1.1 Modulus of classification and inverse problem

Heuristically, if $\Delta $ is a germ that is tangent to the identity and fixes $0\in \mathbb {C}$ ,

then the conformal structure of its orbit space $\Omega $ completely encodes its analytic class up to local changes of analytic coordinate near the fixed point. For the sake of simplicity, assume that $\Delta $ is generic, that is, that it belongs to the space

(here and in everything that follows, $*$ stands for a non-zero complex number).

Figure 1

Fundamental region of the dynamics of a generic parabolic germ (left) and the corresponding orbit space and horn maps (right) (colour online).

It is well known that a fundamental region for the iterative action of $\Delta $ is given by a pair of crescent-shaped regions that are identified by the dynamics near their horns, attached at the fixed point $0$ . More precisely, as dipicted in Figure 1, the orbit space

is obtained as the gluing of two spheres $\overline {\mathbb {C}}$ near $0$ and $\infty $ by germs of a diffeomorphism and , sometimes called horn maps:

When we endow the space with the quotient topology, the two points $0$ and $\infty $ (corresponding to the fixed point of $\Delta $ ) are not separated.

One can choose conformal coordinates on the spheres so that itself becomes tangent to the identity. The only degree of freedom that remains is the linear change of coordinates $h\mapsto ch$ for $c\in \mathbb {C}^{\times }$ applied simultaneously to both spheres. The Birkhoff–Écalle–Voronin (BÉV) theorem (see aforementioned bibliography) states that the map

is well defined and one-to-one. The Écalle–Voronin theorem states that it is also onto, answering the inverse problem.

In the present article, we revisit the question and provide an essentially unique representative $\Delta $ for a given pair , what can be called a normal form. The germ $\Delta $ will be ‘global’ on the sphere $\overline {\mathbb {C}}$ in some sense, which is why we borrow Écalle’s terminology and speak about spherical normal forms.

1.2 Statement of the main results

We answer the inverse problem in the following fashion.

Synthesis Theorem. For fixed $\mu \in \mathbb {C}$ and positive $\unicode{x3bb}>0$ , define the rational vector field (holomorphic near $0$ and $\infty $ ), called here the formal model,

$$ \begin{align*} X_{0}(z) :=\frac{1-z^{2}}{1+z^{2}}\times\frac{\unicode{x3bb} z^{2}}{1+\mu\unicode{x3bb} z-z^{2}}\frac{\partial}{\partial z}, \end{align*} $$

and denote by $f\mapsto X_{0}\cdot f$ the associated Lie (directional) derivative on a power series. Let be given and pick $\mu $ so that .

One can then find an explicit $\unicode{x3bb} (\psi )>0$ such that for each $0<\unicode{x3bb} \leq \unicode{x3bb} (\psi )$ , there exists a unique formal power series $F\in z\mathbb {C}[[z]] $ , depending on $\unicode{x3bb} $ and satisfying all of the following properties.

1. (1) The time- $1$ map $\Delta $ of the formal vector field, said to be in spherical normal form,

   $$ \begin{align*} X_{F}:= \frac{1}{1+X_{0}\cdot F}X_{0}, \end{align*} $$
   is a convergent power series at $0$ (that is, with positive radius of convergence).

2. (2) $\Delta $ belongs to with

   

3. (3) The power series F is $1$ -summable (see below) and its 1-sum can be realized as a pair $(f^{+},f^{-})$ of functions holomorphic and bounded by $1$ on the respective infinite sectors

   $$ \begin{align*} V^{\pm} :=\bigg\{ z\neq0:\lvert\ \arg(\pm z)\rvert<\frac{5\pi}{8}\bigg\} \end{align*} $$
   as in Figure 2.
   Figure 2

   The infinite sectors $V^{\pm }$ and the components , of their intersection $V^{\cap }$ (colour online).

   

A pair $f=(f^{+},f^{-})$ of functions holomorphic on the corresponding sector $V^{+}$ or $V^{-}$ has order-1 flat discrepancy at $0$ whenever

$$ \begin{align*} \limsup_{\begin{array}{l} z\to0\\ z\in V^{\cap} \end{array}}|z|\ln|f^{-}(z)-f^{+}(z)|\in\ ]{-}\infty,0[, \quad V^{\cap}:=V^{+}\cap V^{-}. \end{align*} $$

It follows from a theorem of Ramis and Sibuya [Reference Loday-RichaudLR16, Theorem 5.2.1] that there exists some $F=\sum _{n\geq 0}F_{n}z^{n}\in \mathbb {C}[[z]] $ , which is their common Gevrey-1 asymptotic expansion at $0$ (in the sense of Poincaré):

$$ \begin{align*} (\text{there exists } C>0)&(\text{for all } N\in\mathbb{N}) (\text{for all } z\in V^{\pm})\\&\bigg|f^{\pm}(z)-\sum_{n>0}^{N}F_{n}z^{n}\bigg|\leq C(N+1)!|z|^{N+1}. \end{align*} $$

We then say that F is $1$ -summable with $1$ -sum f. The mapping $f\mapsto F$ is injective (because $V^{\pm }$ is wider than a half-plane). This notion is compatible with the usual arithmetic and differential operations (by taking a narrower subsector if necessary).

We can exhibit a value of the upper bound $\unicode{x3bb} (\psi )$ as follows. There exist $\mathfrak {m}_{\mu },\mathfrak {t}_{\mu }>0$ depending only on $\mu $ (given for instance in equation (4.2)) for which, if we denote by and the respective radii of convergence of the Taylor series at $0$ of and , we may define

and take

where

The functions $z\in V^{\pm }\mapsto f^{\pm }(z)-(({1-z^{2}})/{\unicode{x3bb} z})+\mu \log ({\unicode{x3bb} z}/({1-z^{2}}))$ are Fatou coordinates for $\Delta $ (that is, local conformal coordinates on which $\Delta $ acts as a translation by $1$ , see §3.2). Hence, we parametrize the analytic classes of parabolic germs by their formal Fatou coordinates. By picking them in a class of ‘global’ formal objects, we benefit from a rigidity which makes the parametrization injective.

The power series F is global in the sense that its $1$ -sum $f=(f^{+},f^{-})$ exists on a pair of sectors whose union covers $\mathbb {C}^{\times }$ . Moreover, $\Delta |_{V^{\pm }}$ is the time-1 map of the $1$ -sum of its formal infinitesimal generator

$$ \begin{align*} X_{}^{\pm} :=\frac{1}{1+X_{0}\cdot f^{\pm}}X_{0}, \end{align*} $$

which is a meromorphic vector field on $V^{\pm }$ . A quick computation tells us that the formal model $X_{0}$ is invariant by the involution

$$ \begin{align*} \sigma:\overline{\mathbb{C}} & \longrightarrow\overline{\mathbb{C}}\\ z & \mapsto\frac{-1}{z} \end{align*} $$

fixing $\pm {i}$ , while the sectors $V^{+}$ and $V^{-}$ are swapped, that is,

$$ \begin{align*} \sigma(V^{\pm}) =V^{\mp}. \end{align*} $$

A nice feature of the synthesis is that f not only defines a parabolic germ near $0$ , but it also creates one at $\infty $ , corresponding to the companion dynamics induced there by $X^{\pm }$ , since (§4.3)

(1.1) $$ \begin{align} f^{\pm}\circ\sigma =-f^{\mp}. \end{align} $$

This is why we call these dynamical systems spherical normal forms, a term borrowed from Écalle [Reference EcalleEca05] as discussed in §1.2.2. We can describe completely the dynamics of $\Delta $ and its monodromy as a multivalued map over the whole Riemann sphere $\overline {\mathbb {C}}$ .

It is remarkable that a twin fixed-point is automatically produced at $\infty $ , especially because the modulus there is given by the reciprocal modulus. This has a simple explanation, however: the involution $\sigma $ permutes the sectors $V^{+}$ and $V^{-}$ , while it does not change the infinitesimal generators much; therefore, the identification at the horns is the same while taking place in the reverse direction.

Another remarkable feature is that $\Delta $ has very simple dynamics: it is an injective, holomorphic map on a slit sphere $\mathcal {D}$ that can be forward iterated on the open set $\overline {\mathbb {C}}\backslash \gamma $ , where $\gamma $ is the union of the closure of the stable manifolds of $X^{\pm }$ passing through the poles. On each connected component of $\overline {\mathbb {C}}\backslash \gamma $ , the dynamics of $\Delta $ converges uniformly to a fixed-point (a parabolic point or the attracting point sitting at $\pm 1$ when $\mu \notin {i}\mathbb {R}$ ). The reason behind this stable behaviour is the parameter $\unicode{x3bb} $ , which is chosen so that no chaotic remains of a global dynamics, usually arising from a frontier in the analytic continuation of the modulus [Reference EpsteinEps93], can be seen from the normal form.

The obvious downside of the previous feature is precisely that the dynamics of $\Delta $ is too simple to readily offer an insight regarding the dynamical richness that is displayed by holomorphic iteration on compact curves. What we prove is that any such dynamics, say of a rational mapping with a parabolic basin $\mathcal {B}$ , is locally conjugate by some $\Psi $ to a spherical normal form near the parabolic fixed-point $0$ , but the domain of the conjugacy cannot contain the whole component $\partial \mathcal {B}$ of the Julia set. Indeed, can be pushed forward by the flow of $X^{+}$ to give a compact invariant set J which is locally conformally equivalent to the fractal set $\partial \mathcal {B}$ . However, the one set J has no relevance to the dynamics of $\Delta $ whatsoever, for the latter has only Fatou components with real-analytic boundary, and that regularity increases as $\unicode{x3bb} \to 0$ . In that respect, an interesting question arises: can one recover part of the original dynamics by letting $\unicode{x3bb} $ grow (which may cause $X^{\pm }$ to sport more and more poles, for instance)?

1.2.1 Regarding the formal model

The family of vector fields $X_{0}$ depending on $\unicode{x3bb}>0$ is obtained from the ‘usual’ formal model $({x^{2}}/({1+\mu x}))({\partial }/{\partial x})$ (see Figure 4) by performing the degree-2 pullback

$$ \begin{align*} \Pi:z \mapsto x=x(z):=\frac{\unicode{x3bb} z}{1-z^{2}} \end{align*} $$

(which is invertible near $0$ ). Observe that $\Pi $ is invariant under the involution $\sigma $ , so that $\sigma $ is a symmetry of $X_{0}$ , a fact that has been already remarked. The obvious consequence is that $X_{0}$ has a stationary point at $\infty $ that mirrors the one it admits at $0$ .

Remark 1.3. The only normal form $X_{f}$ that can be factored through $\Pi $ (that is, realized as a global sectorial perturbation of the usual formal model) is the formal model itself ( $f=0$ ) since we must both have $f^{\pm }\circ \sigma =f^{\mp }$ (factorization through $\Pi $ ) and $f^{\pm }\circ \sigma =-f^{\mp }$ as in equation (1.1).

Figure 4

Foliation induced by the real-time flow of $({x^{2}}/({1+\mu x})) {\partial }/{\partial x}$ ((a) $\mu :=0$ ; (b), $\mu :=2$ ). There is one double stationary point at $0$ (green circle), yielding a parabolic germ for the time-1 map, and one saddle point (red square) corresponding to the pole $-{1}/{\mu }$ (colour online).

The time-1 map of $X_{0}$ is a generic parabolic germ (Lemma 2.4)

$$ \begin{align*} \Delta_{0}(z):=\Phi_{X_{0}}^{1}(z) =z+\unicode{x3bb} z^{2}+\unicode{x3bb}^{2}(1-\mu)z^{3}+\mathrm{o}(z^{3}) \end{align*} $$

with trivial BÉV modulus. Any germ with the same $\mu $ is formally conjugate to $\Delta _{0}$ (whatever $\unicode{x3bb} $ ), which explains the terminology formal model.

The sectors and the model $X_{0}$ are tailored in such a way that the orbital region of $\Delta _{0}$ , as seen from the intersection $V^{\cap }$ , is a neighbourhood of $0$ or $\infty $ of size as $\unicode{x3bb} \to 0$ (cf. §4.1). Gaining such a control is essential: eventually the neighbourhood fits within the discs of convergence of the data and keeps lying there even after small perturbations.

Picking $X_{0}$ among all vector fields realizing the trivial pair $({e}^{4\pi ^{2}\mu }\mathrm {Id},\mathrm {Id})$ is arguably canonical since the pullback $z\mapsto {\unicode{x3bb} z}/({1-z^{2}})$ is very simple and the pair of sectors $(V^{+},V^{-})$ is fairly standard.

Remark 1.4. For technical reasons, we also need that $\infty $ and $0$ be mapped to $0$ by the pullback. Thereby, the ‘obvious’ choice of a degree-1 pullback, like $z\mapsto {\unicode{x3bb} z}/({1-z})$ , cannot be used to perform the construction presented here. Apart from this restriction, one can use pretty much any other pullback, provided the ‘sectors’ are tailored conveniently.

Example 1.5. We refer to Figure 5. Assume here that $\mu :=0$ so that

$$ \begin{align*} X_{0}(z) =\frac{\unicode{x3bb} z^{2}}{1+z^{2}}\frac{\partial}{\partial z} \end{align*} $$

has only stationary (double) points at $0$ and $\infty $ , as well as two simple poles at $\pm {i}$ with residue $\pm {\unicode{x3bb} {i}}/{2}$ . These poles spawn the ramification locus of $\Delta _{0}$ , as can be seen from a direct integration of $\dot {z}=X_{0}(z)$ at time $1$ with initial value $z_{*}$ :

$$ \begin{align*} 1 =\int_{z_{*}}^{\Delta_{0}(z_{*})}\frac{1+z^{2}}{\unicode{x3bb} z^{2}}\,\mathit{d}z, \end{align*} $$

which yields

$$ \begin{align*} z_{*}\Delta_{0}(z_{*})^{2}-(z_{*}^{2}+\unicode{x3bb} z_{*}-1)\Delta_{0}(z_{*})-z_{*} =0. \end{align*} $$

Hence, $\Delta _{0}$ is algebraic and its ramification points are the four points z for which

$$ \begin{align*} (z^{2}+\unicode{x3bb} z-1)^{2}+4z^{2} =0. \end{align*} $$

Notice that $\Delta _{0}$ admits a limit at each one of these points z, and $\Delta _{0}(z)$ lies within the polar locus $\{ \pm {i}\} $ of $X_{0}$ . Therefore, $\Delta _{0}$ extends as a multivalued function over the domain $\overline {\mathbb {C}}\backslash \{ z^{2}+\unicode{x3bb} z=1\} $ (which contains $0$ ) with involutive monodromy around any one of the ramification points given by the action of $\sigma $ , which is moreover sent to a pole of $X_{0}$ by the time-1 map. More details can be found in Lemma 2.7.

Figure 5

Foliation induced by the real-time flow of $X_{0}$ for $\mu :=0$ , revealing the double stationary point (green circle) and two simple poles (red squares) (colour online).

Remark 1.6. When $\mu \neq 0$ , the mapping $\Delta _{0}$ cannot be algebraic (see [Reference ÉcalleÉca75]). It seems safe to conjecture that the only algebraic normal form $\Delta $ is $\Delta _{0}$ when $\mu =0$ .

1.3 Parabolic renormalization

Iteration near a parabolic fixed-point is an important topic in dynamical systems, since complicated phenomena arise once a perturbation is made. Some significant consequences include [Reference Avila and LyubichAL22, Reference Buff and ChéritatBC10, Reference ShishikuraShi98], in which the (near-)parabolic renormalization and its fixed-points play an important role.

The parabolic renormalization map

associates the $\infty $ -component of the invariant of a generic parabolic germ $\Delta $ . It may happen that $\mathcal {R}(\Delta )$ is itself generic, in which case, $\mathcal {R}$ can be iterated. As far as I know, a statement of the kind (while not referring specifically to spherical normal forms) was first communicated privately by R. Schäfke.

Parabolic Renormalization Corollary. Take an arbitrary . There exists a bound such that, for all , there exists a unique spherical normal form $\Delta $ satisfying

Again, the bound $\widehat {\unicode{x3bb} }$ can be made explicit, we refer to Corollary 6.3.

1.4 Real case

If is real (that is, commutes with the complex conjugation $\overline {\bullet }$ ), then $\mu \in \mathbb {R}$ and its modulus satisfies the identity:

(◻)

1.5 Structure of the article and summary of the proofs

One of the main concerns of this paper is to provide as explicit bounds as possible, which explains why so much effort went into technical lemmas. The main tools are basic calculus / complex analysis, and the work is necessary in any case to obtain uniform bounds with respect to $\unicode{x3bb} $ .

All the constructions will take place within the space of pairs $f=(f^{+},f^{-})$ with 1-flat difference in $V^{\cap }$ and bounded 1-type, both at $0$ and $\infty $ , as discussed at the beginning of the introduction:

$$ \begin{align*} \mathcal{S} :=\big\{ f\in\mathcal{S}(V^{+})\times\mathcal{S}(V^{-}):\limsup_{z^{\pm1}\to0}|z|^{\pm1}\ln|f^{-}(z)-f^{+}(z)|\leq-5\text{ for }z\in V^{\cap}\big\} , \end{align*} $$

where

is equipped with the canonical product Banach norm. We denote by $\mathcal {B}$ its unit ball.

1. (1) In §2, we give some basic material about vector fields and their time-1 maps, focusing more particularly on the local dynamics near their poles and zeroes, and how they stitch together to organize the global dynamics of the model $X_{0}$ . This will allow us to conjugate

   $$ \begin{align*} X_{f}^{\pm} :=\frac{1}{1+X_{0}\cdot f^{\pm}}X_{0} \end{align*} $$
   to the model $X_{0}$ over the better part of $V^{\pm }$ (effective bounds are described in §5.2). This sectorial mapping takes the form $\Phi _{X_{0}}^{f^{\pm }}$ given by the flow of $X_{0}$ at time $f^{\pm }$ . The dynamics of the formal model $\Delta _{0}$ , studied in §2.3, is then pulled-back by $\Psi ^{\pm }$ to give sectorial counterparts for the time-1 map of $X_{f}^{\pm }$ .

2. (2) Coupled to the Ramis–Sibuya theorem, the above preliminary part allows one to reduce the problem of finding a parabolic germ $\Delta $ with prescribed modulus to that of finding $(f^{+},f^{-})\in \mathcal {S}$ satisfying some (nonlinear) Cousin problem $(\star )$ below; see Proposition 3.3 in §3.

3. (3) We can then tackle the main result (§4). We begin with fixing a formal class $\mu \in \mathbb {C}$ in . Then, must be tangent to the linear map ${e}^{4\pi ^{2}\mu }\mathrm {Id}$ . Being given with holomorphic near $\sharp \in \{ 0,\infty \} $ , we must find $f^{\pm }$ on $V^{\pm }$ such that

   (⋆)

   
   where are the two components of $V^{\cap }$ , and $H_{f}^{+}$ is a primitive first-integral of the sectorial time-1 map $\Delta ^{+}$ of $X_{f}^{+}$ (in other words, $H_{f}^{+}$ provides preferred orbital coordinates over the sector $V^{+}$ ). Condition $(\star )$ implies that $\Delta ^{+}$ coincides with $\Delta ^{-}$ in $V^{\cap }$ and thus is the restriction to $V^{+}$ of a single .

   The nonlinear Cousin problem $(\star )$ is solved using a fixed-point method involving a classical Cauchy–Heine integral transform. This technique was already employed in [Reference Schäfke and TeyssierST15] (and [Reference Rousseau and TeyssierRT21]) to build normal forms for convergent saddle-node foliations in the complex plane (and their unfoldings). More precisely, we build a map (Proposition 4.11):

   
   such that $\Lambda (f)^{-}-\Lambda (f)^{+}=\varphi \circ H_{f}^{+}$ . Therefore, if and only if condition $(\star )$ holds. We prove (Proposition 4.14) that eventually stabilizes the unit ball $\mathcal {B}\subset \mathcal {S}$ . The smaller $\unicode{x3bb} $ , the more contracting , so that eventually a unique fixed-point of exists in that ball (Corollary 4.15).

4. (4) In §5.1, as is discussed above, we establish the Globalization Theorem making use of the involution $\sigma $ and the property that for the fixed-point f of , we have

   $$ \begin{align*} f^{\pm}\circ\sigma =-f^{\mp}. \end{align*} $$
   Prior to that, we describe the dynamical properties of $\Delta $ in §5.3 using a priori bounds on the flow of $X^{\pm }$ .

5. (5) The fixed-points of the parabolic renormalization are also obtained as a holomorphic fixed-point in §6.1. Let $\text {Synth}_{\unicode{x3bb} }:\psi \mapsto \Delta $ be the synthesis map built previously and consider for given , the iterations $\Delta _{0}:=\mathrm {Id}$ and . It so happens that $\text {Synth}_{\unicode{x3bb} }$ cannot shrink too much the radius of convergence of $\Delta _{n}$ , which always remains above ${3}/{16}$ . By taking $\unicode{x3bb} $ smaller, we prove next that $\text {Synth}_{\unicode{x3bb} }$ is a contracting self-map of some closed ball of the Banach space .

6. (6) The real setting is explored in §6.2 (Real Synthesis Corollary). It merely consists in checking that every step leading to the fixed-point of preserves realness, which is straightforward.

1.6 Notation

• • $\dot {z}={\mathit {d}z}/{\mathit {d}t}$ stands for differentiation with respect to the variable $t\in \mathbb {C}$ .

• • For a topological space $\mathcal {M}$ and a point $p\in \mathcal {M}$ , we use $(\mathcal {M},p)$ to stand for a (sufficiently small) neighbourhood of p in $\mathcal {M}$ .

• • The set $\overline {U}$ is the topological closure of $U\subset \mathcal {M}$ .

• • We use the notation $\mathbb {D}$ for the standard open unit disk in $\mathbb {C}$ .

• • $\overline {\mathbb {C}}$ stands for the compactification of the complex line $\mathbb {C}$ as the Riemann sphere.

• • The nth iterate of a diffeomorphism $\Delta $ is written $\Delta ^{\circ n}$ for $n\in \mathbb {Z}$ .

• • $\Phi _{X}^{t}$ designates the flow at time t of the vector field X.

• • If $\Psi :U\to V$ is locally biholomorphic, we recall its action by conjugacy (change of coordinate) on diffeomorphisms:

  $$ \begin{align*} \Psi^{*}\Delta :=\Psi^{\circ-1}\circ\Delta\circ\Psi, \end{align*} $$
  on vector fields:
  $$ \begin{align*} \Psi^{*}X :=\mathrm{D}\Psi^{\circ-1}(X\circ\Psi). \end{align*} $$
  If $\Delta $ is the time-1 map of X, then $\Psi ^{*}\Delta $ is the time-1 map of $\Psi ^{*}X$ .