mathcal L}(\mathbb R)$-formal invariants have the form $(\alpha,m,\rho)$, $\alpha>1$, $m\in\mathbb Z$, $\rho\in\mathbb R$.

Clearly, since $\widehat{\mathcal L}^{\rm inv}(\mathbb R)$ is a group under composition, any two parabolic generalized Dulac series $\widehat f$ and $\widehat g$ with the same formal invariants $(\alpha,m,\rho)$ are formally conjugate in $\widehat{\mathcal L}^{\rm inv}(\mathbb R)$.

Lemma 5.3  (blockwise eliminations). Let $\widehat f\in \widehat{\mathcal L}(\mathbb R)$, $\widehat f(z)=z-z^\alpha\boldsymbol\ell^{m}+z^{\beta_i}\widehat T_i(\boldsymbol\ell)+\mathrm{h.o.b.}$, ¹⁴ where $\alpha>1$, $m\in\mathbb Z$, $\widehat T_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$. The block $z^{\beta_i}\widehat T_i(\boldsymbol\ell)$, with the possible exception of the monomial $z^{2\alpha-1}\boldsymbol\ell^{2m+1}$ in the residual block $\beta_i=2\alpha+1$, can be eliminated by the elementary change of variables $\widehat\varphi_i(z)=z+z^{\gamma_i}\widehat{R}_i(\boldsymbol\ell)$, where $\gamma_i=\beta_i-\alpha+1$, and the series $\widehat R_i(\boldsymbol\ell)\in \mathbb R((\boldsymbol\ell))$ are given by

$$\begin{equation} \begin{cases} \widehat R_i(\boldsymbol\ell)= -\dfrac{1}{z^{\gamma_i-\alpha}\boldsymbol\ell^{-m}} \displaystyle\int z^{\gamma_i-\alpha}\boldsymbol\ell^{-2m-2} \widehat T_i(\boldsymbol\ell)\,d\boldsymbol\ell,& \beta_i>\alpha, \\ \widehat R_i(\boldsymbol\ell)=\dfrac{1}{z} \biggl[\biggl(\displaystyle\int\dfrac{z^{1-\alpha}}{\boldsymbol\ell^{m+2}} \,d\boldsymbol\ell\bigr)^{-1}\circ \biggl(\displaystyle\int\dfrac{z^{1-\alpha}} {\widehat T_0(\boldsymbol\ell)\boldsymbol\ell^2}\, d\boldsymbol\ell\biggr)\biggr]-1,& \beta_i=\alpha, \end{cases} \end{equation} \tag{ 5.2 }$$

with $0$ as the constant of formal integration. Here $\widehat T_0(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$, $\widehat T_0(\boldsymbol\ell)=-\boldsymbol\ell^m+\mathrm{h.o.t.}$, is the whole first block of $\widehat f(z)$, the coefficient of $z^\alpha$. In the case $\beta_i=2\alpha+1$, $\widehat T_i(\boldsymbol\ell)$ is the residual block without the term $\rho z^{2\alpha-1}\boldsymbol\ell^{2m+1}$, $\rho\in\mathbb Z$.

The above integrals are formally integrated by parts, with $dv=e^{(\alpha-\gamma_i)/\boldsymbol\ell}\boldsymbol\ell^{-2}$ and $u$ equal to the rest of the integrand. We always choose $0$ as the constant of formal integration in (5.2), in order that $\widehat \varphi_i(z)$ be an elementary change of variables with only one block. Note that by taking $\beta_1=\alpha$ we eliminate the first block, except for the first term in it. The proof of Lemma 5.3 is in the Appendix (§ 10).

We also state the following generalization of Lemma 5.3, which is proved in the same way as Lemma 5.3, so we omit the proof. We will need Lemma 5.4 in § 6.2 to prove the block iterated integral summability of formal conjugacies of two parabolic generalized Dulac germs.

Lemma 5.4  (formal conjugation of two transseries). Assume that $\widehat f,\widehat g\in \widehat{\mathcal L}(\mathbb R)$ have the same formal invariants $(\alpha,m,\rho)$. For $n\in\mathbb N_0$ let

$$\begin{equation*} \begin{aligned} \, \widehat f(z)&=z-z^\alpha\boldsymbol\ell^{m}+ \sum_{i=1}^{n} z^{\alpha_i} \widehat T_i(\boldsymbol\ell)+ z^{\alpha_{n+1}}\widehat F_{n+1}(\boldsymbol\ell)+\mathrm{h.o.b.}, \\ \widehat g(z)&=z-z^\alpha\boldsymbol\ell^{m}+ \sum_{i=1}^{n} z^{\alpha_i} \widehat T_i(\boldsymbol\ell)+ z^{\alpha_{n+1}}\widehat G_{n+1}(\boldsymbol\ell)+\mathrm{h.o.b.}, \end{aligned} \end{equation*}$$

where $\alpha>1$, $m\in\mathbb Z$, the numbers $\alpha_i\geqslant \alpha$ are strictly increasing as $i\to\infty$, $\widehat T_i\in\mathbb R((\boldsymbol\ell))$, $i=1,\dots,n$, $\widehat F_{n+1},\widehat G_{n+1}\in\mathbb R((\boldsymbol\ell))$, and $\widehat F_{n+1}\ne 0$ $(\widehat G_{n+1}$ can be $0)$. Then

$$\begin{equation*} \widehat\varphi_{n+1}\circ \widehat g\circ \widehat\varphi_{n+1}^{-1}= z-z^\alpha\boldsymbol\ell^{m}+ \sum_{i=1}^{n} z^{\alpha_i}\widehat T_i(\boldsymbol\ell)+ z^{\alpha_{n+1}}\widehat F_{n+1}(\boldsymbol\ell)+\mathrm{h.o.b.}, \end{equation*}$$

where $\widehat{\varphi}_{n+1}(z)=z+z^{\gamma_{n+1}}\widehat R_{n+1}(\boldsymbol\ell)$, with $\gamma_{n+1}=\alpha_{n+1}-\alpha+1$ and the series $\widehat R_{n+1}(\boldsymbol\ell)\in \mathbb R((\boldsymbol\ell))$ given by

$$\begin{equation} \begin{cases} \widehat R_{n+1}(\boldsymbol\ell)= -\dfrac{1}{z^{\gamma_{n+1}-\alpha}\boldsymbol\ell^{-m}} \displaystyle\int z^{\gamma_{n+1}-\alpha}\boldsymbol\ell^{-2m-2} \bigl(\widehat G_{n+1}(\boldsymbol\ell)- \widehat F_{n+1}(\boldsymbol\ell)\bigr)\,d\boldsymbol\ell \\ \textit{if } \alpha_{n+1}>\alpha, \\ \widehat R_{n+1}(\boldsymbol\ell)= \dfrac{1}{z}\biggl[\biggl(\displaystyle\int\dfrac{z^{1-\alpha}} {\widehat F_1(\boldsymbol\ell)\boldsymbol\ell^2}\,d\boldsymbol\ell\biggr)^{-1} \circ \biggl(\displaystyle\int\dfrac{z^{1-\alpha}} {\widehat G_1(\boldsymbol\ell)\boldsymbol\ell^2}\, d\boldsymbol\ell\biggr)\biggr]-1 \\ \textit{if } \alpha_{n+1}=\alpha, \end{cases} \end{equation} \tag{ 5.3 }$$

and with $0$ as the constant of formal integration. Here $\widehat F_1(\boldsymbol\ell)=-\boldsymbol\ell^m+\mathrm{h.o.t.}$ and $\widehat G_1(\boldsymbol\ell)=-\boldsymbol\ell^m+\mathrm{h.o.t.}$, $\widehat F_1,\widehat G_1\in\mathbb R((\boldsymbol\ell))$, are the whole first blocks of $\widehat f$ and $\widehat g$, respectively (that is, blocks that are the coefficients of $z^\alpha)$.

Proof of Proposition 5.2. Let

$$\begin{equation*} \widehat f(z)=z+z^\alpha \widehat T_0(\boldsymbol\ell)+ z^{\alpha_1} \widehat T_1(\boldsymbol\ell)+\cdots, \end{equation*}$$

where $\widehat T_i(\boldsymbol\ell)\in \mathbb R((\boldsymbol\ell))$ for all $i\in\mathbb N_0$, with the leading term of $\widehat T_0$ given by $\mathrm{Lt}(\widehat T_0(\boldsymbol\ell))=-\boldsymbol\ell^m$, $m\in\mathbb Z$.

We proceed as in [16], Theorem A, but eliminate block-by-block instead of term-by-term. We first eliminate the first block $z^\alpha \widehat T_0(\boldsymbol\ell)$, which can be eliminated except for the first term $-z^\alpha\boldsymbol\ell^m$. Indeed, in each step of the elimination procedure we look for a formal change of variables of the form

$$\begin{equation*} \widehat \varphi_i(z)=z+z^{\gamma_i} \widehat R_i(\boldsymbol\ell),\qquad \operatorname{ord}(z^{\gamma_i}\widehat R_i)\succ (1,0),\quad i\in\mathbb N_0, \end{equation*}$$

where $\widehat R_i\in\mathbb R((\boldsymbol\ell))$. Here $\operatorname{ord}(\,\cdot\,)$ denotes the lexicographic ordinal number of the first term of a transseries.

By Lemma 5.3, in order to remove the first block $z^\alpha \widehat T_0(\boldsymbol\ell)$ (that is, the part of it that can be eliminated) we need to take $\gamma_0=1$. But then $\operatorname{ord}_{\boldsymbol\ell}(\widehat R_0)\geqslant 1$. Therefore, the first monomial $-z^\alpha\boldsymbol \ell^{m}$ cannot be eliminated. The remaining terms of the first block are eliminated by the change $\widehat\varphi_0(z)=z+z\widehat R_0(\boldsymbol\ell)$, where $\widehat R_0(\boldsymbol\ell)\in\boldsymbol\ell\,\mathbb R[[\boldsymbol\ell]]$ is given by (5.2) (it is deduced by integration by parts and formal composition).

Furthermore, it is obvious from integration by parts in the explicit formula (5.2) in Lemma 5.3 that if $\gamma_i\ne 1$ and $\gamma_i\ne \alpha$, then $\widehat T_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$ implies that $\widehat R_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$ and no iterated logarithms are generated. Therefore, we do not need iterated logarithms to remove blocks before the residual block, and consequently all the blocks $\widehat T_i(\boldsymbol\ell)$ up to the residual block (inclusive) belong to $\mathbb R((\boldsymbol\ell))$. Thus, $\widehat f_i,\widehat\varphi_i\in\widehat{\mathcal L}(\mathbb R)$ for all steps of the elimination procedure before the residual block, where $\widehat f_i$ denotes the initial transseries after the first $i$ changes of variables, that is,

$$\begin{equation*} \widehat f_i=\Bigl(\,\mathop{\underset{j=0,\dots,i}\circ}\widehat\varphi_j\Bigr)^{-1} \circ\widehat f \circ\Bigl(\,\mathop{\underset{j=0,\dots,i}\circ} \widehat\varphi_j\Bigr),\qquad i\in\mathbb N_0. \end{equation*}$$

If $\beta_i=2\alpha-1$ (elimination of the residual block $z^{2\alpha-1} \widehat T_i(\boldsymbol\ell)$, $\widehat T_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$), then by Lemma 5.3 we choose $\gamma_i=\alpha$ and

$$\begin{equation*} \widehat R_i(\boldsymbol\ell)=-\boldsymbol\ell^{m}\int \boldsymbol\ell^{-2m-2}\widehat T_i(\boldsymbol\ell)\,d\boldsymbol\ell. \end{equation*}$$

It is easy to see that, by an appropriate series $\widehat R_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$, we can eliminate all terms in $\widehat T_i(\boldsymbol\ell)$ except for the residual term $z^{2\alpha-1}\boldsymbol\ell^{2m+1}$. The set of parabolic transseries in $\widehat{\mathcal L}(\mathbb R)$ is a group under composition, so that after eliminating the residual block we have $\widehat f_i:=\widehat \varphi_i^{-1}\circ \widehat f_{i-1}\circ \widehat\varphi_i\in \widehat{\mathcal L}(\mathbb R)$. After this step the other blocks in $\widehat f_i$ do not contain iterated logarithms. By Lemma 5.3, for $\beta_i> 2\alpha-1$ ($\gamma_i>\alpha$) we can remove all the further blocks $z^{\beta_i}\widehat T_i(\boldsymbol\ell)$, $\widehat T_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$, using changes of variables with $\widehat R_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$, as we explained before.

In the inductive procedure, to remove all possible terms we solve a sequence of Lie bracket equations (see [16], Theorem A, for details). But in contrast to [16], where we eliminate term-by-term, here we must solve only countably many equations and not a transfinite number of equations. Indeed, since the powers $\alpha_i$ in $\widehat f$ are finitely generated, the set of all powers $\gamma_i$ needed in blockwise elimination is also finitely generated (see [16] for details). The proposition is proved. $\square$

Remark 5.5.  If we allow formal conjugation in Proposition 5.2 in the larger class $\widehat{\mathcal L}_2(\mathbb R)$, then we can eliminate also the residual term $\rho z^{2\alpha-1}\boldsymbol\ell^{2m+1}$. By a formal diffeomorphism $\widehat\varphi\in\widehat{\mathcal L}_2(\mathbb R)$, a parabolic generalized Dulac series $\widehat f$ in Proposition 5.2 can be reduced to either of the two normal forms

$$\begin{equation*} \begin{aligned} \, f_F(z)&=z-z^\alpha\boldsymbol\ell^{m}, \\ \widehat f_2(z)&=\operatorname{Exp} \biggl(-z^{\alpha}\boldsymbol\ell^m\,\frac{d}{dz}\biggr)\!. \mathrm{id}. \end{aligned} \end{equation*}$$

The $\widehat{\mathcal L}_2(\mathbb R)$-formal invariants are $(\alpha,m)$. The proof is similar to the proof of Proposition 5.2. Now Lemma 5.3 can simply be rewritten with ¹⁵ $\widehat R_i(\boldsymbol\ell),\widehat T_i(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ instead of $\widehat R_i(\boldsymbol\ell),\widehat T_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$. In order to remove the residual term $z^{2\alpha-1}\boldsymbol\ell^{2m+1}$ in some block $\widehat{T}_i(\boldsymbol\ell)$, we need a double logarithm monomial $\boldsymbol\ell^{m}\boldsymbol\ell_2^{-1}$ in the corresponding series $\widehat R_i(\boldsymbol\ell)$ in the formal change of variables $\widehat\varphi_i$. Therefore, the residual elementary change $\widehat \varphi_i$ belongs to $\widehat{\mathcal L}_2(\mathbb R)$. In general, the blocks of $\widehat f_i$ after removal of the residual block also contain double logarithms, that is, $\widehat T_i(\boldsymbol\ell)\in \widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$. By a generalization of Lemma 5.3 to $\widehat{\mathcal L}_2(\mathbb R)$, we can remove them with the help of transseries $\widehat R_i(\boldsymbol\ell)\in \widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ in subsequent changes of variables. The parabolic transseries in $\widehat{\mathcal L}_2(\mathbb R)$ also form a group under composition.

In this section we describe the property of integral summability of blocks in formal conjugacies $\widehat\varphi\in\widehat{\mathcal L}(\mathbb R)$ that reduce a parabolic generalized Dulac germ to its formal normal form, and in its formal Fatou coordinate $\widehat\Psi\in\widehat{\mathcal L}_2(\mathbb R)$. After introducing some necessary definitions, the statement is given in Propositions 6.7 and 6.11 below.

6.1. The notion of integral summability

In this subsection we apply Definition 6.1 of integral summability of length $1$ to blocks in formal Fatou coordinates of a generalized Dulac germ (see Proposition 6.7). Definition 6.1 is a generalization of the notion of integral summability introduced earlier in Definition 3.9 of [17] for the formal Fatou coordinate of a parabolic Dulac germ. For simplicity we will again use the same name. However, in the case of formal conjugacies we will need the integral summability of greater lengths in Definition 6.1 and the notion of block iterated integral summability in Definition 6.8 of § 6.2. This notion is motivated by Lemmas 5.3 and 5.4, which show that there is one integration in each step of the construction of $\widehat\varphi$. In view of Proposition 5.2, up to some initial transformation connected with the elimination of the first block, $\widehat\varphi$ is constructed as a composition of countably many elementary changes $\widehat{\varphi}_i=z+z^{\beta_i}\widehat R_i(\boldsymbol\ell)$, $i\in\mathbb N$, in which the exponents $\beta_i>1$ are strictly increasing. By the formulae (5.2) and (5.3), $\widehat R_1(\boldsymbol\ell)$ is integrally summable of length $1$ , and the series $\widehat R_2(\boldsymbol\ell)$, $\widehat R_3(\boldsymbol\ell),\dots$ are integrally summable of greater lengths than their immediate predecessors $\widehat R_i(\boldsymbol\ell)$, as in the following inductive definition.

The notion of integral summability recalls the notion of block iterated integrals introduced by Chen in [3] and studied extensively in the context of first return maps (see [6] and [14], among others).

Below we will use the notation $\widehat{\mathcal L}_{\boldsymbol\ell}^{\infty}(\mathbb R)$ for the set of transseries in $\boldsymbol\ell$ and $\boldsymbol\ell_2$ with integer exponents and real coefficients:

$$\begin{equation} \widehat F(\boldsymbol\ell)=\sum_{(m,n)\in A\subseteq \mathbb Z\times \mathbb Z}a_{m,n}\boldsymbol\ell^m\boldsymbol\ell_2^n,\qquad a_{m,n}\in\mathbb R, \end{equation} \tag{ 6.1 }$$

where $A\subseteq \mathbb Z\times \mathbb Z$ is well ordered.

Definition 6.1  (integral summability of series of length $k\in\mathbb N_0$). Let $0<\theta\leqslant 2\pi$ and let $S_\theta=\boldsymbol\ell(V_\theta)$ be an $\boldsymbol\ell$-cusp, where $V_\theta$ is a sector or a petal of opening $\theta$. Assume that the series $\widehat F(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ contains at most one term with double logarithm.

1) The series $\widehat F(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ is said to be integrally summable of length $0$ on $S_\theta$ if there exists an $m>\pi/\theta$ such that $\widehat F(\boldsymbol\ell)\in \widehat{\operatorname{LG}}_m(S_{\theta})$, and

$$\begin{equation*} \boldsymbol\ell\mapsto F(\boldsymbol\ell)\in \operatorname{LG}_m(S_\theta) \end{equation*}$$

is called its $0$ -integral sum.

2) The series $\widehat F(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ is said to be integrally summable of length $1$ on $S_\theta$ if $\widehat F(\boldsymbol\ell)$ is not integrally summable of length $0$ on $S_\theta$, and there exist exponents $\alpha_1\in\mathbb R$ and $p_1\in\mathbb Z$ and a series $\widehat R(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ integrally summable of length $0$ on $S_\theta$ (with $0$-integral sum $R$) such that

$$\begin{equation} \frac{d}{d\boldsymbol\ell} \bigl(e^{-\alpha_1/\boldsymbol\ell}\boldsymbol\ell^{p_1} \widehat F(\boldsymbol\ell)\bigr)=e^{-\alpha_1/\boldsymbol\ell} \boldsymbol\ell^{2p_1-2}\widehat R(\boldsymbol\ell). \end{equation} \tag{ 6.2 }$$

The analytic germ

$$\begin{equation*} F(\boldsymbol\ell):= \frac{1}{e^{-\alpha_1/\boldsymbol\ell}\boldsymbol\ell^{p_1}} \int_*^{\boldsymbol\ell} e^{-\alpha_1/\eta}\eta^{2p_1-2} R(\eta)\,d\eta \end{equation*}$$

on $S_\theta$ is called the $1$ -integral sum of $\widehat F$ on $S_\theta$ . Here $*$ is equal to $0$ if $(\alpha_1,\operatorname{ord}(\widehat R)+2p_1-1)\succ (0,0)$, that is, if the integrand is bounded at $0$, and is equal to some $\boldsymbol\ell_0\in S_\theta$ otherwise. This germ is unique up to an additive term $Ce^{\alpha_1/\boldsymbol\ell}\boldsymbol\ell^{-p_1}= Cz^{-\alpha_1}\boldsymbol\ell^{-p_1}$, $C\in\mathbb R$. The pair of exponents $(\alpha_1,p_1)\in(\mathbb R,\mathbb Z)$ is called the exponent of integration of $\widehat F(\boldsymbol\ell)$.

3) In general, a series $\widehat F(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ is said to be integrally summable of length $k\geqslant 2$ on $S_\theta$ with respect to a set of series

$$\begin{equation*} \widehat U:=\big\{\widehat R_i^{j_i}(\boldsymbol\ell)\colon i=1,\dots,k-1,\ j_i=1,\dots,r_i\}\subseteq \widehat{\mathcal L}_{\boldsymbol\ell}^{\infty}(\mathbb R), \end{equation*}$$

where the series $\widehat R_i^{j_i}(\boldsymbol\ell)\in \widehat U$, $j_i=1,\dots,r_i$, are integrally summable of length $i$ on $S_\theta$, $i=1,\dots,k-1$, with respect to the previous set of series $\{\widehat R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,i-1,\ j_n=1,\dots,r_n\}\subseteq \widehat U$ and where all the elements of $\widehat U$ have different exponents of integration, if the following two conditions hold.

• •  
  The series $\widehat F(\boldsymbol\ell)$ is not of the form

  $$\begin{equation*} \widehat F(\boldsymbol\ell)=\widehat S_{\leqslant l}^{\widehat R_1^{1},\dots, \widehat R_1^{r_1};\dots;\widehat R_{l}^{1},\dots, \widehat R_l^{r_l}}(\boldsymbol\ell)\in \widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R), \qquad l\leqslant k-1, \end{equation*}$$

  where $\widehat S_{\leqslant l}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1}; \dots;\widehat R_{l}^{1},\dots,\widehat R_l^{r_l}}(\boldsymbol\ell)\in \widehat {\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ is as follows:

  • (i)  
    for $l=1$, $\widehat S_{\leqslant 1}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1}}(\boldsymbol\ell)$ is a finite algebraic combination (with operations $+$, $\,\cdot\,$, $\,/\,$, $d/d\boldsymbol\ell$) of series that are integrally summable of length $0$ and series $\widehat R_1^{j_1}(\boldsymbol\ell)$, $j_1=1,\dots,r_1$, that are not integrally summable of length $0$;
  • (ii)  
    for $l\geqslant 2$, $\widehat S_{\leqslant l}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1};\dots;\widehat R_{l}^{1},\dots,\widehat R_l^{r_l}}(\boldsymbol\ell)$ is a finite algebraic combination (with operations $+$, $\,\cdot\,$, $\,/\,$, $d/d\boldsymbol\ell$) of series that are integrally summable of length $0$, series $\widehat R_1^{j_1}(\boldsymbol\ell)\in \widehat U$, $j_1=1,\dots,r_1$, and series $\widehat R_i^{j_i}(\boldsymbol\ell)\in \widehat U$, $j_i=1,\dots,r_i$, that are integrally summable of length $i$ with respect to the previous series $\{\widehat R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,i-1,\ j_n=1,\dots,r_n\}\subseteq \widehat U$, $1\leqslant i\leqslant l$, and does not have the form $\widehat S_{\leqslant l-1}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1};\dots;\widehat R_{l-1}^{1},\dots,\widehat R_{l-1}^{r_{l-1}}}(\boldsymbol\ell)$.
• •  
  There exists a pair of exponents $(\alpha_k,p_k)\in(\mathbb R,\mathbb Z)$ such that

  $$\begin{equation} \frac{d}{d\boldsymbol\ell}\bigl(e^{-\alpha_k/\boldsymbol\ell} \boldsymbol\ell^{p_k} \widehat F(\boldsymbol\ell)\bigr)= e^{-\alpha_k/\boldsymbol\ell}\boldsymbol\ell^{2p_k-2} \widehat S_{\leqslant k-1}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1};\dots; \widehat R_{k-1}^{1},\dots,\widehat R_{k-1}^{r_{k-1}}}(\boldsymbol\ell), \end{equation} \tag{ 6.3 }$$

  where $\widehat S_{\leqslant k-1}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1};\dots;\widehat R_{k-1}^{1},\dots,\widehat R_{k-1}^{r_{k-1}}}(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ are of the above described form.

Denote by

$$\begin{equation*} S_{\leqslant k-1}^{R_1^{1},\dots,R_1^{r_1};\dots;R_{k-1}^{1},\dots, R_{k-1}^{r_{k-1}}}(\boldsymbol\ell) \end{equation*}$$

the sum of the series $\widehat S_{\leqslant k-1}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1};\dots;\widehat R_{k-1}^{1},\dots,\widehat R_{k-1}^{r_{k-1}}}(\boldsymbol\ell)$ (computed in the natural way with the corresponding algebraic operations and with some fixed choice of the constants of integration in the integrals for the germs $R_i^{j_i}(\boldsymbol\ell)$, $i=1,\dots,l$, $j_i=1,\dots,r_i$, computed from the previous steps). Then the analytic germ

$$\begin{equation*} F(\boldsymbol\ell):=\frac{1}{e^{-\alpha_k/\boldsymbol\ell} \boldsymbol\ell^{p_k}}\int_*^{\boldsymbol\ell}e^{-\alpha_k/\eta} \eta^{2p_k-2}S_{\leqslant k-1}^{R_1^{1},\dots,R_1^{r_1};\dots; R_{k-1}^{1},\dots,R_{k-1}^{r_{k-1}}}(\eta)\,d\eta \end{equation*}$$

on $S_\theta$ is called the $k$ -integral sum of the series. The choice of a base point $*$ is as in 2). This $k$-sum is unique (after a particular sum $S_{\leqslant k-1}^{R_1^{1},\dots,R_1^{r_1};\dots;R_{k-1}^{1}, \dots,R_{k-1}^{r_{k-1}}}\!\!\!(\boldsymbol\ell)$ of the series $\widehat S_{\leqslant k-1}^{\widehat R_1^{1},\dots, \widehat R_1^{r_1};\dots;\widehat R_{k-1}^{1},\dots, \widehat R_{k-1}^{r_{k-1}}}(\boldsymbol\ell)$ is fixed) up to an additive term $Ce^{\alpha_k/\boldsymbol\ell}\boldsymbol\ell^{-p_k}= Cz^{-\alpha_k}\boldsymbol\ell^{-p_k}$, $C\in\mathbb R$ (due to the possibility of a different choice of the point $\boldsymbol\ell_0\in \boldsymbol\ell(V_j^{\pm})$ if the integrand is not bounded at $0$).

Note that solving equation (6.2) in Definition 6.1 is equivalent to solving the non-homogenous linear ordinary differential equation

$$\begin{equation*} \boldsymbol\ell^2\frac{d}{d\boldsymbol\ell}\widehat F+ (\alpha_1+p_1\boldsymbol\ell)\widehat F=\boldsymbol\ell^{p_1}\widehat R, \end{equation*}$$

and similarly for equation (6.3).

The integration paths in the integrals from $*$ to $\boldsymbol\ell$ do not matter. Indeed, $\boldsymbol\ell(V_\theta)$ is a simply connected cusp with sufficiently small radius (a germ domain) and the integrand is analytic on $\boldsymbol\ell(V_\theta)$.

By the block iterated integral sum

$$\begin{equation*} U:=\{R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,k-1,\ j_n=1,\dots,r_n\} \end{equation*}$$

of a set

$$\begin{equation*} \widehat U=\{\widehat R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,k-1,\ j_n=1,\dots,r_n\} \end{equation*}$$

of integrally summable series $\widehat R_i^r(\boldsymbol\ell)$ with respect to previous series $\widehat R_j^p(\boldsymbol\ell)$, $1\leqslant j<i$, of strictly smaller lengths of summation, as described in Definition 6.1 above, we mean that at each step the constant of integration $\boldsymbol\ell_0$ is fixed, and the integrands are determined by the constants of integration from the previous steps.

Remark 6.2.  If the series $\widehat F$ in (6.2) is already integrally summable of length $0$ on $S_\theta$, then (6.2) is satisfied for every exponent of integration $(\alpha,p)\in(\mathbb R,\mathbb Z)$. Similarly, let $k\geqslant2$. If $\widehat F(\boldsymbol\ell)$ is already a finite algebraic combination of integrally summable series of length $0$, integrally summable series $\widehat R_1^{j_1}(\boldsymbol\ell)$, $j_1=1,\dots,r_1$, of length $1$, and integrally summable series $\widehat R_l^{j_l}(\boldsymbol\ell)$, $j_l=1,\dots,r_l$, of arbitrary lengths $2\leqslant l\leqslant k-1$ with respect to the previous series $\{\widehat R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,l-1,\ j_n=1,\dots,r_n\}$, then (6.3) holds with any exponents of integration $(\alpha,p)\in(\mathbb R,\mathbb Z)$. Then we may take their sums in the natural way, performing the corresponding operations (with ambiguity in the choice of additive terms in each step). That is why, in each step of the inductive Definition 6.1, we exclude such trivial cases, in order to have uniqueness for the exponents of integration.

Proposition 6.3  (uniqueness of integral sums of length $k\geqslant 1$). Let the series $\widehat F(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ be integrally summable of length $1$ on an $\boldsymbol\ell$-cusp $S_\theta$, or integrally summable of length $k\geqslant 2$ on $S_\theta$ with respect to the series $\{\widehat R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,k-1$, $j_n=1,\dots,r_n\}\subset\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$, which are themselves integrally summable of lengths $1,\dots, k-1$, respectively, on $S_\theta$, corresponding to Definition 6.1. Then the exponent of integration $(\alpha,p)\in(\mathbb R,\mathbb Z)$ of $\widehat F$ is uniquely determined.

The proof, similar to that in [17], is in the Appendix (§ 10). Proposition 6.3 states that the $1$-integral sums and the $k$-integral sums with respect to given integrally summable series of smaller length are unique, up to the choice of the constants of integration in the inductive Definition 6.1.

In Definition 6.1 we also implicitly assume that if $\widehat S_{\leqslant k}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1};\dots;\widehat R_{k}^{1},\dots,\widehat R_k^{r_k}}(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ can be represented as a finite algebraic combination (with the operations $+$, $\,\cdot\,$, $\,/\,$, $d/d\boldsymbol\ell$) of integrally summable series $\widehat R_i^{j_i}(\boldsymbol\ell)$, $j_i=1,\dots,r_i$, of lengths $i\in\{0,1,\dots,k\}$, respectively, with respect to the previous series $\{\widehat R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,i-1$, $j_n=1,\dots,r_n\}$, then it can be given a unique sum (up to addition of the constants of integration). We prove this in Proposition 6.5 below. In the proof we use the following auxiliary lemma, Lemma 6.4, whose proof is in the Appendix (§ 10).

Lemma 6.4.  Any finite algebraic combination

$$\begin{equation*} \widehat S_{\leqslant k}^{\widehat R_1^{1},\dots,\widehat R_1^{r_1};\dots; \widehat R_{k}^{1},\dots,\widehat R_k^{r_k}}(\boldsymbol\ell)\in \widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R),\qquad k\in\mathbb N, \end{equation*}$$

with respect to the operations $+$, $\,\cdot\,$, $\,/\,$, $d/d\boldsymbol\ell$, where the series $\widehat R_i^{j_i}(\boldsymbol\ell)$, $j_i=1,\dots,r_{i}$, are integrally summable of length $i$, $i=1,\dots,k$, as in Definition 6.1, is equal ¹⁶ to a rational function of the series $\widehat R_n^{j_n}(\boldsymbol\ell)$, $n=1,\dots,k$, $j_n=1,\dots,r_n$, and of integrally summable series of length $0$.

Proposition 6.5.  Let

$$\begin{equation*} \widehat U=\{\widehat R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,k,\ j_n=1,\dots,r_n\}\subseteq \widehat{\mathcal L}_{\boldsymbol\ell}^{\infty}(\mathbb R) \end{equation*}$$

be a set of consecutively integrally summable series with respect to previous series of strictly smaller lengths, according to Definition 6.1, with a corresponding sequence of strictly increasing exponents of integration $\{\beta_1,\beta_2,\dots,\beta_{r_1+\dots+r_k}\}$. Here the subscripts of elements of the set $\widehat U$ denote the length of integral summability of the corresponding transseries.

Let $\{R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,k,\ j_n=1,\dots,r_n\}$ be a fixed choice of their integral sums, as described after Definition 6.1. Let $\widehat S\in\widehat{\mathcal L}_{\boldsymbol\ell}^{\infty}(\mathbb R)$ be a transseries that can be represented as a rational function of elements of the set $\widehat U$ and of integrally summable series of length $0$. Then this representation is unique. ¹⁷ Therefore, the sum of the series $S$, after the corresponding algebraic operations, is unique with respect to fixed sums $\{R_n^{j_n}(\boldsymbol\ell)\colon n=1,\dots,k,\ j_n=1,\dots,r_n\}$ of elements of the set $\widehat U$.

Note that, by Lemma 6.4, Proposition 6.5 states that, if the series $\widehat S(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ is representable as a finite algebraic combination of elements of $\widehat U$ and of integrally summable series of length $0$, then its sum, given the corresponding algebraic operations, is uniquely determined with respect to fixed sums of the series in $\widehat U$.

Proof.  Suppose that there exists a rational function of elements in $\widehat U$ and of 0- integrally summable series that equals $0$. Then its numerator is a polynomial in integrally summable series of length 0 and in series in $\widehat U$. We prove that, up to some operations on integrally summable series of length $0$, the algebraic expression of this polynomial, as described in footnote 17, is equal to 0. In this case the sum, taken in the natural way after the algebraic operations, is zero, and the assertion is proved.

Suppose that the length $0\leqslant l_0\leqslant k$ is the maximal length of integral summability of series in this polynomial. We now express the monomial of greatest length in the variables $\widehat R_{l_0}^{j_{l_0}}(\boldsymbol\ell)$, $j_{l_0}=1,\dots,r_{l_0}$, as a rational function of monomials of different length ($\leqslant$) in the variables $\widehat R_{l_0}^{j_{l_0}}(\boldsymbol\ell)$, $j_{l_0}=1,\dots,r_{l_0}$, and of monomials formed from series integrally summable of length $0$ and series $\widehat R_n^{j_n}(\boldsymbol\ell)$, $n=1,\dots,l_0-1$, $j_n=1,\dots,r_{l_0-1}$, summable of strictly smaller lengths. In the denominator there are only monomials in integrally summable series of length $0$ and in series $\widehat R_n^{j_n}(\boldsymbol\ell)$, $n=1,\dots,l_0-1$, $j_n=1,\dots,r_{l_0-1}$, summable of strictly smaller lengths. We decrease the length by multiplying this maximal-length monomial by suitable factors of the form $e^{-\alpha/\boldsymbol\ell}\boldsymbol\ell^p$ (as many factors as there are series in this monomial) and differentiating. Then the length of the monomials on the left-hand side has been reduced by $1$. We multiply by the denominator of the right-hand side and repeat the process for the monomial of greatest length in the variables $\widehat R_{l_0}^{j_{l_0}}(\boldsymbol\ell)$, $j_{l_0}=1,\dots,r_{l_0}$, whose length is now equal or strictly smaller. In each step, we simplify the algebraic expression as described in footnote 17, up to operations on integrally summable series of length $0$. If in some step we get a trivial algebraic combination, then the initial rational function was trivial, and the procedure stops (meaning that the initial polynomial numerator was equal to zero, which is what we wanted to prove); if not, then we proceed. After repeating the procedure finitely many times, if the procedure did not stop before, then we end up with a simple monomial $\widehat R_{l_0}^r(\boldsymbol\ell)$, $r\in\{1,\dots,r_{l_0}\}$, of length of summation $l_0$ expressed as a polynomial in integrally summable series of strictly smaller lengths and in one other monomial $\widehat R_{l_0}^{r_1}(\boldsymbol\ell)$, $r_1\ne r$, of length of summation $l_0$, divided by a polynomial in integrally summable series of lengths strictly smaller than $l_0$. Since $\widehat R_{l_0}^{r_{1}}(\boldsymbol\ell)$ and $\widehat R_{l_0}^r(\boldsymbol\ell)$ have different exponents of integration by assumption, by one more multiplication by a suitable factor of the form $e^{-\alpha/\boldsymbol\ell}\boldsymbol\ell^p$ and differentiation we end up with a series $\widehat R_{l_0}^{r_1}(\boldsymbol\ell)$ summable of length $l_0$ that is equal to a rational function of series of strictly smaller lengths, which is a contradiction. $\square$

We remark that item 2) of Definition 6.1, integral summability of length $1$, corresponds to Definition 3.9 in [17], since $\dfrac{d}{dz}=\dfrac{\boldsymbol\ell^2}{z}\,\dfrac{d}{d\boldsymbol\ell}$ .

Remark.  It can be checked directly using (6.2) that the sum of two integrally summable series $\widehat F(\boldsymbol\ell)$ and $\widehat G(\boldsymbol\ell)$ of length $1$ with the same exponents of integration $(\alpha,m)$ can either be integrally summable of length $0$ or integrally summable of length $1$ with the same exponent of integration $(\alpha,m)$.

The sum of two integrally summable series $\widehat F(\boldsymbol\ell)$ and $\widehat G(\boldsymbol\ell)$ of length $k\geqslant 2$ with respect to the same integrally summable series of smaller lengths, where the exponents of integration of $\widehat F$ and $\widehat G$ are not necessarily equal, is equal to some algebraic combination of the series $\widehat S_{\leqslant l}(\boldsymbol\ell)$, for some $l\leqslant k$, with respect to the same integrally summable series of smaller lengths. Indeed, one can show by differentiation and using the formulae (6.2) and (6.3) that the sum satisfies the formula (6.3) with an algebraic combination of the series $\widehat S_{\leqslant l}(\boldsymbol\ell)$, $l\leqslant k$, on the right-hand side and with any exponent of integration. Let $(\alpha,m)$ and $(\beta,n)$ be the exponents of integration of $\widehat F(\boldsymbol\ell)$ and $\widehat G(\boldsymbol\ell)$, respectively (in general not equal). For every $(\gamma,q)\in\mathbb R$

$$\begin{equation} \begin{aligned} \, \nonumber &\frac{d}{d\boldsymbol\ell}\bigl(e^{-\gamma/\boldsymbol\ell}\boldsymbol\ell^q \bigl(\widehat F(\boldsymbol\ell)+\widehat G(\boldsymbol\ell)\bigr)\bigr) \\ \nonumber &\qquad=\frac{d}{d\boldsymbol\ell} \bigl(e^{-\alpha/\boldsymbol\ell}\boldsymbol\ell^m\widehat F(\boldsymbol\ell) \cdot e^{-(\gamma-\alpha)/\boldsymbol\ell}\boldsymbol\ell^{q-m}+ e^{-\beta/\boldsymbol\ell}\boldsymbol\ell^n\widehat G(\boldsymbol\ell)\cdot e^{-(\gamma-\beta)/\boldsymbol\ell}\boldsymbol\ell^{q-n}\bigr) \\ \nonumber &\qquad=e^{-\alpha/\boldsymbol\ell}\boldsymbol\ell^{2m-2} \widehat R_1(\boldsymbol\ell)\cdot e^{-(\gamma-\alpha)/\boldsymbol\ell} \boldsymbol\ell^{q-m} \\ \nonumber &\qquad\qquad+e^{-\alpha/\boldsymbol\ell}\boldsymbol\ell^m \widehat F(\boldsymbol\ell)\cdot e^{-(\gamma-\alpha)/\boldsymbol\ell} \boldsymbol\ell^{q-m-2}\bigl((q-m)\boldsymbol\ell+(\gamma-\alpha)\bigr) \\ \nonumber &\qquad\qquad+e^{-\beta/\boldsymbol\ell}\boldsymbol\ell^{2n-2} \widehat R_2(\boldsymbol\ell)\cdot e^{-(\gamma-\beta)/\boldsymbol\ell} \boldsymbol\ell^{q-n} \\ \nonumber &\qquad\qquad+e^{-\beta/\boldsymbol\ell}\boldsymbol\ell^n \widehat F(\boldsymbol\ell)\cdot e^{-(\gamma-\beta)/\boldsymbol\ell} \boldsymbol\ell^{q-n-2}\bigl((q-n)\boldsymbol\ell+(\gamma-\beta)\bigr) \\ \nonumber &\qquad=e^{-\gamma/\boldsymbol\ell}\boldsymbol\ell^{2q-2} \bigl[\widehat R_1(\boldsymbol\ell)\cdot\boldsymbol\ell^{m-q}+ \boldsymbol\ell^{-q}\cdot \bigl((q-m)\boldsymbol\ell+ (\gamma-\alpha)\bigr)\cdot \widehat F(\boldsymbol\ell) \\ &\qquad\qquad+\widehat R_2(\boldsymbol\ell)\cdot\boldsymbol\ell^{n-q}+ \boldsymbol\ell^{-q}\cdot \bigl((q-n)\boldsymbol\ell+ (\gamma-\beta)\bigr)\cdot \widehat G(\boldsymbol\ell)\bigr]. \end{aligned} \end{equation} \tag{ 6.4 }$$

Here, since $\widehat F(\boldsymbol\ell)$ and $\widehat G(\boldsymbol\ell)$ are integrally summable of length $k$, the series $\widehat R_1(\boldsymbol\ell)$ and $\widehat R_2(\boldsymbol\ell)$ are both of type $\widehat S_{\leqslant (k-1)}(\boldsymbol\ell)$. The term in square brackets on the right-hand side is then obviously of type $\widehat S_{\leqslant l}(\boldsymbol\ell)$ for some $l\leqslant k$ (due to possible cancellations). In particular, if $(\alpha,m)=(\beta,n)$, then $\widehat F(\boldsymbol\ell)+\widehat G(\boldsymbol\ell)$ is by (6.4) integrally summable of length $k$ with the same exponent of integration $(\gamma,q):=(\alpha,m)=(\beta,n)$.

By (6.4), for any exponents of integration $(\gamma,q)\ne(\delta,p)$ the series

$$\begin{equation*} \frac{d}{d\boldsymbol\ell}\bigl(e^{-\gamma/\boldsymbol\ell} \boldsymbol\ell^q \bigl(\widehat F(\boldsymbol\ell)+ \widehat G(\boldsymbol\ell)\bigr)\bigr)\quad\text{and}\quad \frac{d}{d\boldsymbol\ell}\bigl(e^{-\delta/\boldsymbol\ell} \boldsymbol\ell^p \bigl(\widehat F(\boldsymbol\ell)+ \widehat G(\boldsymbol\ell)\bigr)\bigr) \end{equation*}$$

are of type $\widehat S_{\leqslant l_1}(\boldsymbol\ell)$ and $\widehat S_{\leqslant l_2}(\boldsymbol\ell)$, respectively, for some $l_1,l_2\leqslant k$. Then

$$\begin{equation} \begin{aligned} \, \nonumber &e^{-\gamma/\boldsymbol\ell}\boldsymbol\ell^{2q-2} \widehat R_3(\boldsymbol\ell)=\frac{d}{d\boldsymbol\ell} \bigl(e^{-\gamma/\boldsymbol\ell}\boldsymbol\ell^q \bigl(\widehat F(\boldsymbol\ell)+\widehat G(\boldsymbol\ell)\bigr)\bigr) \\ \nonumber &\qquad=\frac{d}{d\boldsymbol\ell}\bigl(e^{-\delta/\boldsymbol\ell} \boldsymbol\ell^p \bigl(\widehat F(\boldsymbol\ell)+ \widehat G(\boldsymbol\ell)\bigr)\cdot e^{-(\gamma-\delta)/\boldsymbol\ell} \boldsymbol\ell^{q-p}\bigr) \\ \nonumber &\qquad=e^{-\delta/\boldsymbol\ell}\boldsymbol\ell^{2p-2} \widehat R_4(\boldsymbol\ell)\cdot e^{-(\gamma-\delta)/\boldsymbol\ell} \boldsymbol\ell^{q-p} \\ \nonumber &\qquad\qquad+e^{-\delta/\boldsymbol\ell}\boldsymbol\ell^p \bigl(\widehat F(\boldsymbol\ell)+\widehat G(\boldsymbol\ell)\bigr) \cdot e^{-(\gamma-\delta)/\boldsymbol\ell}\boldsymbol\ell^{q-p-2} \bigl((q-p)\boldsymbol\ell+(\gamma-\delta)\bigr) \\ &\qquad=e^{-\gamma/\boldsymbol\ell}\bigl[\boldsymbol\ell^{p+q-2} \widehat R_4(\boldsymbol\ell)+\boldsymbol\ell^{q-2} \bigl(\widehat F(\boldsymbol\ell)+\widehat G(\boldsymbol\ell)\bigr) \bigl((q-p)\boldsymbol\ell+(\gamma-\delta)\bigr)\bigr]. \end{aligned} \end{equation} \tag{ 6.5 }$$

Here $\widehat R_3(\boldsymbol\ell)$ is of type $\widehat S_{\leqslant l_1}(\boldsymbol\ell)$, and $\widehat R_4(\boldsymbol\ell)$ is of type $\widehat S_{\leqslant l_2}(\boldsymbol\ell)$. Comparing the sides in (6.5), we conclude that the sum $\widehat F(\boldsymbol\ell)+\widehat G(\boldsymbol\ell)$ is of type $\widehat S_{\leqslant l}(\boldsymbol\ell)$ for some $l\leqslant\max\{l_1,l_2\}$. Note that $\max\{l_1,l_2\}\leqslant k$, so that $\widehat F(\boldsymbol\ell)+\widehat G(\boldsymbol\ell)$ is of type $S_{\leqslant l}$ for some $l\leqslant k$.

A similar result holds for products $\widehat F(\boldsymbol\ell)\cdot \widehat G(\boldsymbol\ell)$. In the same way as for (6.4) and (6.5) above, it can be proved that if both $\widehat F(\boldsymbol\ell)$ and $\widehat G(\boldsymbol\ell)$ are integrally summable of length $k\geqslant 2$ with respect to the same integrally summable series of smaller lengths, then their product is of the same type $\widehat S_{\leqslant l}(\boldsymbol\ell)$, $l\leqslant k$, with respect to the same integrally summable series of smaller lengths. This holds independently of the exponents of integration of $\widehat F(\boldsymbol\ell)$ and $\widehat G(\boldsymbol\ell)$.

Proposition 6.7  $\!\!$ (formal Fatou coordinate for a parabolic generalized Dulac germ). Let $f$ be a parabolic generalized Dulac germ on a standard quadratic domain, and let $\widehat f(z)=z-z^\alpha\widehat R_1(\boldsymbol\ell)+\mathrm{h.o.b.}$, $\alpha>1$, $m\in\mathbb Z$, be its generalized Dulac expansion. Then there exists a unique, up to an additive constant, formal Fatou coordinate $\widehat\Psi\in\widehat{\mathfrak L}^\infty(\mathbb R)$. Moreover, $\widehat\Psi\in\widehat{\mathcal L}_2^\infty(\mathbb R)$ and it has the form

$$\begin{equation*} \widehat\Psi(z)=\sum_{i\in\mathbb N}z^{\beta_i}\widehat T_i(\boldsymbol\ell), \end{equation*}$$

where the $\widehat T_i(\boldsymbol\ell)\in\widehat{\mathcal L}_{\boldsymbol\ell}^\infty(\mathbb R)$ are integrally summable of length $1$ and have exponent of integration $(\beta_i,0)$ in the sense of Definition 6.1 on $\boldsymbol\ell$-cusps $\boldsymbol\ell(V_{\pm}^j)$ of petals ¹⁸ $V_{\pm}^j$ ($j\in\mathbb Z$) with opening $2\pi/(\alpha-1)$, and the exponents $(\beta_i)_i\in\mathbb R$ are finitely generated and strictly increasing to $+\infty$, finitely many of them are negative, and $\beta_1=-\alpha+1$.

Proof.  The proof is similar to the proof of the theorem in [17] about the formal Fatou coordinate for a parabolic Dulac germ, with use of the Abel difference equation and blockwise construction of the formal Fatou coordinate. The only difference is that, instead of polynomials, in each block of $\widehat f$ we have $\log$ -Gevrey series in $\boldsymbol\ell$, which nevertheless are canonically summable in view of § 4. Accordingly, our Definition 6.1 of series integrally summable of length $1$ is a generalization of the notion of integrally summable series in Definition 3.8 of [17], which was used in constructing the formal Fatou coordinate of parabolic Dulac germs.

The proof follows the same lines as the proof of Theorem THA, (ii). Let

$$\begin{equation*} \widehat g(z)=\widehat f(z)-\operatorname{id}= z^{\alpha}\widehat R_1(\boldsymbol\ell)+z^{\alpha_1} \widehat R_2(\boldsymbol\ell)+\mathrm{h.o.b.},\qquad \alpha_1>\alpha>1, \end{equation*}$$

where $\widehat R_1(\boldsymbol\ell)$ and $\widehat R_2(\boldsymbol\ell)$ are $\log$-Gevrey series of order strictly greater than $(\alpha-1)/2$ on $\boldsymbol\ell(V_\pm^{j})$. We construct the formal Fatou coordinate block-by-block, using the formal Taylor expansion of the Abel equation $\widehat\Psi(\widehat f)-\widehat\Psi=1$:

$$\begin{equation} \widehat\Psi'(z)\widehat g(z)+ \frac{1}{2!}\widehat \Psi''(z)\widehat g^2(z)+\cdots=1. \end{equation} \tag{ 6.6 }$$

Let $\widehat\Psi_1$ be the lowest-order block of $\widehat\Psi$. Since the orders (in $z$) of the terms in the Taylor expansion are strictly increasing,

$$\begin{equation*} \widehat\Psi_1'(z)\cdot z^\alpha\widehat R_1(\boldsymbol\ell)=1. \end{equation*}$$

Because $\widehat R_1(\boldsymbol\ell)$ is a $\log$-Gevrey series of order strictly greater than $(\alpha-1)/2$ on $\boldsymbol\ell(V_\pm^j)$ by the definition of generalized Dulac expansions, and because by Proposition 4.9 the series $1/\widehat R_1(\boldsymbol\ell)$ is also $\log$-Gevrey of order strictly greater than $(\alpha-1)/2$ on $\boldsymbol\ell(V_\pm^j)$, it follows that the series $z^{\alpha-1}\widehat \Psi_1(z)= z^{\alpha-1}\displaystyle\int\dfrac{z^{-\alpha}\,dz} {\widehat R_1(\boldsymbol\ell)}$ is, by Definition 6.1, integrally summable of length $1$ in the variable $\boldsymbol\ell$ with exponent of integration $(-\alpha+1,0)$ on $\boldsymbol\ell(V_\pm^j)$ ($j\in\mathbb Z$). We have $\beta_1:=-\alpha+1$.

To find the second block $\widehat\Psi_2(\boldsymbol\ell)$, we put $\widehat\Psi(z)=\widehat\Psi_1(z)+\widehat \Psi_2(z)+\mathrm{h.o.b.}$ in the equation (6.6). After cancellations, and after comparing the lowest-order blocks on the right-hand and left-hand side as above, we get that

$$\begin{equation} \widehat\Psi_2'(z)\cdot z^\alpha\widehat R_1(\boldsymbol\ell)=\begin{cases} z^{\alpha_1-\alpha}\dfrac{\widehat R_2(\boldsymbol\ell)} {\widehat R_1(\boldsymbol\ell)}\,, &\alpha_1<2\alpha-1, \\ z^{\alpha-1}\biggl(-\dfrac{1}{2}\biggr)(\alpha \widehat R_1+ \widehat R_1'\cdot\boldsymbol\ell^2), &\alpha_1>2\alpha-1, \\ \text{the sum of both},&\alpha_1=2\alpha-1. \end{cases} \end{equation} \tag{ 6.7 }$$

In the denominator we can get only powers of the series $\widehat R_1(\boldsymbol\ell)\ne 0$. Since the $\log$- Gevrey expansions are closed with respect to the algebraic operations and differentiation (Propositions 4.5–4.7), we conclude that the series $z^{-\min\{\alpha-\alpha_1+1,-\alpha+2\}}\widehat\Psi_2(z)$ is again integrally summable of length $1$ on $\boldsymbol\ell(V_\pm^j)$, with exponent of integration $(\min\{\alpha-\alpha_1+1,-\alpha+2\},0)$. Let $\beta_2:=\min\{\alpha-\alpha_1+1,-\alpha+2\}$. We continue the construction by induction. More precisely, in the $i$th step of the induction ($i\in\mathbb N$) we suppose that the series $z^{-\beta_1}\widehat\Psi_1,\dots,z^{-\beta_i}\widehat\Psi_i$, are integrally summable of length $1$ on $\boldsymbol\ell(V_\pm^j)$ with exponents of integration $(\beta_1,0),\dots,(\beta_i,0)$, respectively. Therefore, the series $\widehat\Psi_1'(z),\dots,\widehat \Psi_i'(z)$ are products of some power of $z$ by some $\log$-Gevrey series of order strictly greater than $(\alpha-1)/2$ on $\boldsymbol\ell(V_\pm^j)$. Due to the closedness of $\log$-Gevrey series with respect to the algebraic operations, all their derivatives are of the same form. Moreover, all the blocks in $\widehat g^k(z)$ ($k\in\mathbb N$), are products of powers of $z$ by $\log$-Gevrey series of order strictly greater than $(\alpha-1)/2$, since $\widehat f$ is a generalized Dulac expansion. Now it follows from (6.6) that $\widehat \Psi_{i+1}'(z)$ is again a product of some power of $z$ by a $\log$-Gevrey series of order strictly greater than $(\alpha-1)/2$ on $\boldsymbol\ell(V_\pm^j)$, which is well defined since only powers of $\widehat R_1(\boldsymbol\ell)\ne 0$ appear in the denominator. Therefore, there exists a unique exponent $\beta_{i+1}$ such that $z^{-\beta_{i+1}}\widehat \Psi_{i+1}(z)$ is integrally summable of length $1$ in $\boldsymbol\ell$ on the domain $\boldsymbol\ell(V_{\pm}^j)$. The exponent of integration is $(\beta_{i+1},0)$, and $\beta_{i+1}>\beta_i$ by our algorithm. $\square$

6.2. The notion of block iterated integral summability

Definition 6.8 below of block iterated integral summability is adapted to formal normalizations of parabolic generalized Dulac germs. The technical complication in the definition of the final composition with the transformation $\widehat h_0^{-1}$ comes from the elimination of the first block in reducing a parabolic generalized Dulac transseries to its normal form. This first change of variables is more complicated in form than the changes considered in Lemma 5.3 by which we eliminate higher-order blocks. By removing the first block in the first step, we would essentially change the type of blocks in the initial parabolic generalized Dulac expansion (they would no longer be $\log$-Gevrey, that is, integrally summable of length $0$). This would involve technical difficulties in describing subsequent changes using the notion of integral summability of length $k\geqslant 1$ introduced in Definition 6.1. We avoid these difficulties by first reducing the initial parabolic generalized Dulac transseries to a parabolic generalized Dulac transseries with the same initial block, which we call an auxiliary normal form. Finally, applying one additional change of variables, $\widehat h_0^{-1}$, we reduce this transseries to a normal form in (5.1). This will be clarified in the proof of Proposition 6.10 below.

Definition 6.8  (block iterated integrally summable transseries). Let $\widehat{\varphi}\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$, $\widehat\varphi(z)=z+z\widehat R_0(\boldsymbol\ell)+\mathrm{h.o.b.}$, where $\widehat R_0(\boldsymbol\ell)\in\boldsymbol\ell\mathbb R[[\boldsymbol\ell]]$, be a parabolic transseries. The case $\widehat R_0=0$ is also allowed.

We say that the series $\widehat\varphi$ is block iterated integrally summable with parameters $(\alpha,m,\rho)\in\mathbb R_{>1}\times \mathbb Z\times \mathbb R$ on a petal $V$ of opening $2\pi/(\alpha-1)$ if the following conditions 1) and 2) hold.

1) There exists a series $\widehat T_0(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$, $\widehat T_0(\boldsymbol\ell)=-\boldsymbol\ell^m+\mathrm{h.o.t.}$, that is $\log$-Gevrey summable of order strictly greater than $(\alpha-1)/2$ on the $\boldsymbol\ell$-cusp $\boldsymbol\ell(V)$ and such that

$$\begin{equation} \biggl(-\int\frac{z^{1-\alpha}} {\boldsymbol\ell^{m+2}}\,d\boldsymbol\ell\biggr)\circ \bigl(z+z\widehat R_0(\boldsymbol\ell)\bigr)= \int\frac{z^{1-\alpha}}{\widehat T_0(\boldsymbol\ell)\boldsymbol\ell^2}\, d\boldsymbol\ell \end{equation} \tag{ 6.8 }$$

(with $0$ as the constant of formal integration).

2) If $\widehat h_0\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a transseries given by the formula (with the constant of formal integration equal to $0$)

$$\begin{equation} \widehat h_0(z):=\biggl(-\int \frac{z^{1-\alpha}}{\boldsymbol\ell^{m+2}}\, d\boldsymbol\ell-\frac{\alpha}{2}\boldsymbol\ell^{-1}+ \biggl(\frac{m}{2}+\rho\biggr)\boldsymbol\ell_2^{-1}\biggr)^{-1}\circ \biggl(\,\int\frac{z^{-\alpha+1}}{\widehat T_0(\boldsymbol\ell) \boldsymbol\ell^2}\,d\boldsymbol\ell-\frac{\alpha}{2}\boldsymbol\ell^{-1}+ b\boldsymbol\ell_2^{-1}\biggr), \end{equation} \tag{ 6.9 }$$

where $b\in\mathbb R$ is uniquely determined by the condition that $\widehat h_0(z)$ does not contain iterated logarithms, then the composition $\widehat\varphi\circ \widehat h_0$ can be decomposed into a sequence of compositions ¹⁹ $\widehat\varphi\circ \widehat h_0=\mathop{\underset{i\in\mathbb N}\circ}\widehat\varphi_i$ of elementary changes of variables of the form

$$\begin{equation*} \widehat\varphi\circ \widehat h_0= \mathop{\underset{i\in\mathbb N}\circ}\widehat\varphi_i,\qquad \widehat\varphi_i(z)=z+z^{\beta_i}\widehat R_i(\boldsymbol\ell), \end{equation*}$$

where $(\beta_i)_{i\in\mathbb N}$, $\beta_i>1$, is a strictly increasing sequence of real numbers, either finite or tending to $+\infty$, and the transseries $\widehat R_i(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$, $\widehat R_i\ne 0$, $i\in\mathbb N$, form a sequence

$$\begin{equation} \widehat{\mathcal H}=\{\widehat R_i(\boldsymbol\ell)\colon i\in\mathbb N\}\subseteq \mathbb R((\boldsymbol\ell)) \end{equation} \tag{ 6.10 }$$

with the following properties:

• (a)  
  each series $\widehat R_i(\boldsymbol\ell)\in\widehat{\mathcal H}$ is either integrally summable of some length $n_i\in\mathbb N_0$, $n_i\leqslant i$, with respect to all the previous elements of $\widehat{\mathcal H}$ that are integrally summable of strictly smaller lengths, or an algebraic combination of all the previous series $\widehat R_1(\boldsymbol\ell),\dots,\widehat R_{i-1}(\boldsymbol\ell)$;
• (b)  
  the lengths $i\mapsto n_i$ are not necessarily increasing.

Note that we can also present the block iterated integrally summable transseries $\widehat \varphi$ in Definition 6.8 as the composition

$$\begin{equation} \widehat \varphi=\Bigl(\,\mathop{\underset{i\in\mathbb N}\circ} \widehat\varphi_i\Bigr)\circ\widehat h_0^{-1}, \end{equation} \tag{ 6.11 }$$

where $\widehat h_0$ and $\widehat\varphi_i$ ($i\in\mathbb N$) are as described in that definition.

We remark also that we can associate with $\widehat{\mathcal H}$ a collection (non-unique, depending on a choice of consecutive constants of integration) of analytic integral sums of its terms on $\boldsymbol\ell(V)$:

$$\begin{equation} \mathcal{H}^{h_0}=\{R_i(\boldsymbol\ell)\colon i\in\mathbb N\}. \end{equation} \tag{ 6.12 }$$

The sums in $\mathcal H^{h_0}$ also depend on the choice of the constant in the definition of $h_0$. Here the $R_i(\boldsymbol\ell)$ are the integral sums of given length of the series $\widehat R_i(\boldsymbol\ell)$ ($i\in\mathbb N$) on $\boldsymbol\ell(V)$ with respect to previous integral sums, as in Definition 6.1. They differ by the choice of the constants of integration in inductive definitions of the integral sums. The series $\widehat R_i(\boldsymbol\ell)$ are their common asymptotic expansions.

Proposition 6.9  (uniqueness of the decomposition). Let

$$\begin{equation*} \widehat{\varphi}\in \widehat{\mathcal L}^{\operatorname{id}}(\mathbb R),\quad \widehat\varphi(z)= z+z\widehat R_0(\boldsymbol\ell)+\mathrm{h.o.b.}, \quad \widehat R_0(\boldsymbol\ell)\in \boldsymbol\ell\mathbb R[[\boldsymbol\ell]], \end{equation*}$$

be a block iterated integrally summable transseries with parameters $(\alpha,m,\rho)$ on a petal $V$ of opening $2\pi/(\alpha-1)$. Then a decomposition of it

$$\begin{equation*} \widehat\varphi=\Bigl(\,\mathop{\underset{i\in\mathbb N}\circ} \widehat\varphi_i\Bigr)\circ \widehat h_0^{-1} \end{equation*}$$

with $\widehat h_0(z)$ and elementary changes $\widehat\varphi_i(z)$ as described in Definition 6.8 is unique.

Proof.  Since $\alpha$, $m$, and $\widehat R_0(\boldsymbol\ell)$ are given, the transseries $\widehat T_0(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$ is uniquely determined from equation (6.8), and it has the form $\widehat T_0(\boldsymbol\ell)=-\boldsymbol\ell^m+\mathrm{h.o.t.}$ Then $\widehat h_0(z)$ is uniquely given by (6.9), where $0$ is taken as the constant of formal integration. We note that $\widehat h_0\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is parabolic, so $\widehat\varphi\circ\widehat h_0(z)$ is also parabolic. Let us single out its first block: $\widehat\varphi\circ\widehat h_0(z)=z+z^{\beta_1}\widehat R_1(\boldsymbol\ell)+\mathrm{h.o.b.}$, where $\widehat R_1(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$. Further, since $\widehat\varphi\circ\widehat h_0(z)$ admits a decomposition $\mathop{\underset{i\in\mathbb N}\circ}\widehat\varphi_i$, where the exponents $\beta_i>1$ are strictly increasing, the first change of variables $\widehat\varphi_1$ should have the form $\widehat\varphi_1(z)=z+z^{\beta_1}\widehat R_1(\boldsymbol\ell)$, and moreover, the series $\widehat R_1(\boldsymbol\ell)\in\mathbb R((\boldsymbol\ell))$ is integrally summable of length $0$ or $1$. We now compose on the left with the inverse change $\widehat{\varphi}_1^{-1}$ and, continuing by induction, uniquely determine the changes $\widehat\varphi_i$ ($i\in\mathbb N$). $\square$

Proposition 6.10  (block iterated integral summability of formal reduction to the formal normal form). Let $\widehat f(z)=z-z^\alpha\boldsymbol\ell^m+\mathrm{h.o.t.}$, where $\alpha>1$ and $m\in\mathbb Z$, be a normalized parabolic generalized Dulac transseries belonging to the formal class $(\alpha,m,\rho)$, $\alpha>1$, $m\in\mathbb Z$, $\rho\in\mathbb R$. Let $V$ be a petal of opening $2\pi/(\alpha-1)$. Let $\widehat f_1$ be the formal normal form of this transseries defined in (5.1). Then:

• (i)  
  there exists a formal normalization $\widehat\varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ which reduces $\widehat f$ to the form $\widehat f_1$ and which is block iterated integrally summable with parameters $(\alpha,m,\rho)$ on $V$;
• (ii)  
  every formal normalization $\widehat \varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ reducing $\widehat f$ to $\widehat f_1$ is, up to a precomposition with $\widehat f_c$ ($c\in\mathbb R$), block iterated integrally summable with parameters $(\alpha,m,\rho)$ on $V$, where $f_c=\operatorname{Exp}\biggl(c X_1\dfrac{d}{dz}\biggr).\mathrm{id}$ ($c\in\mathbb R$), and the vector field $X_1$ is given in (5.1).

Proof.  (i) We construct $\widehat\varphi$ algorithmically, eliminating block after block from $\widehat f$, and choosing $0$ as the constant of formal integration for each block. Let $\widehat f=z+z^\alpha \widehat T_0(\boldsymbol\ell)+\mathrm{h.o.b.}$, where $\widehat T_0(\boldsymbol\ell)=-\boldsymbol\ell^m+\mathrm{h.o.t.}$ is a $\log$-Gevrey series of order strictly greater than $(\alpha- 1)/2$ on $V$. We first reduce $\widehat f$ to an auxiliary normal form $\widehat f_2$, given as the time-one map along a trajectory of the formal vector field:

$$\begin{equation} \widehat f_2(z)=\operatorname{Exp} \biggl(\frac{z^\alpha\widehat T_0(\boldsymbol\ell)} {1+(\alpha/2)z^{\alpha-1}\widehat T_0(\boldsymbol\ell)+ b z^{\alpha-1}\widehat T_0(\boldsymbol\ell)\boldsymbol\ell}\, \frac{d}{dz}\biggr).\mathrm{id}. \end{equation} \tag{ 6.13 }$$

It can easily be checked that $\widehat f_2$ is again a parabolic generalized Dulac transseries, since all blocks in it are $\log$-Gevrey of order strictly greater than $(\alpha-1)/2$ on the $\boldsymbol\ell$-cusp $\boldsymbol\ell(V)$ in view of the closedness of the $\log$-Gevrey classes with respect to addition, multiplication, and differentiation (Propositions 4.5–4.7). Moreover, $\widehat f_2$ has the unchanged initial block $z^\alpha \widehat T_0(\boldsymbol\ell)$, and $b\in\mathbb R$ is chosen (uniquely) so that $\widehat f_2$ has the same formal invariants $(\alpha,m,\rho)$ as $\widehat f_1$. Also, its formal Fatou coordinate is given by the simple formula

$$\begin{equation*} \widehat\Psi_{2}= \int\frac{z^{-\alpha+1}}{T_0(\boldsymbol\ell)\boldsymbol\ell^2}\, d\boldsymbol\ell-\frac{\alpha}{2}\boldsymbol\ell^{-1}+ b \boldsymbol\ell_2^{-1}. \end{equation*}$$

Therefore, the auxiliary normal form $\widehat f_2$ can be reduced to the normal form $\widehat f_1$ of $\widehat f$ given in (5.1) simply by one additional parabolic change of variables $\widehat h_0^{-1}\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ that can be explicitly expressed as

$$\begin{equation} \begin{aligned} \, \nonumber \widehat h_0^{-1}&:=\widehat \Psi_{2}^{-1}\circ \widehat\Psi_{0} =\biggl(\,\int\frac{z^{-\alpha+1}} {\widehat T_0(\boldsymbol\ell)\boldsymbol\ell^2}\,d\boldsymbol\ell- \frac{\alpha}{2}\boldsymbol\ell^{-1}+b\boldsymbol\ell_2^{-1}\biggr)^{-1} \\ &\qquad\circ\,\biggl(-\int\frac{z^{1-\alpha}}{\boldsymbol\ell^{m+2}}\, d\boldsymbol\ell- \frac{\alpha}{2}\boldsymbol\ell^{-1}+ \biggl(\frac{m}{2}+\rho\biggr)\boldsymbol\ell_2^{-1}\biggr). \end{aligned} \end{equation} \tag{ 6.14 }$$

Here we fix the constant of formal integration to be $0$. Note that $\widehat h_0^{-1}$ belongs to $\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$, since $b$ is chosen so that $\widehat f_2$ and $\widehat f_1$ belong to the same $\widehat{\mathcal L}$-formal class. We now reduce $\widehat f$ to its auxiliary normal form $\widehat f_2$ by a parabolic reduction $\widehat h\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$. Since all blocks in $\widehat f$ and $\widehat f_2$ are integrally summable of length $0$ (are $\log$-Gevrey) and we do not eliminate the first block, it is obvious from the construction of the blockwise changes of variables in the algorithm for eliminating blocks in Lemma 5.4 and from Definition 6.8 that $\widehat h=\mathop{\underset{i\in\mathbb N}\circ}\widehat \varphi_i$, where the $\widehat\varphi_i(z)=z+z^{\beta_i}\widehat R_i(\boldsymbol\ell)$ are elementary changes of variables, the exponents $\beta_i$ are strictly increasing, and the transseries $\widehat R_i(\boldsymbol\ell)$ in the expressions for $\widehat\varphi_i$ ($i\in\mathbb N$) belong to a sequence $\widehat{\mathcal H}$ of integrally summable series and algebraic combinations of them of the form (6.10). We can reach this conclusion by carefully keeping account of the blocks in the reduced transseries $\widehat f$ after each elimination of a block, and also by using the formula (5.3) in Lemma 5.4 for elimination of blocks. It is important to note that in using (5.3) to get elementary changes of variables (containing only one block after $z$) we choose $0$ as the constant of formal integration. Note also by (5.3) that the exponents $p$ in Definition 6.1 of integral summability of the series $\widehat R_i(\boldsymbol\ell)$ are equal to $m$ for all $\widehat R_i(\boldsymbol\ell)\in\widehat{\mathcal H}$ that are integrally summable of some length.

Finally, we reduce $\widehat f_2$ to $\widehat f_1$ by the transformation $\widehat h_0^{-1}$ given above. The transformation $\widehat \varphi:=\widehat h\circ\widehat h_0^{-1}$ reducing $\widehat f$ to $\widehat f_1$ is then block iterated integrally summable in the sense of Definition 6.8. Indeed, let $\widehat R_0(\boldsymbol\ell)\in\boldsymbol\ell R[[\boldsymbol\ell]]$ be the first block of $\widehat\varphi$, $\widehat\varphi(z)=z+z\widehat R_0(\boldsymbol\ell)+\mathrm{h.o.b.}$ It eliminates the first block of $\widehat f$ and is therefore, by Lemma 5.3, connected with the first block $\widehat T_0(\boldsymbol\ell)$ of $\widehat f$ by the formula (6.8).

(ii) In Proposition 6.11 below we prove that if $\widehat \varphi$ is a formal normalization of $\widehat f$, then any other formal normalization of $\widehat f$ has the form $\widehat\varphi\circ \widehat f_c$ ($c\in\mathbb R$). Therefore, any formal normalization $\widehat\varphi$ of $\widehat f$ admits the decomposition (6.11) as its unique decomposition, up to precomposition with $\widehat f_c$. $\square$

The algorithm in Lemma 5.3 for reducing generalized Dulac series to $\widehat{\mathcal L}(\mathbb R)$-normal form $\widehat f_1$ by blockwise eliminations is not unique. The following proposition shows, however, that the formal changes of variables differ from one another in a controlled way.

Proposition 6.11.  Let $\widehat f(z)=z-z^\alpha\boldsymbol\ell^m+\dots$, $\alpha>1$, $m\in\mathbb Z$, be a normalized parabolic generalized Dulac series, and let $\widehat \Psi\in \widehat{\mathcal L}_2^\infty(\mathbb R)$ be its formal Fatou coordinate ²⁰ (with constant term equal to $0)$. Let $\widehat{f}_1(z)$ be the normal form in (5.1), and let

$$\begin{equation*} \widehat\Psi_0(z)=-\int\frac{dz}{z^\alpha\boldsymbol\ell^m}+ \frac{\alpha}{2}\log z+\biggl(\frac{m}{2}+\rho\biggr)\boldsymbol\ell_2^{-1} \end{equation*}$$

be its formal Fatou coordinate (with constant term equal to $0)$. Then:

• (i)  
  the formal change of variables $\widehat\varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ such that $\widehat f=\widehat \varphi\circ \widehat f_1\circ \widehat{\varphi}^{\,-1}$ is uniquely determined up to precomposition with $\widehat f_c$ ( $c\in\mathbb R$ ), where $\widehat f_c=\operatorname{Exp}\biggl(c X_1\dfrac{d}{dz}\biggr).\mathrm{id}$ ($c\in\mathbb R$), and $X_1$ is defined in (5.1);
• (ii)  
  for any formal change of variables $\widehat\varphi$ reducing $\widehat f$ to $\widehat f_1$ there exists a constant $C\in\mathbb R$ such that $\widehat \Psi+C=\widehat \Psi_0\circ \widehat\varphi^{\,-1}$.

The proof is well known and can be found in the Appendix (§ 10).

6.3. The notion of generalized block iterated integral summability

The following definition is introduced for the purpose of describing formal conjugacies between two parabolic generalized Dulac transseries in the same formal class, and not just formal reductions to normal forms (see Proposition 6.14 below).

Definition 6.12.  Let $\widehat \varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$, let $(\alpha,m,\rho)\in\mathbb R_{>1}\times\mathbb Z\times\mathbb R$, and let $V$ be a petal of opening $2\pi/(\alpha-1)$. We say that the transseries $\widehat\varphi$ is generalized block iterated integrally summable on $V$ with parameters $(\alpha,m,\rho)$ if there exists a decomposition

$$\begin{equation} \widehat\varphi=\widehat\varphi_1\circ \widehat\varphi_2^{\,-1}, \end{equation} \tag{ 6.15 }$$

where $\widehat\varphi_{1}$ and $\widehat\varphi_{2}$ are, up to precompositions with $\widehat f_c$ ($c\in\mathbb R$), block iterated integrally summable on $V$ with parameters $(\alpha,m,\rho)$. Here $f_c$ ($c\in\mathbb R$) is the time-$c$ map along trajectories of the $(\alpha,m,\rho)$-model field $X_1$ as in (5.1).

Note that we do not ask for uniqueness of this decomposition.

Proposition 6.13.  Two (normalized) parabolic generalized Dulac germs $f$ and $g$ are $\widehat{\mathcal L}(\mathbb R)$-formally conjugate ²¹ if and only if they belong to the same $\widehat{\mathcal L}(\mathbb R)$-formal class $(\alpha,m,\rho)$.

Proof.  Let $f_1$ be the $(\alpha,m,\rho)$-normal form as in (5.1). Let $\widehat \varphi_1$ be a formal normalization of $f$ that reduces it to $f_1$: $\widehat\varphi_1^{\,-1}\circ \widehat f\circ\widehat\varphi_1=\widehat f_1$. Putting this into the equality $\widehat g=\widehat\varphi^{\,-1}\circ\widehat f\circ\widehat\varphi$, we get that $\widehat \varphi_2:=\widehat \varphi^{\,-1}\circ\widehat\varphi_1\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a normalization of $\widehat g$ to the same normal form $f_1$. On the other hand, if $\widehat \varphi_1\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a formal normalization of $f$ and $\widehat\varphi_2\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a formal normalization of $g$, then $\widehat\varphi:=\widehat\varphi_1\circ\widehat\varphi_2^{\,-1}\in \widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a formal conjugacy of $f$ to $g$. $\square$

Proposition 6.14.  Let $\widehat f$ and $\widehat g$ be two normalized parabolic generalized Dulac transseries belonging to the formal class $(\alpha,m,\rho)$, $\alpha>1$, $m\in\mathbb Z$, $\rho\in\mathbb R$. Let $V$ be a petal of opening $2\pi/(\alpha-1)$. Then:

• (i)  
  there exists a parabolic formal change of variables $\widehat\varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ conjugating ²² $\widehat f$ to $\widehat g$ which is generalized block iterated integrally summable on $V$ with parameters $(\alpha,m,\rho)$;
• (ii)  
  every conjugacy $\widehat \varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ conjugating $\widehat f$ to $\widehat g$ is generalized block iterated integrally summable.

Proof.  (i) Put $\widehat\varphi:=\widehat \varphi_1\circ\widehat\varphi_2^{\,-1}$, where $\widehat\varphi_1\in \widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a reduction of $\widehat f$ to its $(\alpha,m,\rho)$-normal form $\widehat f_1$ and $\widehat\varphi_2\in \widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a reduction of $\widehat g$ to its $(\alpha,m,\rho)$-normal form $\widehat f_1$, and both are block iterated integrally summable with parameters $(\alpha,m,\rho)$ on $V$ by Proposition 6.10.

(ii) Every such element $\widehat\varphi$ can be decomposed as

$$\begin{equation*} \widehat\varphi=\widehat\varphi_1\circ \widehat \varphi_2^{\,-1}, \end{equation*}$$

where $\widehat\varphi_2$ is a reduction of $\widehat g$ to $\widehat f_1$ and $\widehat\varphi_1$ is a reduction of $\widehat f$ to $\widehat f_1$. Indeed, take $\widehat\varphi_1$ to be any normalization of $\widehat f$. By Proposition 6.10, up to a precomposition with $\widehat f_c$ it is block iterated integrally summable with parameters $(\alpha,m,\rho)$. Then $\widehat\varphi^{\,-1}\circ\widehat\varphi_1$ is a normalization of $g$. By Proposition 6.10, up to a precomposition with $\widehat f_c$, it is also block iterated integrally summable with parameters $(\alpha,m,\rho)$. $\square$

The proof of Proposition 6.14 gives us an exact description of possible non- uniqueness of the conjugacy $\widehat\varphi$ of $\widehat g$ and $\widehat f$, namely,

$$\begin{equation*} \widehat\varphi_1\circ \widehat f_c\circ\widehat\varphi_2^{\,-1}, \qquad c\in\mathbb R, \end{equation*}$$

where $\widehat \varphi_1$ and $\widehat\varphi_2$ are any normalizations of $\widehat f$ and $\widehat g$, respectively. Indeed, by Proposition 6.11, such normalizations are unique up to precomposition with $\widehat f_c$ ($c\in\mathbb R$).

Note also that the decomposition (6.15) of a generalized block iterated integrally summable change of variables $\widehat\varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ may not be unique. Indeed, suppose that $\widehat\varphi$ is a conjugacy between two parabolic generalized Dulac transseries that is generalized block iterated integrally summable. In general, there may exist different pairs of parabolic generalized Dulac transseries $\widehat f$, $\widehat g$ and $\widehat f_1$, $\widehat g_1$ that are conjugate by $\widehat\varphi$.

We denote the class of generalized block iterated integrally summable parabolic transseries with parameters $(\alpha,m,\rho)$ on some petal ²³ $V$ of opening $2\pi/(\alpha-1)$ by

$$\begin{equation*} \widehat{\mathcal S}^{\alpha,m,\rho}(V)\subset \widehat{\mathcal L}^{\,\operatorname{id}}(\mathbb R). \end{equation*}$$

For simplicity of notation, below we will not specify the sector $V$, although the notion of (generalized, iterated) integral summability is always relative to some sector.

As a trivial case of Proposition 6.10, $\widehat\varphi:=\mathrm{id}$ is trivially block iterated integrally summable for any triple of parameters and on any petal $V$. Therefore, the block iterated integrally summable parabolic transseries with parameters $(\alpha,m,\rho)$ form a subclass of the class of generalized block iterated integrally summable parabolic transseries with parameters $(\alpha,m,\rho)$.

Moreover, parabolic generalized Dulac series $\widehat f(z)=z-z^{\alpha}\boldsymbol\ell^m+\dotsb$ belong to the class $\widehat{\mathcal S}^{\alpha,m,\rho}(V)$, where $V$ is any petal of opening strictly less than $2\pi/(\alpha-1)$, for any triple of parameters $(\alpha,*,*)$. This can be proved by decomposition into a sequence of elementary changes whose blocks are integrally summable of length $0$, using Proposition 8.2.

Remark 6.15.  We note that $\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ is a group under composition. However, we are not able to prove the group property for the class $\widehat{\mathcal S}^{\alpha,m,\rho}(V)$ of all parabolic generalized block iterated integrally summable series on $V$ with parameters $(\alpha,m,\rho)$. We are not even able to prove the group property for the subclass of all parabolic transseries that are realizable as formal conjugacies of two parabolic generalized Dulac transseries in the formal class $(\alpha,m,\rho)$.

Indeed, even if $\widehat \varphi\in\widehat{\mathcal S}^{\alpha,m,\rho}(V)$ can be realized as a conjugacy between two parabolic generalized Dulac series $\widehat f$ and $\widehat g$ in the class $(\alpha,m,\rho)$, and $\widehat \eta\in \widehat{\mathcal S}^{\alpha,m,\rho}(V)$ can be realized as a conjugacy between another two parabolic generalized Dulac series $\widehat h$ and $\widehat k$ in the same formal class $(\alpha,m,\rho)$, we see no reason in general to believe that the composition $\widehat \eta\circ\widehat\varphi$ will be a conjugacy between two parabolic generalized Dulac series if the pairs are not chainable. But if they do form a chain, that is, if $\widehat g=\widehat h$, then $\widehat\varphi\circ\widehat\eta$ is again a conjugacy between $\widehat f$ and $\widehat k$, and by Proposition 6.14, this composition is indeed generalized block iterated integrally summable with parameters $(\alpha,m,\rho)$.

Moreover, we do not assert that the formal inverse of a parabolic generalized block iterated integrally summable series is again a parabolic generalized block iterated integrally summable series. However, if $\widehat\varphi$ is a parabolic generalized block iterated integrally summable series with parameters $(\alpha,m,\rho)$ that can be realized as a formal conjugacy between two parabolic generalized Dulac series $\widehat f$ and $\widehat g$, then its formal inverse $\widehat\varphi^{\,-1}$ is a formal conjugacy between $\widehat g$ and $\widehat f$. Furthermore, these Dulac series belong to the formal class $(\alpha,m,\rho)$. Therefore, by Proposition 6.14, $\widehat\varphi^{\,-1}$ is again generalized block iterated integrally summable with parameters $(\alpha,m,\rho)$.

Let $f$ be a parabolic generalized Dulac germ and let

$$\begin{equation*} \widehat f(z)=z-z^\alpha\boldsymbol\ell^m+\mathrm{h.o.t.},\qquad \alpha>1,\quad m\in\mathbb Z, \end{equation*}$$

be its generalized Dulac expansion. Let $\widehat \Psi\in\widehat{\mathcal L}_2^\infty(\mathbb R)$ be the formal Fatou coordinate for $f$ and let $\widehat \varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ be a formal change of variables that reduces it to its $\widehat{\mathcal L}(\mathbb R)$-formal normal form $\widehat f_1(z)=\operatorname{Exp}(X_1).\mathrm{id}$, where $X_1$ is the vector field in (5.1). In this section we suppose that we have already proved the existence of sectorially analytic Fatou coordinates $\Psi_{j}^\pm$ on the petals $V_j^{\pm}$ ($j\in\mathbb Z$) of a standard quadratic domain $\mathcal R_C$. In fact, this will be proved by a construction in the proof of Theorem THA, (i), in § 8. For this we use only the assumptions that $f$ is analytic on $\mathcal R_C$ and that it admits a uniform estimate (2.5) in the first two terms. It is well known that the relation between the reduction to normal form and the Fatou coordinate is given through the Fatou coordinate $\Psi_0$ of the formal normal form $\widehat f_1$, which is defined and analytic on the whole of $\mathcal R_C$ (up to an additive constant in $\Psi$, that is, up to a precomposition of $\varphi$ with the map $f_c=\operatorname{Exp}(cX_1).\mathrm{id}$, $c\in\mathbb R$):

$$\begin{equation*} \widehat\Psi=\Psi_0\circ\widehat\varphi,\quad \Psi_j^{\pm}=\Psi_0\circ\varphi^{\pm}_j\quad\text{on } V^{\pm}_j,\quad j\in\mathbb Z. \end{equation*}$$

Sectorial analyticity of the Fatou coordinate therefore immediately implies sectorial analyticity of the changes of variables $\varphi_{\pm}^j$ on the sectors $V_j^{\pm}$ ($j\in\mathbb Z$).

In Propositions 7.1 and 7.4 in §§ 7.1 and 7.2 we state that by the choice of integral sections the formal Fatou coordinate and the formal conjugacy can be made the unique sectional asymptotic expansions of the sectorial Fatou coordinates and conjugacies, respectively. For more on section functions and sectional transserial expansions, see [17] and Definition 10.1 in the Appendix (in the complex version).

7.1. Sectorially analytic Fatou coordinates

In Proposition 7.1 below, by integral asymptotic expansion we mean a sectional asymptotic expansion where section functions at limit ordinal steps (that is, sums of series in $\boldsymbol\ell$ in each block) are chosen as integral sums of length $1$ , as defined in Definition 6.1, 2).

Proposition 7.1  (integral asymptotic expansion of the Fatou coordinate of a parabolic generalized Dulac germ). Let $f$ be a parabolic generalized Dulac germ, and let $\widehat f(z)=z-z^\alpha\boldsymbol\ell^{m}+\mathrm{h.o.t.}$, $\alpha>1$, $m\in\mathbb Z$, be its generalized Dulac expansion. Let ²⁴ $\Psi_j^{\pm}(z)\sim_{z\to 0}- \dfrac{1}{\alpha-1}z^{-\alpha+1}\boldsymbol\ell^{-m}$ be its analytic Fatou coordinates on the petals $V_{\pm}^j$ ($j\in\mathbb Z$), which are constructed in Theorem THA.

(i) Up to a constant, the Fatou coordinates $\Psi_{\pm}^j$ admit a common integral asymptotic expansion $\widehat\Psi$ as $z\to 0$ on $V_{\pm}^j$.

(ii) Let $\Phi_{\pm}^j$ be any other Fatou coordinates for $f$ on petals $V_{\pm}^j$ such that

$$\begin{equation*} \Phi_j^{\pm}(z)\sim_{z\to 0}-(\alpha-1)^{-1}z^{-\alpha+1}\boldsymbol\ell^{-m}. \end{equation*}$$

Then on every petal $V_{\pm}^j$ the Fatou coordinate $\Phi_{\pm}^j$ differs from $\Psi_{\pm}^j$ only by an additive constant. Up to a constant, the coordinates $\Phi_{\pm}^j$ also admit $\widehat\Psi$ as their integral asymptotic expansion as $z\to 0$ on $V_{\pm}^j$.

Proof.  (i) The proof is similar to the proof of the theorem for parabolic Dulac germs in [17]. The only difference is that the coefficients now are not polynomials in $\boldsymbol\ell^{-1}$, but $\log$-Gevrey sums on $\boldsymbol\ell$-cusps ($\boldsymbol\ell$-images of petals). Suppose that $f(z)=z-z^{\alpha} R_1^{j,\pm}(\boldsymbol\ell)+O(z^{\alpha+\delta})$, $z\in V_j^{\pm}$, $\alpha>1$, $\delta>0$. Here the $R_1^{j,\pm}(\boldsymbol\ell)$ are analytic $\log$-Gevrey sums of orders strictly greater than $(\alpha-1)/2$ of their formal analogue, the series $\widehat R_1(\boldsymbol\ell)$, on the $\boldsymbol\ell$-cusps $\boldsymbol\ell(V_j^{\pm})$, where the $V_{j}^{\pm}$ are petals of opening $2\pi/(\alpha-1)$. Thus, for the Fatou coordinate of a generalized Dulac germ the integral sums of length $1$ on $\boldsymbol\ell$-cusps $\boldsymbol\ell(V_{j}^{\pm})$ in Definition 6.1 (and not the integral sums considered earlier) turn out to be the right choice of sections in each block. By Propositions 4.5–4.7 in § 4 the class of all $\log$-Gevrey summable series of order strictly greater than some fixed $r>0$ on $\boldsymbol\ell(V_{j}^{\pm})$ remains closed with respect to addition, multiplication, and differentiation. Note also that the germ $\boldsymbol\ell\mapsto R_1^{j,\pm}(\boldsymbol\ell)$ in the first block of $f$ does not have an accumulation of singularities at $0$ in $\boldsymbol\ell(V_j^{\pm})$ ($j\in\mathbb Z$). Indeed, if it had, then its asymptotic expansion $\widehat R_1(\boldsymbol\ell)$ in the power-logarithmic scale on $\boldsymbol\ell(V_{j}^{\pm})$ would be $0$. Therefore, the algorithm for constructing the Fatou coordinate block-by-block by Taylor expansion of the Abel equation works as in the proof of the theorem in [17].

(ii) Fix one petal, say, $V_j^{+}$ ($j\in\mathbb Z$). Then

$$\begin{equation} \Phi_j^+\circ (\Psi_j^+)^{-1}(w)=w+o(w),\qquad w\in\Psi_j^+(V_j^+),\quad |w|\to\infty. \end{equation} \tag{ 7.1 }$$

Indeed, $\Psi_+^j(z)$ is injective on $V_{+}^j$ sufficiently close to $0$. This can easily be checked in the logarithmic chart $w=\log z$, where the petal $V_+^j$ is a convex set. Suppose that $\widetilde \Psi_+^j(w):=\Psi_+^j(e^w)$ is not injective in the image of $V_j^+$ in the logarithmic chart in a neighbourhood of the set $\{\operatorname{Re}w=-\infty\}$. By the complex Rolle's theorem, there exist sequences $(w_n)$ and $(v_n)$ such that $\operatorname{Re}w_n,\operatorname{Re}v_n\to-\infty$ and such that $\operatorname{Re}\bigl((\widetilde \Psi_j^+)'(w_n)\bigr)=\operatorname{Im}\bigl((\widetilde \Psi_j^+)'(v_n)\bigr)=0$. By the continuity of the real and imaginary parts of the Fatou coordinate, we have $(\Psi_+^j)'(0)=0$. This contradicts the condition $(\Psi_+^j)'(z)\sim z^{-\alpha}\boldsymbol\ell^{-m}$ as $z\to 0$ on $V_+^j$, which holds by the construction of the Fatou coordinate $\Psi_+^j$ in Theorem THA.

Now from the asymptotic behaviour of $\Psi_+^j(z)$ we easily get the exact asymptotic behaviour of the inverse map $(\Psi_+^j)^{-1}(w)$ as $|w|\to\infty$, $w\in\Psi_+^j(V_j^+)$. It can be computed directly by letting

$$\begin{equation*} w=\Psi_j^+(z)=-(\alpha-1)^{-1}z^{-\alpha+1}\boldsymbol\ell^{-m} \bigl(1+o(1)\bigr), \end{equation*}$$

taking the logarithm, and expressing $z$ in terms of $w$. Then the assertion (7.1) follows simply by composing with

$$\begin{equation*} \Phi_j^+(z)=-(\alpha-1)^{-1}z^{-\alpha+1}\boldsymbol\ell^{-m}+ o(z^{-\alpha+1}\boldsymbol\ell^{-m}),\qquad z\to 0. \end{equation*}$$

Since $\Psi_+^j(V_j^+)$ contains $\{w\colon\operatorname{Re}w>R\}$ for some $R>0$, and since, as in the proof of Lemma 7.5 below, the map $\Phi_+^j\circ (\Psi_+^j)^{-1}-\mathrm{id}$ is periodic on this domain, we can extend it by periodicity to the whole of $\mathbb C$ and conclude, using Liouville's theorem, that $\Phi_+^j-\Psi_+^j=C_j$, $C_j\in\mathbb R$. $\square$

7.2. Sectorially analytic normalizations

Let

$$\begin{equation*} \widehat\varphi\in\widehat{\mathcal S}^{\alpha,m,\rho}(V),\qquad \widehat\varphi(z)=z+z\widehat R_0(\boldsymbol\ell)+\mathrm{h.o.b.}, \quad\text{where}\quad \widehat R_0(\boldsymbol\ell)\in\boldsymbol\ell\mathbb R[[\boldsymbol\ell]], \end{equation*}$$

be a block iterated integrally summable series with parameters $(\alpha,m,\rho)\in\mathbb R_{>1}\times\mathbb Z\times\mathbb R$ on a petal $V$ of opening $2\pi/(\alpha-1)$ in the sense of Definition 6.8. Let $\widehat h_0(z)$ be as defined from $\widehat R_0(\boldsymbol\ell)$ in (6.9) in Definition 6.8. Let $h_0(z)$ be the analytic germ on $V$ given by the analytic variant of the formula (6.9) for $\widehat h_0(z)$:

$$\begin{equation} \begin{aligned} \, \nonumber h_0(z)&:=\biggl(-\int_{\boldsymbol\ell_0}^{\boldsymbol\ell} \frac{e^{(\alpha-1)/\eta}}{\eta^{m+2}}\,d\eta- \frac{\alpha}{2}(\boldsymbol\ell^{-1}-\boldsymbol\ell_0^{-1})+ \biggl(\frac{m}{2}+\rho\biggr)(\boldsymbol\ell_2^{-1}+ \log\boldsymbol\ell_0)\biggr)^{-1} \\ &\qquad\circ \biggl(\,\int_{\boldsymbol\ell_0}^{\boldsymbol\ell} \frac{e^{(\alpha-1)/\eta}}{T_0(\eta)\eta^{2}}\,d\eta- \frac{\alpha}{2}(\boldsymbol\ell^{-1}-\boldsymbol\ell_0^{-1})+ b(\boldsymbol\ell_2^{-1}+\log\boldsymbol\ell_0)\biggr), \qquad z\in V. \end{aligned} \end{equation} \tag{ 7.2 }$$

Here $T_0$ is the $\log$-Gevrey sum of the expansion $\widehat T_0$ in (6.9) on the cusp $\boldsymbol\ell(V)$, and $\boldsymbol\ell_0\in\boldsymbol\ell(V)$. Note that $h_0$ is unique up to the choice of a constant of integration $\boldsymbol\ell_0\in\boldsymbol\ell(V)$.

Let $\widehat\varphi\circ\widehat h_0$ decompose as in Definition 6.8, as

$$\begin{equation} \widehat\varphi\circ\widehat h_0= \mathop{\underset{i\in\mathbb N}\circ}\widehat\varphi_i,\qquad \widehat\varphi_i(z)=z+z^{\beta_i}\widehat R_i(\boldsymbol\ell), \end{equation} \tag{ 7.3 }$$

where the $\widehat\varphi_i(z)$ are elementary changes of coordinates, the exponents $\beta_i>1$ are strictly increasing, either a finite sequence or tending to $+\infty$, and the series $\widehat R_i(\boldsymbol\ell)$ ($i\in\mathbb N$) belong to a sequence $\widehat{\mathcal H}$ of the form (6.10) of integrally summable series and algebraic combinations of them.

Definition 7.2  (block iterated integral asymptotic expansion). Let $\varphi(z)=z+o(z)$ be analytic on some open petal $V$ of opening $2\pi/(\alpha-1)$ for some $\alpha>1$, and let the series $\widehat\varphi\in \widehat{\mathcal S}^{\alpha,m,\rho}(V)$ be block iterated integrally summable on $V$ with respect to parameters $(\alpha,m,\rho)$. Let $\widehat h_0\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$, and let $h_0$ be the analytic transformation on $V$ defined in (7.2). Let $\widehat{\mathcal H}$ be the sequence described above in the decomposition $\widehat\varphi\circ\widehat h_0$.

We say that the germ $\varphi$, up to a precomposition with $f_c$ ($c\in\mathbb R$), admits $\widehat\varphi$ as its block iterated integral asymptotic expansion on $V$ with parameters $(\alpha,m,\rho)$ if, for every constant of integration $\boldsymbol\ell_0\in\boldsymbol\ell(V)$ in (7.2) for $h_0$, and for every integral sum $\mathcal H^{h_0}=\{R_i(\boldsymbol\ell)\colon i\in\mathbb N\}$ of the sequence $\widehat {\mathcal H}$ on $\boldsymbol\ell(V)$, there exists a $c\in\mathbb R$ such that $(\varphi\circ f_c)\circ h_0$ can be written as the following infinite asymptotic ²⁵ composition:

$$\begin{equation} (\varphi\circ f_c)\circ h_0\sim \mathop{\underset{i\in\mathbb N}\circ}\varphi_i,\qquad z\in V,\quad z\to 0, \end{equation} \tag{ 7.4 }$$

where ²⁶ $\varphi_i(z)=z+z^{\beta_i}R_i(\boldsymbol\ell)$. Here the phrase asymptotic composition $\sim$ in (7.4) means that there exists a sequence $(\gamma_n)_n$ of strictly increasing positive numbers tending to $+\infty$ such that, for every $n\in\mathbb N$,

$$\begin{equation} \begin{gathered} \, (\varphi\circ f_c)\circ h_0=\varphi_1\circ\cdots\circ \varphi_n+o(z^{\gamma_n}), \\ \text{that is,}\quad\varphi_n^{-1}\circ\cdots\circ \varphi_1^{-1}\circ\bigl((\varphi\circ f_c)\circ h_0\bigr)= z+o(z^{\gamma_n}),\quad z\in V,\quad z\to 0. \end{gathered} \end{equation} \tag{ 7.5 }$$

For the change $\varphi_i(z)$ the series $\widehat\varphi_i(z)=z+z^{\beta_i}\widehat R_i(\boldsymbol\ell)$ ($i\in\mathbb N$) are the Poincaré asymptotic expansions, since $R_i(\boldsymbol\ell)$ is an integral sum of $\widehat R_i(\boldsymbol\ell)$ of some length. This can be checked using Definition 6.1.

Remark 7.3.  We do not claim uniqueness of a block iterated integral asymptotic expansion of a general analytic germ $\varphi$ on $V$. However, if $\varphi$ is a normalization on $V$ of a parabolic Dulac germ $f$ in a formal class $(\alpha,m,\rho)$ that admits a block iterated integral asymptotic expansion, then its block iterated integral asymptotic expansion on $V$ with parameters $(\alpha,m,\rho)$ (if it exists) is unique up to a precomposition with $\widehat f_c$, where $f_c$ is the time-$c$ map along trajectories of the $(\alpha,m,\rho)$-model field $X_1$ in (5.1). Indeed, every block iterated integral asymptotic expansion of an analytic normalization on $V$ is a formal normalization of the series $\widehat f$, by (7.5) and since $\varphi_i$ admits $\widehat\varphi_i$ as its Poincaré asymptotic expansion. On the other hand, a formal normalization of $\widehat f$ is by Proposition 6.11 unique up to a precomposition with $\widehat f_c$ ($c\in\mathbb R$).

Proposition 7.4  (block iterated integral asymptotic expansions of analytic reductions to the formal normal form). Let $f(z)=z-z^\alpha \boldsymbol\ell^m+o(z^\alpha \boldsymbol\ell^m)$, $\alpha>1$, $m\in\mathbb Z$, be a parabolic generalized Dulac germ on $\mathcal R_C$ in the formal class $(\alpha,m,\rho)$. Let $\widehat f$ be its generalized Dulac expansion, and let $V_j^{\pm}$ ($j\in\mathbb Z$) be the petals of $f$ on $\mathcal R_C$. Let $\widehat\varphi\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ be the formal change of variables that reduces $\widehat f$ to its $\widehat{\mathcal L}(\mathbb R)$-normal form $\widehat f_1$ and is block iterated integrally summable (see Proposition 6.10).

(i) There exist on the open petals $V_j^{\pm}$ analytic changes of variables $\varphi_{\pm}^j(z)=z+o(z)$ ($j\in\mathbb Z$) conjugating ²⁷ $f$ to $f_1$ which, up to a precomposition with $f_c$ ($c\in\mathbb R$), admit the formal change of variables $\widehat\varphi$ as their block iterated integral asymptotic expansion with parameters $(\alpha,m,\rho)$ as $z\to 0$ on $V_j^\pm$. Here different choices of the constant $c$ are connected with different choices of the constants in integral sums.

(ii) Let $\eta_j^{\pm}$ be any other analytic changes of variables on $V_j^{\pm}$ conjugating $f$ to $f_1$, with $\eta_j^{\pm}(z)=z+o(z)$, $z\in V_j^{\pm}$. Then there exist constants $c_j$ ($j\in\mathbb Z$) such that $\eta_j^{\pm}=\varphi_j^{\pm}\circ f_{c_j}$. Moreover, up to a precomposition with $f_c$ ($c\in\mathbb R$) the $\eta_j^{\pm}$ admit $\widehat\varphi$ as a block iterated integral asymptotic expansion with parameters $(\alpha,m,\rho)$ as $z\to 0$ on $V_j^{\pm}$.

Due to the importance (in view of Definition 2.4) of analytic normalization of parabolic Dulac germs on a standard quadratic domain, we reformulate the assertion (ii) of Proposition 7.4 as the separate Lemma 7.5 (whose proof is in the Appendix, § 10), which shows that in Definition 2.4 we do not need any assumption about the asymptotic expansion of the normalization in the power-logarithmic scale, since it necessarily follows from the construction.

Lemma 7.5  (analytic normalizations tangent to the identity map for parabolic Dulac germs necessarily admit power-logarithmic asymptotic expansions). Let $f$ be a normalized parabolic generalized Dulac germ in the formal class $(\alpha,m,\rho)$ and let $f_1$ be its $\widehat{\mathcal{L}}^{\,\mathrm{id}}(\mathbb R)$-formal normal form, with formal conjugacy ²⁸ $\widehat\varphi\in\widehat{\mathcal{L}}^{\,\mathrm{id}}(\mathbb R)$. Let $V_j^{\pm}$ be petals of $f$, as in Theorem THA. Let $h_j^{\pm}$ be analytic conjugacies of $f$ to $f_1$ on $V_j^{\pm}$ ($j\in\mathbb Z$), with $h_j^{\pm}(z)=z+o(z).$ ²⁹ Then the germs $h_j^{\pm}$ admit the formal conjugacy $\widehat{\varphi}\in\widehat{\mathcal L}^{\,\mathrm{id}}(\mathbb R)$ as their common block iterated integral asymptotic expansion with parameters $(\alpha,m,\rho)$ as $z\to 0$ on $V_j^{\pm}$, up to a precomposition with $f_{c_j}$, $c_j\in\mathbb R$ ($j\in\mathbb Z$).

The same statement holds for a $\widehat{\mathcal L}_2^{\,\mathrm{id}}(\mathbb R)$-formal normal form, with formal conjugacy $\widehat\varphi\in\widehat{\mathcal L}_2^{\,\mathrm{id}}(\mathbb R)$.

Proof of Proposition 7.4. We prove (i).

Step 1: existence of a petalwise analytic normalization $\varphi_j^\pm$ on $V_j^\pm$. The existence of analytic conjugating changes $\varphi_j^{\pm}$ on petals $V_j^{\pm}$ ($j\in\mathbb Z$) of opening $2\pi/(\alpha-1)$, where the petals are as in Theorem THA, is proved by the existence (by construction) of an analytic Fatou coordinate $\Psi_j^{\pm}$ on $V_j^{\pm}$ in the proof of Theorem THA in § 8. Let $\Psi_0$ be the Fatou coordinate of the normal form $f_1$ on $V_j^{\pm}$, and for $f$ let $\Psi_j^{\pm}$ be the analytic Fatou coordinate on $V_j^{\pm}$ ($j\in\mathbb Z$) constructed in the proof of Theorem THA. Then

$$\begin{equation*} \Psi_j^{\pm}(z),\Psi_0(z)\sim_{z\to 0}-\frac{1}{\alpha-1} z^{-\alpha+1}\boldsymbol\ell^{-m}. \end{equation*}$$

Consequently, the map $\varphi_j^\pm$ defined by

$$\begin{equation} \varphi_j^\pm(z):=(\Psi_j^{\pm})^{-1}\circ \Psi_0(z)=z+o(z),\qquad z\in V_j^{\pm}, \end{equation} \tag{ 7.6 }$$

is a petalwise analytic normalization of $f$.

In the definition (7.6), we might have chosen another Fatou coordinate of $f$ differing from $\Psi_j^\pm$ by some additive constants on petals. As in the final step of the proof of Lemma 7.5, this gives a conjugacy $\widetilde\varphi_j^\pm=\varphi_j^\pm\circ f_{c_j}$ ($j\in\mathbb Z$) that differs from $\varphi_j^\pm$ by a precomposition with $f_{c_j}$, $c_j\in\mathbb R$.

Step 2: proof that there are block iterated integral asymptotic expansions of analytic normalizations. We now prove that any conjugacy $\varphi_j^{\pm}$ tangent to the identity admits the block iterated integrally summable formal conjugacy $\widehat\varphi\in\widehat S^{\alpha,m,\rho}(V_j^\pm)$ as its block iterated integral asymptotic expansion on $V_j^\pm$ with parameters $(\alpha,m,\rho)$, in the sense of Definition 7.2.

Recall that $\widehat\varphi$ decomposes uniquely (see (7.3)) as:

$$\begin{equation*} \widehat\varphi\circ\widehat h_0= \mathop{\underset{i\in\mathbb N}\circ}\widehat\varphi_i, \end{equation*}$$

where $\widehat h_0^{-1}$ is a formal normalization of $\widehat f_2$ (defined in (6.13)) given by (6.14), with $0$ taken as the constant of formal integration for uniqueness and with $b\in\mathbb R$ such that the expression for $\widehat h_0^{-1}$ does not contain iterated logarithms. Moreover, the

$$\begin{equation} \widehat\varphi_i(z)=z+z^{\beta_i}\widehat R_i(\boldsymbol\ell),\qquad i\in\mathbb N, \end{equation} \tag{ 7.7 }$$

with strictly increasing exponents $\beta_i>1$, are blockwise elementary changes reducing $\widehat f$ to $\widehat f_2$ and obtained as in Lemma 5.4, with $0$ taken as the constant of formal integration in (5.3) for $\widehat R_i(\boldsymbol\ell)$, in order to really get elementary changes of variables. Let $\widehat{\mathcal H}=\{\widehat R_i(\boldsymbol\ell)\colon i\in\mathbb N\}$ be as described in (7.3).

On the other hand, let $(h_0^{j,\pm})^{-1}(z)$, given by the formula (7.2) for a fixed choice of the constant of integration $\boldsymbol\ell_0\in\boldsymbol\ell(V_j^\pm)$, be an analytic normalization of the germ $f_2^{j,\pm}$ on $V_j^\pm$. Compared to the formal formula (6.14), it is an analytic analogue of $\widehat h_0^{-1}(z)$. Here $f_2^{j,\pm}(z)$, $z\in V_j^{\pm}$, are defined by (6.13):

$$\begin{equation*} f_2^{j,\pm}(z)= \operatorname{Exp}\biggl(\frac{z^\alpha T_0^{j,\pm}(\boldsymbol\ell)} {1+(\alpha/2)z^{\alpha-1}T_0^{j,\pm}(\boldsymbol\ell)+ bz^{\alpha-1}T_0^{j,\pm}(\boldsymbol\ell)\boldsymbol\ell}\, \frac{d}{dz}\biggr).\mathrm{id}, \end{equation*}$$

where $\widehat T_0(\boldsymbol\ell)$ is the first block of the generalized Dulac expansion $\widehat f$, and the $T_0^{j,\pm}(\boldsymbol\ell)$ are its $\log$-Gevrey sums on $\boldsymbol\ell(V_j^{\pm})$ ($j\in\mathbb Z$). Note that $f_2^{j,\pm}$ is not necessarily a parabolic generalized Dulac germ, but only a collection of germs on petals which do not necessarily glue together on intersections of the petals.

Let $\mathcal H^{h_0^{j,\pm}}:=\{R_i^{j,\pm}(\boldsymbol\ell)\colon i\in\mathbb N\}$ be a fixed sequence of integral sums on the cusps $\boldsymbol\ell(V_j^{\pm})$ for a fixed choice of the constants of integration $\boldsymbol\ell_0\in\boldsymbol\ell(V_j^{\pm})$ in the expressions for $h_0^{j,\pm}(z)$, and for fixed choices of the constants of integration $\boldsymbol\ell_0\in\boldsymbol\ell(V_j^{\pm})$ in the successive integral sums (in each analytic integral in the formula (5.3) defining the germs $R_i^{j,\pm}(\boldsymbol\ell)$; see Definition 6.1). We now let

$$\begin{equation} \varphi_i^{j,\pm}(z):=z+z^{\beta_i}\widehat R_i(\boldsymbol\ell),\qquad i\in\mathbb N,\quad z\in V_j^{\pm}. \end{equation} \tag{ 7.8 }$$

The procedure is explained in more detail in the following algorithm for block-by- block changes reducing $\widehat f$ to $\widehat f_2$, carried out in each step formally and analytically in parallel.

Description of the block-by-block algorithm. For simplicity we do not indicate here the indices of petals, but just let $V:=V_j^{\pm}$, for any petal. Similarly, $f_2:=f_2^{j,\pm}$, $\varphi_i:=\varphi_i^{j,\pm}$, $h_0:=h_0^{j,\pm}$ are analytic on $V$.

We construct the sequence $(\widehat\varphi_i)_{i\in\mathbb N}$ in (7.7) of formal elementary changes of variables and simultaneously the sequence $(\varphi_i)_{i\in\mathbb N}$ in (7.8) of analytic changes of variables on $V$ by blockwise eliminations that transform $f$ into $f_2$ on $V$.

0. Put $\widehat {^1\!f}:=\widehat f$ and $^1\!f:=f$.

1. Let $i\in\mathbb N$. By Lemma 5.4 we find the $i$th elementary change $\widehat\varphi_{i}(z)=z+z^{\beta_{i}} \widehat R_{i}(\boldsymbol\ell)$, where

$$\begin{equation} \widehat R_i(\boldsymbol\ell)=-e^{-(\beta_i-\alpha)/\boldsymbol\ell} \boldsymbol\ell^{m}\int e^{(\beta_i-\alpha)/\boldsymbol\ell} \boldsymbol\ell^{-2m-2}\widehat T_i(\boldsymbol\ell)\,d\boldsymbol\ell. \end{equation} \tag{ 7.9 }$$

In the first step ($i=1$), $\widehat T_1(\boldsymbol\ell)$ is the difference of the series in $\boldsymbol\ell$ in the second blocks (which are $\log$-Gevrey on $V$) of $\widehat f$ and of $\widehat f_2$. Further, for $i\in\mathbb N$, $i>1$, we compute

$$\begin{equation*} \widehat {^i\!f}(z):=\widehat \varphi_{i-1} \circ \widehat {^{i-1}\!f} \circ \widehat \varphi_{i-1}^{\,-1}(z)=\widehat f_2(z)+ \bigl(z^{\gamma_i}\widehat T_i(\boldsymbol\ell)+\mathrm{h.o.b.}\bigr). \end{equation*}$$

Here the $\gamma_i> \alpha$ strictly increase to $+\infty$, and $\widehat T_i(\boldsymbol\ell)$ (like every other coefficient of the new series $\widehat {^i\!f}$) is an algebraic combination (with the operations $+$, $\,\cdot\,$, $d/d\boldsymbol\ell$) of series integrally summable of length strictly less than $i$ (that is, an algebraic combination of series $\widehat R_j(\boldsymbol\ell)$ appearing in previous elementary changes, $1\leqslant j<i$, of coefficients of the series $\widehat {^{i-1}\!f}$, and of coefficients of the series $\widehat f_2$). The constant of formal integration is always taken to be $0$, in order to get elementary, one-block changes of variables $\widehat\varphi_i$.

We put the sequence of $\widehat R_i(\boldsymbol\ell)$ obtained in this formal step-by-step construction in the set $\widehat{\mathcal H}$. The corresponding integral exponents are by (7.9) equal to $\beta_i$, and they are strictly increasing to $+\infty$ by the algorithm.

2. On the other hand, (7.9) can be simultaneously solved analytically on the $\boldsymbol\ell$-cusp $\boldsymbol\ell(V)$,

$$\begin{equation} R_i(\boldsymbol\ell)=-e^{-(\beta_i-\alpha)/\boldsymbol\ell} \boldsymbol\ell^{m}\int_{*}^{\boldsymbol\ell} e^{(\beta_i-\alpha)/\eta} \eta^{-2m-2}\,T_i(\eta)\,d\eta,\qquad \boldsymbol\ell\in\boldsymbol\ell(V). \end{equation} \tag{ 7.10 }$$

Here $*$ is $0$ if $(\beta_i-\alpha,\operatorname{ord}(\widehat T_i)-2m-1)\succ (0,0)$ (lexicographically) in this step, and $\boldsymbol\ell_0\in\boldsymbol\ell(V)$ otherwise (if the integrand is not bounded at $0$). We remark that in the first finitely many steps we may choose $\boldsymbol\ell_0$ arbitrarily, which changes the constant of integration. The integration path is not important as long as it stays in $\boldsymbol\ell(V)$, since the integrand is analytic on $\boldsymbol\ell(V)$ and the cusp is simply connected.

Since $\widehat T_i(\boldsymbol\ell)$ is an algebraic expression (with respect to the operations $+$, $\,\cdot\,$, $d/d\boldsymbol\ell$) of integrally summable series in $\widehat{\mathcal H}$ of lengths strictly smaller than $i$ (of coefficients of $\widehat {^{i-1}\!f}$, of $\widehat f_2$, and of the series $\widehat R_j(\boldsymbol\ell)$ in all the previous elementary changes, $1\leqslant j< i$), after we have chosen the constants of integration in the previous steps, it is uniquely summable by Proposition 6.5. Under the assumption that we have chosen some constants of integration for the germs $R_j(\boldsymbol\ell)$ in (7.10) in all the previous steps $j<i$, this sum is uniquely determined and analytic on $V$, and we denote it by $T_i(\boldsymbol\ell)$. Now we again freely chose the $i$th constant of integration in (7.10) in the $i$th step. This free choice is possible in the first finitely many steps, as long as $(\beta_i-\alpha,\operatorname{ord}(\widehat T_i)-2m- 1)\prec (0,0)$; in later steps we always choose $0$. So the sum $R_i(\boldsymbol\ell)$ of the series $\widehat R_i(\boldsymbol\ell)$ is unique up to a choice of the constants of integration in (7.10) in all the previous steps up to and including the $i$th. As in (6.12), we denote by $\mathcal H^{h_0}=\{R_i(\boldsymbol\ell)\colon i\in\mathbb N\}$ the sequence corresponding to one such choice of the constants of integration in all steps. It thus becomes one integral sum of $\widehat {\mathcal H}$.

We now let

$$\begin{equation*} \varphi_i(z):=z+z^{\beta_i}R_i(\boldsymbol\ell). \end{equation*}$$

The function $\varphi_i$ is analytic on $V$. In this way we get analytic changes $\varphi_{j,\pm}^i(z)$ on each petal $V_j^\pm$ ($j\in\mathbb Z$).

Note that the integration path in (7.10) does not matter. For any petal $V=V_j^{\pm}$ and any step of iteration, we may take any path lying inside the petal. Indeed, since $f$ is a generalized Dulac germ, all function coefficients in blocks of $f$ and $f_2$ are analytic functions on the corresponding $\boldsymbol\ell$-cusps $\boldsymbol\ell(V)$. Moreover, according to our algorithm the operations in the algebraic expression $\widehat T_i(\boldsymbol\ell)$ do not include division. Therefore, the integrand in the formula (7.10) for $R_i(\boldsymbol\ell)$, $i\in\mathbb N$, does not have any singularities on the $\boldsymbol\ell$-cusps $\boldsymbol\ell(V)$, which are simply connected.

The $\varphi_{j,\pm}^i$, $i\in\mathbb N$, in (7.8) obtained by the above algorithm are analytic on the petals $V_j^\pm$ ($j\in\mathbb Z$), but in general they do not glue together to form globally defined analytic germs $\varphi^i(z)$ on $\mathcal R_C$. We also do not assert that the $^i\!f(z)$, for any $i>1$, are analytic on $\mathcal R_C$ (they are at best analytic petalwise).

Every elementary change $\varphi_i(z)$ is an analytic function on $V$, by (7.10). For $i$ such that $(\beta_i-\alpha,\operatorname{ord}(\widehat T_i)-2m-1)\succ(0,0)$ we choose a canonical method of integration $\displaystyle\int_0^{\boldsymbol\ell}*\,dw$, so that the integral is a unique analytic function on $V$. Otherwise, we arbitrarily choose the initial point $\boldsymbol\ell_0\in\boldsymbol\ell(V)$ in the integral $\displaystyle\int_{\boldsymbol\ell_0}^{\boldsymbol\ell}*\,dw$, so the integral is unique only up to a term of the form $C_i e^{-(\alpha-\beta_i)/\boldsymbol\ell}\boldsymbol\ell^{m}$, $C_i\in\mathbb R$. This ambiguity corresponds to adding an exponentially small term of the form $C_i e^{-(\alpha-\beta_i)/\boldsymbol\ell}\boldsymbol\ell^{m}$ to $\varphi_i(z)$, $C_i\in\mathbb R$, in all changes of variables until we come to the residual block ($\beta_i\leqslant \alpha$).

Again let $V:=V_j^{\pm}$. Fix an arbitrary choice of the constant of integration $\boldsymbol\ell_0\in \boldsymbol\ell(V)$ in $h_0(z)$ (that is, take any analytic normalization $h_0^{-1}$ of $f_2$ on $V$) and any choice of integral sums $\mathcal H^{h_0}=\{R_i(\boldsymbol\ell)\colon i\in\mathbb N\}$ of $\widehat {\mathcal H}$ (choices of the constants of integration $\boldsymbol\ell_0\in\boldsymbol\ell(V)$), in the above algorithm. In the algorithm steps the germs $R_i(\boldsymbol\ell)\in\mathcal H^{h_0}$, $i\in\mathbb N$, are found by solving the corresponding Lie bracket equation for block-by-block eliminations. Now let

$$\begin{equation*} \varphi_i(z):=z+z^{\beta_i}R_i(\boldsymbol\ell),\qquad i\in\mathbb N. \end{equation*}$$

Since in the algorithm we eliminate block after block, there exists a strictly increasing sequence of exponents $(\gamma_n)_n$, $\gamma_n>1$, tending to $+\infty$ such that for every $n\in\mathbb N$

$$\begin{equation} \begin{aligned} \, \nonumber &(\varphi_1^{-1}\circ\dots\circ\varphi_n^{-1})\circ f\circ (\varphi_1\circ\dots\circ\varphi_n)=f_2+o(z^{\gamma_n}) \\ &\qquad=h_0^{-1}\circ f_1\circ h_0+o(z^{\gamma_n}),\qquad z\to 0, \end{aligned} \end{equation} \tag{ 7.11 }$$

on $V$. On the other hand, for any analytic normalization $\varphi$ of $f$ on $V$

$$\begin{equation} \varphi^{-1}\circ f\circ\varphi=f_1. \end{equation} \tag{ 7.12 }$$

Putting (7.12) in (7.11) and defining $T_n:=\varphi^{-1}\circ (\varphi_1\circ\dots\circ\varphi_n)$, we now get that

$$\begin{equation*} T_n^{-1}\circ f_1\circ T_n=h_0^{-1}\circ f_1\circ h_0+o(z^{\gamma_n}),\qquad z\to 0, \end{equation*}$$

so that

$$\begin{equation} h_0\circ T_n^{-1}\circ f_1\circ T_n\circ h_0^{-1}=f_1+o(z^{\gamma_n}),\qquad z\to 0. \end{equation} \tag{ 7.13 }$$

Using the equality $f_c\circ f_1=f_1\circ f_c$ ($c\in\mathbb R$), and defining $r_n:=T_n\circ h_0^{-1}-f_c$, $r_n(z)=O(z^{>1})$, $n\in\mathbb N$, we get from (7.13) that

$$\begin{equation*} f_1\circ (f_c+r_n)=(r_n+f_c)\circ f_1+o(z^{\gamma_n}), \end{equation*}$$

so that

$$\begin{equation*} f_1\circ (f_c+r_n)-f_1\circ f_c=r_n\circ f_1+o(z^{\gamma_n}). \end{equation*}$$

Comparing the leading terms on both sides in the last equality, we conclude that $r_n(z)=o(z^{\gamma_n-\alpha})$, $z\to 0$, $n\in\mathbb N$. Thus,

$$\begin{equation*} \varphi\circ f_c=(\varphi_1\circ\cdots\circ\varphi_n)\circ h_0^{-1}+ o(z^{\gamma_n-\alpha}). \end{equation*}$$

By Definition 7.2, this proves that $\widehat\varphi$ is the block iterated integral asymptotic expansion of any analytic normalization $\varphi$ of $f$ on $V$.

Proof of (ii): see Lemma 7.5.

Proposition 7.4 is proved. $\square$

We note that the existence of a power-logarithmic asymptotic expansion is not obvious, neither for a sectorial Fatou coordinate nor for a sectorial conjugacy of a parabolic generalized Dulac germ, as the following Example 7.6 shows. However, it can be verified if we assume the same asymptotic behaviour as in Propositions 7.1 and 7.4, (ii). See also Lemma 7.5.

Example 7.6  (a Fatou coordinate for a parabolic generalized Dulac germ which does not admit a sectional asymptotic expansion in the class $\widehat{\mathfrak L}^\infty(\mathbb R)$). In Theorem THA and its proof in § 8 we construct a holomorphic Fatou coordinate $\Psi$ of a parabolic generalized Dulac germ $f$ on an invariant petal $V$ of opening $2\pi/(\alpha-1)$, with $\Psi$ admitting an integral sectional asymptotic expansion equal to the formal Fatou coordinate $\widehat \Psi\in\widehat{\mathcal L}_2^\infty(\mathbb R)$ up to a constant. Here we define another holomorphic Fatou coordinate $\Psi_1$ on $V$ by

$$\begin{equation*} \Psi_1(z):=\Psi(z)+g_0(e^{2\pi \mathrm{i}\Psi(z)}),\qquad z\in V, \end{equation*}$$

where $g_0$ is any analytic germ on the doubly punctured sphere (without the poles $0$ and $\infty$). For example, we can take

$$\begin{equation*} \Psi_1(z):=\Psi(z)+\sin (2\pi \Psi(z)),\qquad z\in V, \end{equation*}$$

or

$$\begin{equation*} \Psi_2(z):=\Psi(z)+ce^{2\pi \mathrm{i}\Psi(z)},\qquad c\in\mathbb R,\quad z\in V. \end{equation*}$$

Due to the unbounded exponential term $\sin (2\pi \Psi(z))$ or $ce^{2\pi \mathrm{i}\Psi(z)}$, the functions $\Psi_{1,2}$ are Fatou coordinates of $f$ that do not admit sectional asymptotic expansions in the class $\widehat{\mathfrak L}^\infty(\mathbb R)$ on $V$ as $z\to 0$.