Shingo Kamimoto

David Sauzin

Iterated convolutions and endless Riemann surfaces

Keywords:

may come

Introduction

In this article, we deal with the following version of Écalle's definition of resurgence: Definition 1.1. A convergent power series φ ∈ C{ζ} is said to be endlessly continuable if, for every real L > 0, there exists a finite subset F L of C such that the holomorphic germ at 0 defined by φ can be analytically continued along every Lipschitz path γ : [0, 1] → C of length smaller than L such that γ(0) = 0 and γ (0, 1] ⊂ C \F L . We denote by R ⊂ C{ζ} the space of endlessly continuable functions. (j-1)! is an endlessly continuable function.

In other words, the space of resurgent series is

R := B -1 (C δ ⊕ R) ⊂ C[[z -1 ]],
where

B : C[[z -1 ]] → C δ ⊕ C[[ζ]
] is the formal Borel transform, defined by B φ := ϕ 0 δ + φ(ζ) in the notation of Definition 1.2.

We will also treat the more general case of functions which are "endlessly continuable w.r.t. bounded direction variation": we will define a space R dv containing R and, correspondingly, a space R dv containing R, but for the sake of simplicity, in this introduction, we stick to the simpler situation of Definitions 1.1 and 1.2.

Note that the radius of convergence of an element of R may be 0. As for the elements of R, we will usually identify a convergent power series and the holomorphic germ that it defines at the origin of C, as well as the holomorphic function which is thus defined near 0. Holomorphic germs with meromorphic or algebraic analytic continuation are examples of endlessly continuable functions, but the functions in R can have a multiple-valued analytic continuation with a rich set of singularities.

The convolution product is defined as the Borel image of multiplication and denoted by the symbol * : for φ, ψ ∈ C [[ζ]], φ * ψ := B(B -1 φ • B -1 ψ), and δ is the convolution unit (obtained from (C [[ζ]], * ) by adjunction of unit). As is well known, for convergent power series, convolution admits the integral representation Our aim is to study the analytic continuation of the convolution product of an arbitrary number of endlessly continuable functions, to check its endless continuability, and also to provide bounds, so as to be able to deal with nonlinear operations on resurgent series. A typical example of nonlinear operation is the substitution of one or several series without constant term φ1 , . . . , φr into a power series F (w 1 , . . . , w r ), defined as (1.2) F ( φ1 , . . . , φr ) :=

k∈N r c k φk 1 1 • • • φkr r for F = k∈N r c k w k 1 1 • • • w kr r .
One of our main results is Theorem 1.3. Let r ≥ 1 be an integer. Then, for any convergent power series F (w 1 , . . . , w r ) ∈ C{w 1 , . . . , w r } and for any resurgent series φ1 , . . . , φr without constant term, F ( φ1 , . . . , φr ) ∈ R.

The proof of this result requires suitable bounds for the analytic continuation of the Borel transform of each term in the right-hand side of (1.2). Along the way, we will study the Riemann surfaces generated by endlessly continuable functions. We will also prove similar results for the larger spaces R dv and R dv .

Resurgence theory was developed in the early 1980s, with [START_REF] Écalle | Les fonctions résurgentes[END_REF] and [START_REF] Écalle | Les fonctions résurgentes[END_REF], and has many mathematical applications in the study of holomorphic dynamical systems, analytic differential equations, WKB analysis, etc. (see the references e.g. in [START_REF]Nonlinear analysis with resurgent functions[END_REF]). More recently, there has been a burst of activity on the use of resurgence in Theoretical Physics, in the context of matrix models, string theory, quantum field theory and also quantum mechanics-see e.g. [START_REF] Aniceto | Nonperturbative ambiguities and the reality of resurgent transseries[END_REF], [START_REF] Aniceto | The resurgence of instantons in string theory[END_REF], [START_REF] Argyres | The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects[END_REF], [START_REF] Cherman | Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles[END_REF], [START_REF] Couso-Santamaría | Finite N from resurgent large N[END_REF], [START_REF] Dunne | Resurgence and trans-series in quantum field theory: the CP N -1 model[END_REF], [START_REF] Dunne | Uniform WKB, multi-instantons, and resurgent trans-series[END_REF], [START_REF] Garay | Resurgent deformation quantisation[END_REF], [START_REF] Mariño | Lectures on non-perturbative effects in large N gauge theories, matrix models and strings[END_REF]. In almost all these applications, it is an important fact that the space of resurgent series be stable under nonlinear operations: such stability properties are useful, and at the same time they account for the occurrence of resurgent series in concrete problems. These stability properties were stated in a very general framework in [START_REF] Écalle | Les fonctions résurgentes[END_REF], but without detailed proofs, and the part of [START_REF] Candelpergher | Approche de la résurgence[END_REF] which tackles this issue contains obscurities and at least one mistake. It is thus our aim in this article to provide a rigorous treatment of this question, at least in the slightly narrower context of endless continuability. The definitions of resurgence that we use for R and R dv are indeed more restrictive than Écalle's most general definition [START_REF] Écalle | Les fonctions résurgentes[END_REF]. In fact, our definition of R dv is almost identical to the one used by Pham et al. in [START_REF] Candelpergher | Approche de la résurgence[END_REF], and our definition of R is essentially equivalent to the definition used in [START_REF] Deleabaere | Endless continuability and convolution product[END_REF], but the latter preprint has flaws which induced us to develop the results of the present paper. These versions of the definition of resurgence are sufficient for a large class of applications, which virtually contains all the aforementioned ones-see for instance [START_REF] Kamimoto | Resurgence of formal series solutions of nonlinear differential and difference equations[END_REF] for the details concerning the case of nonlinear systems of differential or difference equations. The advantage of the definitions based on endless continuability is that they allow for a description of the location of the singularities in the Borel plane by means of discrete filtered sets or discrete doubly filtered sets (defined in Sections 2.1 and 2.5); the notion of discrete (doubly) filtered set, adapted from [START_REF] Candelpergher | Approche de la résurgence[END_REF] and [START_REF] Deleabaere | Endless continuability and convolution product[END_REF], is flexible enough to allow for a control of the singularity structure of convolution products.

A more restrictive definition is used in [START_REF]Nonlinear analysis with resurgent functions[END_REF] and [START_REF] Mitschi | Divergent Series, Summability and Resurgence[END_REF] (see also [START_REF] Écalle | Les fonctions résurgentes[END_REF]): Definition 1.4. Let Σ be a closed discrete subset of C. A convergent power series φ is said to be Σ-continuable if it can be analytically continued along any path which starts in its disc of convergence and stays in C \Σ. The space of Σ-continuable functions is denoted by RΣ . This is clearly a particular case of Definition 1.1: any Σ-continuable function is endlessly continuable (take

F L = { ω ∈ Σ | |ω| ≤ L }). It is proved in [MS16] that, if Σ ′ and Σ ′′ are closed discrete subsets of C, and if also Σ := {ω ′ + ω ′′ | ω ′ ∈ Σ ′ , ω ′′ ∈ Σ ′′ } is closed and discrete, then φ ∈ RΣ ′ , ψ ∈ RΣ ′′ ⇒ φ * ψ ∈ RΣ .
This is because in formula (1.1), heuristically, singular points tend to add to create new singularities; so, the analytic continuation of φ * ψ along a path which does not stay close to the origin is possible provided the path avoids Σ. In particular, if a closed discrete set Σ is closed under addition, then RΣ is closed under convolution; moreover, in this case, bounds for the analytic continuation of iterated convolutions φ1 * • • • * φn are given in [START_REF]Nonlinear analysis with resurgent functions[END_REF], where an analogue of Theorem 1.3 is proved for Σ-continuable functions.

The notion of Σ-continuability is sufficient to cover interesting applications, e.g. differential equations of the saddle-node singularity type or difference equations like Abel's equation for one-dimensional tangent-to-identity diffeomorphisms, in which cases one may take for Σ a one-dimensional lattice of C. However, reflecting for a moment on the origin of resurgence in differential equations, one sees that one cannot handle situations beyond a certain level of complexity without replacing Σ-continuability by a more general notion like endless continuability.

(z) + b 1 (z)ϕ + b 2 (z)ϕ 2 + • • • with b(z, w) = b m (z)w m ∈ z -1 C{z -1
, w} given, we may expect a formal solution whose Borel transform φ has singularities at ζ = -nλ, n ∈ Z >0 (because, as an effect of the nonlinearity, the singular points tend to add), i.e. φ will be Σ-continuable with Σ = {-λ, -2λ, . . .} (see [START_REF]Mould expansions for the saddle-node and resurgence monomials[END_REF] for a rigorous proof of this), but in the multidimensional case, for a system of r coupled equations with left-hand sides of the form dϕ j dz λ j ϕ j with λ 1 , . . . , λ r ∈ C * , we may expect that the Borel transforms φj of the components of the formal solution have singularities at the points

ζ = -(n 1 λ 1 + • • • + n r λ r ), n ∈ Z r >0
; this set of possible singular points may fail to be closed and discrete (depending on the arithmetical properties of (λ 1 , . . . , λ r )), hence, in general, we cannot expect these Borel transforms to be Σ-continuable for any Σ. Still, this does not prevent them from being always endlessly continuable, as proved in [START_REF] Kamimoto | Resurgence of formal series solutions of nonlinear differential and difference equations[END_REF].

-Another illustration of the need to go beyond Σ-continuability stems from parametric resurgence [START_REF] Écalle | Cinq applications des fonctions résurgentes[END_REF]. Suppose that we are given a holomorphic function b(t) globally defined on C, with isolated singularities ω ∈ S ⊂ C, e.g. a meromorphic function, and consider the differential equation

(1.3) dϕ dt -zλϕ = b(t),
where λ ∈ C * is fixed and z is a large complex parameter with respect to which we consider perturbative expansions. It is easy to see that there is a unique solution which is formal in z and analytic in t, namely φ(z,

t) := -∞ k=0 λ -k-1 z -k-1 b (k) (t)
, and its Borel transform φ(ζ, t) = -λ -1 b(t + λ -1 ζ) is singular at all points of the form ζ t,ω := λ(-t + ω), ω ∈ S. Now, if we add to the right-hand side of (1.3) a perturbation which is nonlinear in ϕ, we can expect to get a formal solution whose Borel transform possesses a rich set of singular points generated by the ζ t,ω 's, which might easily be too rich to allow for Σ-continuability with any Σ; however, we can still hope endless continuability.

These are good motivations to study endless continuable functions. As already alluded to, we will use discrete filtered sets (d.f.s. for short) to work with them. A d.f.s. is a family of sets Ω = (Ω L ) L∈R ≥0 , where each Ω L is a finite set; we will define Ω-continuability when Ω is a d.f.s., thus extending Definition 1.4, and the space of endlessly continuable functions will appear as the totality of Ω-continuable functions for all possible d.f.s. This was already the approach of [START_REF] Candelpergher | Approche de la résurgence[END_REF], and it was used in [START_REF] Deleabaere | Endless continuability and convolution product[END_REF] to prove that the convolution product of two endlessly continuable functions is endlessly continuable, hence R is a subring of C[[z -1 ]]. However, to reach the conclusions of Theorem 1.3, we will need to give precise estimates on the convolution product of an arbitrary number of endlessly continuable functions, so as to prove the convergence of the series of holomorphic functions

c k φ * k 1 1 * • • • * φ * kr r
(Borel transform of the right-hand side of (1.2)) and to check its endless continuability. We will proceed similarly in the case of endless continuability w.r.t. bounded direction variation, using discrete doubly filtered sets.

Notice that explicit bounds for iterated convolutions can be useful in themselves; in the context of Σ-continuability, such bounds were obtained in [START_REF]Nonlinear analysis with resurgent functions[END_REF] and they were used in [K 3 16] in a study in WKB analysis, where the authors track the analytic dependence upon parameters in the exponential of the Voros coefficient.

As another contribution to the study of endlessly continuable functions, we will show how to contruct, for each discrete filtered set Ω, a universal Riemann surface X Ω whose holomorphic functions are in one-to-one correspondence with Ω-continuable functions.

The plan of the paper is as follows.

-Section 2 introduces discrete filtered sets, the corresponding Ω-continuable functions and their Borel images, the Ω-resurgent series, and discusses their relation with Definitions 1.1 and 1.2. The case of discrete doubly filtered sets and the spaces R dv and R dv is in Section 2.5.

-Section 3 discusses the notion of Ω-endless Riemann surface and shows how to construct a universal object X Ω (Theorem 3.2).

-In Section 4, we state and prove Theorem 4.8 which gives precise estimates for the convolution product of an arbitrary number of endlessly continuable functions. We also show the analogous statement for functions which are endlessly continuable w.r.t. bounded direction variation.

-Section 5 is devoted to applications of Theorem 4.8: the proof of Theorem 1.3 and even of a more general and more precise version, Theorem 5.2, and an implicit resurgent function theorem, Theorem 5.3. Some of the results presented here have been announced in [START_REF] Kamimoto | Nonlinear analysis with endlessly continuable functions[END_REF].

2 Discrete filtered sets and Ω-continuability

In this section, we review the notions concerning discrete filtered sets (usually denoted by the letter Ω), the corresponding Ω-allowed paths and Ω-continuable functions. The relation with endless continuability is established, and sums of discrete filtered sets are defined in order to handle convolution of enlessly continuable functions.

Discrete filtered sets

We first introduce the notion of discrete filtered sets which will be used to describe singularity structure of endlessly continuable functions (the first part of the definition is adapted from [START_REF] Candelpergher | Approche de la résurgence[END_REF] and [START_REF] Deleabaere | Endless continuability and convolution product[END_REF]):

Definition 2.1. We use the notation

R ≥0 = {λ ∈ R | λ ≥ 0}. 1) A discrete filtered set, or d.f.s. for short, is a family Ω = (Ω L ) L∈R ≥0 where i) Ω L is a finite subset of C for each L, ii) Ω L 1 ⊆ Ω L 2 for L 1 ≤ L 2 ,
iii) there exists δ > 0 such that Ω δ = Ø.

2) Let Ω and Ω ′ be d.f.s. We write Ω ⊂ Ω ′ if Ω L ⊂ Ω ′ L for every L. 3) We call upper closure of a d.f.s. Ω the family of sets Ω = ( Ω) L∈R ≥0 defined by

(2.1) ΩL := ε>0 Ω L+ε for L ∈ R ≥0 .
It is easy to check that Ω is a d.f.s. and Ω ⊂ Ω.

Example 2.2. Given a closed discrete subset Σ of C, the formula

Ω(Σ) L := { ω ∈ Σ | |ω| ≤ L } for L ∈ R ≥0
defines a d.f.s. Ω(Σ) which coincides with its upper closure.

From the definition of d.f.s., we find the following Lemma 2.3. For any d.f.s. Ω, there exists a real sequence (L n ) n≥0 such that 0 = L 0 < L 1 < L 2 < • • • and, for every integer n ≥ 0,

L n < L < L n+1 ⇒ ΩLn = ΩL = Ω L .
Proof. First note that (2.1) entails

(2.2) ΩL := ε>0 ΩL+ε for every L ∈ R ≥0

(because Ω L+ε ⊂ ΩL+ε ⊂ ΩL+2ε ). Consider the weakly order-preserving integer-valued function L ∈ R ≥0 → N (L) := card ΩL . For each L the sequence k → N (L + 1 k ) must be eventually constant, hence there exists ε L > 0 such that, for all L ′ ∈ (L, L + ε L ], N (L ′ ) = N (L + ε L ), whence ΩL ′ = ΩL+ε L , and in fact, by (2.2), this holds also for L ′ = L. The conclusion follows from the fact that R ≥0 = k∈Z N -1 (k) and each non-empty N -1 (k) is convex, hence an interval, which by the above must be left-closed and right-open, hence of the form [L, L ′ ) or [L, ∞). Given a d.f.s. Ω, we set

(2.3) S Ω := (λ, ω) ∈ R × C | λ ≥ 0 and ω ∈ Ω λ
and denote by S Ω the closure of S Ω in R × C. We then call (2.4)

M Ω := R × C \ S Ω (open subset of R × C)
the allowed open set associated with Ω.

Lemma 2.4. One has S Ω = S Ω and M Ω = M Ω.

Proof. Suppose (λ, ω)

∈ S Ω. Then ω ∈ Ω λ+1/k for each k ≥ 1, hence (λ + 1 k , ω) ∈ S Ω , whence (λ, ω) ∈ S Ω .
Suppose (λ, ω) ∈ S Ω . Then there exists a sequence (λ k , ω k ) k≥1 in S Ω which converges to (λ, ω).

If ε > 0, then λ k ≤ λ + ε for k large enough, hence ω k ∈ Ω λ+ε , whence ω ∈ Ω λ+ε (because a finite set is closed); therefore (λ, ω) ∈ S Ω.
Therefore, S Ω = S Ω = S Ω and M Ω = M Ω .

Ω-allowed paths

When dealing with a Lipschitz path γ : [a, b] → C, we denote by L(γ) its length.

We denote by Π the set of all Lipschitz paths γ : [0, t * ] → C such that γ(0) = 0, with some real t * ≥ 0 depending on γ. Given such a γ ∈ Π and t ∈ [0, t * ], we denote by

γ |t := γ| [0,t] ∈ Π
the restriction of γ to the interval [0, t]. Notice that L(γ |t ) is also Lipschitz continuous on [0, t * ] since γ ′ exists a.e. and is essentially bounded by Rademacher's theorem.

Definition 2.5. Given a d.f.s. Ω, we call Ω-allowed path any γ ∈ Π such that γ(t) := L(γ |t ), γ(t) ∈ M Ω for all t.

We denote by Π Ω the set of all Ω-allowed paths.

Notice that, given t * ≥ 0,

(2.5) if t ∈ [0, t * ] → γ(t) = λ(t), γ(t) ∈ M Ω is a piecewise C 1 path such that γ(0) = (0, 0) and λ ′ (t) = |γ ′ (t)| for a.e. t, then γ ∈ Π Ω .
In view of Lemmas 2.3 and 2.4, we have the following characterization of Ω-allowed paths:

Lemma 2.6. Let Ω be a d.f.s. Then Π Ω = Π Ω and, given γ ∈ Π, the followings are equivalent:

1) γ ∈ Π Ω , 2) γ(t) ∈ C \ ΩL(γ |t ) for every t,
3) for every t, there exists n such that L(γ |t ) < L n+1 and γ(t) ∈ C \ ΩLn (using the notation of Lemma 2.3).

Proof. Obvious.

Notation 2.7. For L, δ > 0, we set

M δ,L Ω := (λ, ζ) ∈ R × C | dist (λ, ζ), S Ω ≥ δ and λ ≤ L , (2.6) Π δ,L Ω := γ ∈ Π Ω | L(γ |t ), γ(t) ∈ M δ,L Ω for all t , (2.7) where dist(• , •) is the Euclidean distance in R × C ≃ R 3 . Note that M Ω = δ,L>0 M δ,L Ω , Π Ω = δ,L>0 Π δ,L Ω .

Ω-continuable functions and Ω-resurgent series

Definition 2.8. Given a d.f.s. Ω, we call Ω-continuable function a holomorphic germ φ ∈ C{ζ} which can be analytically continued along any path γ ∈ Π Ω . We denote by RΩ the set of all Ω-continuable functions and define

RΩ := B -1 C δ ⊕ RΩ ⊂ C[[z -1 ]]
to be the set of Ω-resurgent series.

Remark 2.9. Given a closed discrete subset Σ of C, the Σ-continuability in the sense of Definition 1.4 is equivalent to the Ω(Σ)-continuability in the sense of Definition 2.8 for the d.f.s. Ω(Σ) of Example 2.2.

Remark 2.10. Observe that Ω ⊂ Ω ′ implies S Ω ⊂ S Ω ′ , hence M Ω ′ ⊂ M Ω and Π Ω ′ ⊂ Π Ω , therefore Ω ⊂ Ω ′ ⇒ RΩ ⊂ RΩ ′ .
Remark 2.11. Notice that, for the trivial d.f.s. Ω = Ø, RØ = O(C), hence O(C) ⊂ RΩ for every d.f.s. Ω, i.e. entire functions are always Ω-continuable. Consequently, convergent series are always Ω-resurgent: C{z -1 } ⊂ RΩ . However, RΩ = O(C) does not imply Ω = Ø (consider for instance the d.f.s. Ω defined by Ω L = Ø for 0 ≤ L < 2 and Ω L = {1} for L ≥ 2). In fact, one can show

RΩ = O(C) ⇔ ∀L > 0, ∃L ′ > L such that Ω L ′ ⊂ { ω ∈ C | |ω| < L }.
Remark 2.12. In view of Lemma 2.6, we have RΩ = RΩ . Therefore, when dealing with Ωresurgence, we can always suppose that Ω coincides with its upper closure (by replacing Ω with Ω).

We now show the relation between resurgence in the sense of Definition 1.2 and Ω-resurgence in the sense of Definition 2.8.

Theorem 2.13. A formal series φ ∈ C[[z -1 ]] is resurgent if and only if there exists a d.f.s. Ω such that φ is Ω-resurgent. In other words, (2.8) R = Ω d.f.s. RΩ , R = Ω d.f.s.

RΩ .

Before proving Theorem 2.13, we state a technical result.

Lemma 2.14. Suppose that we are given a germ φ ∈ C{ζ} that can be analytically continued along a path γ : [0, t * ] → C of Π, and that F is a finite subset of C. Then, for each ε > 0, there exists a path γ

* : [0, t * ] → C of Π such that • γ * (0, t * ) ⊂ C \F , • L(γ * ) < L(γ) + ε,
• γ * (t * ) = γ(t * ), the germ φ can be analytically continued along γ * and the analytic continuations along γ and γ * coincide.

Proof of Lemma 2.14. Without loss of generality, we can assume that γ [0, t * ] is not reduced to {0} and that t → L(γ |t ) is strictly increasing.

The analytic continuation assumption allows us to find a finite subdivision 0

= t 0 < • • • < t m = t * of [0, t * ] together with open discs ∆ 0 , . . . , ∆ m so that, for each k, γ(t k ) ∈ ∆ k , the analytic continuation of φ along γ |t k extends holomorphically to ∆ k , and γ [t k , t k+1 ] ⊂ ∆ k if k < m. For each k ≥ 1, let us pick s k ∈ (t k-1 , t k ) such that γ [s k , t k ] ⊂ ∆ k-1 ∩ ∆ k ; increasing the value of s k if necessary, we can assume γ(s k ) / ∈ F .
Let us also set s 0 := 0 and s m+1 := t * , so that

0 ≤ k ≤ m ⇒                γ [s k , s k+1 ] ⊂ ∆ k , the analytic continuation of φ along γ |s k is holomorphic in ∆ k γ(s k ) / ∈ F except maybe if k = 0, γ(s k+1 ) / ∈ F except maybe if k = m.
We now define γ * by specifying its restriction γ * | [s k ,s k+1 ] for each k so that it has the same endpoints as γ| [s k ,s k+1 ] and,

-if the open line segment S := γ(s k ), γ(s k+1 ) is contained in C \F , then we let γ * | [s k ,s k+1 ] start at γ(s k ) and end at γ(s k+1 ) following S, by setting

γ * (t) := γ(s k ) + t-s k s k+1 -s k γ(s k+1 ) -γ(s k ) for t ∈ [s k , s k+1 ],
-if not, then S ∩ F = {ω 1 , . . . , ω ν } with ν ≥ 1 (depending on k); we pick ρ > 0 small enough so that

πρ < min 1 2 |ω i -γ(s k )|, 1 2 |ω i -γ(s k+1 )|, 1 2 |ω j -ω i |, ε ν(m+1) | 1 ≤ i, j, ≤ ν, i = j and we let γ * | [s k ,s k+1 ] follow S except that it circumvents each ω i by following a half-circle of radius ρ contained in ∆ k .
This way, γ * | [s k ,s k+1 ] stays in ∆ k ; the resulting path γ * : [0, t * ] → C is thus a path of analytic continuation for φ and the analytic continuations along γ and γ * coincide. On the other hand, the length of

γ * | [s k ,s k+1 ] is < |γ(s k ) -γ(s k+1 )| + ε m+1 , whereas the length of γ| [s k ,s k+1 ] is ≥ |γ(s k ) -γ(s k+1 )|, hence L(γ * ) < L(γ) + ε.
Proof of Theorem 2.13. Suppose first that Ω is a d.f.s. and φ ∈ RΩ . Then, for every L > 0, φ meets the requirement of Definition 1.1 with F L = ΩL , hence φ ∈ R. Thus RΩ ⊂ R, which yields one inclusion in (2.8).

Suppose now φ ∈ R. In view of Definition 1.1, the radius of convergence δ of φ is positive and, for each positive integer n, we can choose a finite set F n such that (2.9) the germ φ can be analytically continued along any path γ :

[0, 1] → C of Π such that L(γ) < (n + 1)δ and γ (0, 1] ⊂ C \F n .
Let F 0 := Ø. The property (2.9) holds for n = 0 too. For every real L ≥ 0, we set

Ω L := n k=0 F k with n := ⌊L/δ⌋.
One can check that Ω := (Ω L ) L∈R ≥0 is a d.f.s. which coincides with its upper closure. We will show that φ ∈ RΩ .

Pick an arbitrary γ :

[0, 1] → C such that γ ∈ Π Ω .
It is sufficient to prove that φ can be analytically continued along γ. Our assumption amounts to γ(t)

∈ C \Ω L(γ |t ) for each t ∈ [0, 1].
Without loss of generality, we can assume that γ [0, 1] is not reduced to {0} and that t → L(γ |t ) is strictly increasing. Let N := ⌊L(γ)/δ⌋.

We define a subdivision 0 = t 0 < t 1 < • • • < t N ≤ 1 by the requirement L(γ |tn ) = nδ and set

I n := [t n , t n+1 ) for 0 ≤ n < N , I N := [t N , 1].
For each integer n such that 0

≤ n ≤ N , (2.10) t ∈ I n ⇒ nδ ≤ L(γ |t ) < (n + 1)δ, thus Ω L(γ |t ) = n k=0 F k , in particular (2.11) t ∈ I n ⇒ γ(t) ∈ C \F n .
Let us check by induction on n that φ can be analytically continued along γ |t for any t ∈ I n .

If t ∈ I 0 , then γ |t has length < δ and the conclusion follows from (2.9).

Suppose now that 1 ≤ n ≤ N and that the property holds for n -1. Let t ∈ I n . By (2.10)-(2.11), we have L(γ |t ) < (n + 1)δ and γ [t n , t] ⊂ C \F n .

-If γ (0, t n ) ∩ F n is empty, then the conclusion follows from (2.9).

-If not, then, since C \F n is open, we can pick t * < t n so that γ [t * , t] ⊂ C \F n , and the induction hypothesis shows that φ can be analytically continued along γ |t * . We then apply Lemma 2.14 to γ |t * with F = F n and ε = (n + 1)δ -L(γ |t ): we get a path γ * : [0, t * ] → C which defines the same analytic continuation for φ as γ |t * , avoids F n and has length < L(γ |t * ) + ε.

The concatenation of γ * with γ| [t * ,t] is a path γ * * of length < (n + 1)δ which avoids F n , so it is a path of analytic continuation for φ because of (2.9), and so is γ itself.

Sums of discrete filtered sets

It is easy to see that, if Ω and Ω ′ are d.f.s., then the formula

(2.12) (Ω * Ω ′ ) L := { ω 1 + ω 2 | ω 1 ∈ Ω L 1 , ω 2 ∈ Ω ′ L 2 , L 1 + L 2 = L } ∪ Ω L ∪ Ω ′ L for L ∈ R ≥0
defines a d.f.s. Ω * Ω ′ . We call it the sum of Ω and Ω ′ .

The proof of the following lemma is left to the reader.

Lemma 2.15. The law * on the set of all d.f.s. is commutative and associative. The formula

Ω * n := Ω * • • • * Ω n times
(for n ≥ 1) defines an inductive system, which gives rise to a d.f.s.

Ω * ∞ := lim -→ n Ω * n .
As shown in [START_REF] Candelpergher | Approche de la résurgence[END_REF] and [START_REF] Deleabaere | Endless continuability and convolution product[END_REF], the sum of d.f.s. is useful to study the convolution product:

Theorem 2.16 ([OD15]
). Assume that Ω and Ω ′ are d.f.s. and φ ∈ RΩ , ψ ∈ RΩ ′ . Then the convolution product φ * ψ is Ω * Ω ′ -continuable.

Remark 2.17. Note that the notion of Σ-continuability in the sense of Definition 1.4 does not give such flexibility, because there are closed discrete sets Σ and Σ ′ such that Ω(Σ) * Ω(Σ ′ ) = Ω(Σ ′′ ) for any closed discrete Σ ′′ (take e.g. Σ = Σ ′ = (Z >0 √ 2) ∪ Z <0 ), and in fact there are Σ-continuable functions φ such that φ * φ is not Σ ′′ -continuable for any Σ ′′ .

In view of Theorem 2.13, a direct consequence of Theorem 2.16 is that the space of endlessly continuable functions R is stable under convolution, and the space of resurgent formal series R is a subring of the ring of formal series

C[[z -1 ]]. Given φ ∈ RΩ ∩ z -1 C[[z -1 ]],
Theorem 2.16 guarantees the Ω * k -resurgence of φk for every integer k, hence its Ω * ∞ -resurgence. This is a first step towards the proof of the resurgence of F ( φ) for F (w) = c k w k ∈ C{w}, i.e. Theorem 1.3 in the case r = 1, however some analysis is needed to prove the convergence of c k φk in some appropriate topology. What we need is a precise estimate for the convolution product of an arbitrary number of endlessly continuable functions, and this will be the content of Theorem 4.8. In Section 5, the substitution problem will be discussed in a more general setting, resulting in Theorem 5.2, which is more general and more precise than Theorem 1.3.

Discrete doubly filtered sets and a more general definition of resurgence

We now define the spaces R dv and R dv which were alluded to in the introduction. We first require the notion of "direction variation" of a C 1+Lip path.

We denote by Π dv the set of all C 1 paths γ belonging to Π, such that γ ′ is Lipschitz and never vanishes. By Rademacher's theorem, γ ′′ exists a.e. on the interval of definition [0, t * ] of γ and is essentially bounded. We can thus define the direction variation V (γ) of γ ∈ Π dv by t) with a real-valued Lipschitz function θ, and then Im γ ′′ (t) γ ′ (t) = θ ′ , hence V (γ) is nothing but the length of the path θ). Note that the function t → V (γ |t ) is Lipschitz.

V (γ) := t * 0 Im γ ′′ (t) γ ′ (t) dt (notice that one can write γ ′ (t) = |γ ′ (t)| e iθ(
Definition 2.18. A convergent power series φ ∈ C{ζ} is said to be endlessly continuable w.r.t. bounded direction variation (and we write φ ∈ R dv ) if, for every real L, M > 0, there exists a finite subset F L,M of C such that φ can be analytically continued along every path γ :

[0, 1] → C such that γ ∈ Π dv , L(γ) < L, V (γ) < M , and γ (0, 1] ⊂ C \F L,M . We also set R dv := B -1 (C δ ⊕ R dv ). Note that R ⊂ R dv ⊂ C{ζ} and R ⊂ R dv ⊂ C[[z -1 ]].
Definition 2.19. A discrete doubly filtered set, or d.d.f.s. for short, is a family Ω

= (Ω L,M ) L,M ∈R ≥0 that satisfies i) Ω L,M is a finite subset of C for each L and M , ii) Ω L 1 ,M 1 ⊆ Ω L 2 ,M 2 when L 1 ≤ L 2 and M 1 ≤ M 2 ,
iii) there exists δ > 0 such that Ω δ,M = Ø for all M ≥ 0.

Notice that a d.f.s. Ω can be regarded as a d.d.f.s. Ω dv by setting Ω dv L,M := Ω L for L, M ≥ 0. For a d.d.f.s. Ω, we set

S Ω := (µ, λ, ω) ∈ R 2 × C | µ ≥ 0, λ ≥ 0 and ω ∈ Ω λ,µ and M Ω := R 2 × C \ S Ω , where S Ω is the closure of S Ω in R 2 × C. We call Ω-allowed path any γ ∈ Π dv such that (2.13) γ dv (t) := V (γ |t ), L(γ |t ), γ(t) ∈ M Ω for all t.
We denote by Π dv Ω the set of all Ω-allowed paths. Finally, the set of Ω-continuable functions (resp. Ω-resurgent series) is defined in the same way as in Definition 2.8, and denoted by R dv Ω (resp. R dv Ω ). Arguing as for Theorem 2.13, one obtains

(2.14) R dv = Ω d.d.f.s. R dv Ω , R dv = Ω d.d.f.s. R dv Ω .
The sum Ω * Ω ′ of two d.d.f.s. Ω and Ω ′ is the d.d.f.s. defined by setting, for L, M ∈ R ≥0 ,

(2.15)

(Ω * Ω ′ ) L,M := { ω 1 + ω 2 | ω 1 ∈ Ω L 1 ,M , ω 2 ∈ Ω ′ L 2 ,M , L 1 + L 2 = L } ∪ Ω L,M ∪ Ω ′ L,M .
3 The endless Riemann surface associated with a d.f.s.

We introduce the notion of Ω-endless Riemann surfaces for a d.f.s. Ω as follows:

Definition 3.1. We call Ω-endless Riemann surface any triple (X, p, 0) such that X is a connected Riemann surface, p : X → C is a local biholomorphism, 0 ∈ p -1 (0), and any path γ

: [0, 1] → C of Π Ω has a lift γ : [0, 1] → X such that γ(0) = 0. A morphism of Ω-endless Riemann surfaces is a local biholomorphism q : (X, p, 0) → (X ′ , p ′ , 0 ′ ) that makes the following diagram commutative: (X, 0) (X ′ , 0 ′ ) (C, 0) G G 0 0 ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ Ð Ð ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ q p p ′
In this section, we prove the existence of an initial object (X Ω , p Ω , 0 Ω ) in the category of Ω-endless Riemann surfaces: Theorem 3.2. There exists an Ω-endless Riemann surface (X Ω , p Ω , 0 Ω ) such that, for any Ωendless Riemann surface (X, p, 0), there is a unique morphism q : (X Ω , p Ω , 0 Ω ) → (X, p, 0).

The Ω-endless Riemann surface (X Ω , p Ω , 0 Ω ) is unique up to isomorphism and X Ω is simply connected.

Construction of X Ω

We first define "skeleton" of Ω:

Definition 3.3. Let V Ω ⊂ ∞ n=1 (C × Z) n be the set of vertices v := ((ω 1 , σ 1 ), • • • , (ω n , σ n )) ∈ (C × Z) n
that satisfy the following conditions:

1) (ω 1 , σ 1 ) = (0, 0) and (ω j , σ j ) ∈ C ×(Z \{0}) for j ≥ 2, 2) ω j = ω j+1 for j = 1, • • • , n -1, 3) ω j ∈ ΩL j (v) with L j (v) := j-1 i=1 |ω i+1 -ω i | for j = 2, • • • , n. Let E Ω ⊂ V Ω × V Ω be the set of edges e = (v ′ , v
) that satisfy one of the following conditions:

i) v = ((ω 1 , σ 1 ), • • • , (ω n , σ n )) and v ′ = ((ω 1 , σ 1 ), • • • , (ω n , σ n ), (ω n+1 , ±1)), ii) v = ((ω 1 , σ 1 ), • • • , (ω n , σ n )) and v ′ = ((ω 1 , σ 1 ), • • • , (ω n , σ n + 1)) with σ n ≥ 1, iii) v = ((ω 1 , σ 1 ), • • • , (ω n , σ n )) and v ′ = ((ω 1 , σ 1 ), • • • , (ω n , σ n -1)) with σ n ≤ -1.
We denote the directed tree diagram (V Ω , E Ω ) by Sk Ω and call it skeleton of Ω.

Notation 3.4. For v ∈ V Ω ∩ (C × Z) n , we set ω(v) := ω n and L(v) := L n (v).
From the definition of Sk Ω , we find the following

Lemma 3.5. For each v ∈ V Ω \ {(0, 0)}, there exists a unique vertex v ↑ ∈ V Ω such that (v, v ↑ ) ∈ E Ω .
To each v ∈ V Ω we assign a cut plane U v , defined as the open set

U v := C \ C v ∪ v ′ → i v C v ′ →v , where v ′ → i v
is the union over all the vertices v ′ ∈ V Ω that have an edge

(v ′ , v) ∈ E Ω of type i), ω( ) ω( ) ω( 1 ) ′ ⭡ ω( 2 ) ′ ω( 3 ) ′ 1 → ´ 2 → ´ 3 → ´Figure 1: The set U v . C v := Ø when v = (0, 0), {ω n -s(ω n -ω n-1 ) | s ∈ R ≥0 } when v = (0, 0), C v ′ →v := {ω n+1 + s(ω n+1 -ω n ) | s ∈ R ≥0 }.
We patch the U v 's along the cuts according to the following rules:

Suppose first that (v ′ , v) is an edge of type i), with v ′ = (v, (ω n+1 , σ n+1 )) ∈ V Ω . To it, we assign a line segment or a half-line ℓ v ′ →v as follows: If there exists u = (v, (ω

′ n+1 , ±1)) ∈ V Ω such that ω ′ n+1 ∈ C v ′ →v \ {ω n+1 }, take u (0) = (v, (ω (0)
n+1 , ±1)) ∈ V Ω so that |ω

(0) n+1 - ω n+1 ) | s ∈ (0, 1)} to (v ′ , v). Otherwise, we assign the open half-line ℓ v ′ →v := C v ′ →v \ {ω n+1 } to (v ′ , v). Since each Ω L (L ≥ 0) is finite, we can take a connected neighborhood U v ′ →v of ℓ v ′ →v so that (3.1) U v ′ →v \ ℓ v ′ →v = U + v ′ →v ∪ U - v ′ →v and U ± v ′ →v ⊂ U v ∩ U v ′ , where U ± v ′ →v := {ζ ∈ U v ′ →v | ±Im(ζ • ζ ′ ) > 0 for ζ ′ ∈ ℓ v ′ →v }. Then, if σ n+1 = 1, we glue U v and U v ′ along U - v ′ →v , whereas if σ n+1 = -1 we glue them along U + v ′ →v . Suppose now that (v ′ , v
) is an edge ot type ii) and iii). As in the case of i), if there exists

u = (v, (ω ′ n+1 , ±1)) ∈ V Ω such that ω ′ n+1 ∈ C v \ {ω n }, then we take u (0) = (v, (ω (0) n+1 , ±1)) ∈ V Ω so that |ω (0) n+1 -ω n | is minimum and assign ℓ v ′ →v := {ω n + s(ω (0) n+1 -ω n ) | s ∈ (0, 1)} to (v ′ , v).
Otherwise, we assign ℓ v ′ →v := C v \{ω n } to (v ′ , v). Then, we take a connected neighborhood U v ′ →v of ℓ v ′ →v satisfying (3.1), and glue U v and U v ′ along U - v ′ →v in case ii), and along U + v ′ →v in case iii). Patching the U v 's and the U v ′ →v 's according to the above rules, we obtain a Riemann surface X Ω , in which we denote by 0 Ω the point corresponding to 0 ∈ U (0,0) . The map p Ω : X Ω → C is naturally defined using local coordinates

U v and U v ′ →v . ω( ) ′ ω( ) (0) → + → -- Figure 2: The set U v ′ →v .
Let U e , ℓ e (e ∈ E Ω ) and U v (v ∈ V Ω ) respectively denote the subsets of X Ω defined by U e , ℓ e and U v . Notice that each ζ ∈ X Ω belongs to one of the ℓ e 's or U v 's (e ∈ E Ω or v ∈ V Ω ). Therefore, we have the following decomposition of X Ω :

X Ω = v∈V Ω U v ⊔ e∈E Ω ℓ e .
Definition 3.6. We define a function L : X Ω → R ≥0 by the following formula:

L(ζ) := L(v) + |p(ζ) -ω(v)| when ζ ∈ U v ⊔ ℓ v→v ↑ .
We call L(ζ) the canonical distance of ζ from 0 Ω . We obtain from the construction of L the following Lemma 3.7. The function L : X Ω → R ≥0 is continuous and satisfies the following inequality for every γ ∈ Π Ω :

L(γ(t)) ≤ L(γ |t ) for t ∈ [0, 1].
We now show the fundamental properties of X Ω .

Lemma 3.8. The Riemann surface X Ω constructed above is simply connected.

Proof. We first note that, since Sk Ω is connected, X Ω is path-connected. Let γ : [0, 1] → X Ω be a path such that γ(0) = γ(1). Since the image of γ is a compact set in X Ω , we can take finite number of vertices {v j } p j=1 ⊂ V Ω and {e j } q j=1 ⊂ E Ω so that v 1 = (0, 0) and the image of γ is covered by {U v j } p j=1 and {U e j } q j=1 . Since each of {v j } p j=2 and {e j } q j=1 has a path to v 1 that contains it, interpolating finite number of the vertices and the edges if necessary, we may assume that the diagram Sk defined by {v j } p j=1 and {e j } q j=1 are connected in Sk Ω . Now, let U be the union of {U v j } p j=1 and {U e j } q j=1 . Since all of the open sets are simply connected and Sk is acyclic, we can inductively confirm using the van Kampen's theorem that U is simply connected. Therefore, the path γ is contracted to the point 0 Ω . It proves the simply connectedness of X Ω .

Lemma 3.9. The Riemann surface X Ω constructed above is Ω-endless.

Proof. Take an arbitrary Ω-allowed path γ and δ, L > 0 so that γ

∈ Π δ,L Ω . Let V δ,L Ω denote the set of vertices v = ((ω 1 , σ 1 ), • • • , (ω n , σ n )) ∈ V Ω that satisfy L δ (v) := L n (v) + n j=2 (|σ j | -1)δ ≤ L and set E δ,L Ω := {(v, v ↑ ) ∈ E Ω | v ∈ V δ,L Ω }. Notice that V δ,L Ω and E δ,L Ω are finite. We set for ε > 0 and v ∈ V δ,L Ω U δ,L,ε v := {ζ ∈ U v | inf (v ′ ,v)∈E Ω |ζ -ω(v ′ )| ≥ δ, D ε ζ ⊂ U v } ∩ D L-L δ (v) ω(v)
,

where D r ζ := { ζ ∈ C | | ζ -ζ| ≤ r} for ζ ∈ C, r > 0. We also set for ε > 0 and (v, v ↑ ) ∈ E δ,L Ω U δ,L,ε v→v ↑ := {ζ ∈ U v→v ↑ | min j=1,2 |ζ -ωj | ≥ δ, inf ζ∈ℓv→v ↑ |ζ -ζ| ≤ ε} ∩ D L-L δ (v ↑ ) ω(v)
, where ω1 := ω(v) and ω2 is the other endpoint of ℓ v→v ↑ if it exists and ω2 := ω(v) otherwise. Since E δ,L Ω are finite set, we can take ε > 0 sufficiently small so that

D ε ζ ⊂ U v→v ↑ for all ζ ∈ U δ,L,ε v→v ↑ and (v, v ↑ ) ∈ E δ,L
Ω . We fix such a number ε > 0. Now, let I be the maximal interval such that the restriction of γ to I has a lift γ on X Ω . Obviously, I = Ø and I is open. Assume that I = [0, a) for a ∈ (0, 1]. We take b ∈ (0, a) so that L(γ |a ) -L(γ |b ) < ε. Then, notice that, since γ ∈ Π δ,L Ω and γ |b has a lift on

X Ω , γ(b) is in U δ,L,ε v for v ∈ V δ,L Ω or U δ,L,ε e for e ∈ E δ,L Ω . Since D ε γ(b) ⊂ U v (resp., D ε γ(b) ⊂ U e ) when γ(b) ∈ U δ,L,ε v (resp., γ(b) ∈ U δ,L,ε e ),

Proof of Theorem 3.2

We first show the following: Lemma 3.10. For all ε > 0 and ζ ∈ X Ω , there exists an Ω-allowed path γ such that L(γ) < L(ζ) + ε and its lift γ on X Ω satisfies γ(0) = 0 Ω and γ(1) = ζ.

Proof. Let ζ ∈ U v for v = ((ω 1 , σ 1 ), • • • , (ω n , σ n ))
. We consider a polygonal curve P 0 ζ obtained by connecting line segments [ω j , ω j+1 ] (j = 1, • • • , n), where we set ω n+1 := p Ω (ζ) for the sake of notational simplicity. Now, collect all the points ω j,k on (ω j , ω j+1 ) such that (L j,k , ω) ∈ S Ω , where

L j,k := L j (v) + |ω j,k -ω j |. Since (3.2) S Ω ∩ {λ ∈ R ≥0 | |λ| ≤ L} × C is written for each L > 0 by the union of finite number of line segments of the form {λ ∈ R ≥0 | L ≤ λ ≤ L} × {ω} ( L > 0, ω ∈ C),
such points are finite. We order ω j and ω j,k so that L j (v) and L j,k increase along the order and denote the sequence by

(ω ′ 1 , ω ′ 2 , • • • , ω ′ n ′ ). We set L ′ j := j-1 i=1 |ω ′ i+1 -ω ′ i |. We extend v to v ′ = ((ω ′ 1 , σ ′ 1 ), • • • , (ω ′ n ′ , σ ′ n ′ )) by setting σ ′ j = 1 (resp., σ ′ j = -1) when (ω ′ j , L ′ j ) = (ω i,k , L i,k
) for some i, k and σ i+1 ≥ 1 (resp., σ i+1 ≤ -1). Then, in view of (3.2), we can take δ > 0 so that

{(L ′ j + |ζ ′ -ω ′ j | + δ, ζ ′ ) | ζ ′ ∈ (ω ′ j , ω ′ j+1 )} ∩ S Ω = Ø, {(L ′ j + δ, ζ ′ ) | 0 < |ζ ′ -ω ′ j | < δ} ∩ S Ω = Ø hold for j = 1, • • • , n ′ . Let ω ′ j,-(resp., ω ′ j,+ ) be the intersection point of [ω ′ j-1 , ω ′ j ] (resp., [ω ′ j , ω ′ j+1 ]) and C ε ′ ω ′ j := {ζ ′ ∈ C | |ζ ′ -ω ′ j | = ε ′ } for sufficiently small ε ′ > 0. We replace the part [ω ′ j,-, ω ′ j ] ∪ [ω ′ j , ω ′ j,+
] of ℓ with a path that goes anti-clockwise (resp., clockwise) along

C ε ′ ω ′ j from ω ′ j,-to ω ′ j,+
and turns around ω ′ j (|σ ′ j | -1)-times when σ ′ j ≥ 1 (resp., when σ ′ j ≤ -1). Let P ε ′ ζ denote a path ζ defines an Ω-allowed path and its lift P ε ′ ζ on X Ω satisfies the conditions. Further, by taking ε ′ sufficiently small so that 2πε

′ n ′ j=2 |σ ′ j | < ε, we find L(P ε ′ ζ ) < L(ζ) + ε, hence one can take γ = P ε ′ ζ . When ζ ∈ ℓ e for an edge e = (v, v ↑ ) ∈ E Ω , we can construct such a path P ε ′ ζ ∈ Π Ω
Q ε ′ ζ,ζ ′ ∈ Π Ω (ζ ′ ∈ U ζ ) of the path γ = P ε ′ ζ constructed in the proof of Lemma 3.10 such that L(Q ε ′ ζ,ζ ′ ) < L(ζ ′ ) + 2ε for each ζ ′ ∈ U ζ and the lift Q ε ′ ζ,ζ ′ on X Ω satisfies Q ε ′ ζ,ζ ′ (0) = 0 and Q ε ′ ζ,ζ ′ (1) = ζ ′ .
Indeed, the deformation of P ε ′ ζ is concretely given as follows:

-

When ζ ∈ U v for v ∈ V Ω , taking a neighborhood U ζ ⊂ U v of ζ
sufficiently small, we find that the family of the paths

P ε ′ ζ ′ (ζ ′ ∈ U ζ )
constructed in the proof of Lemma 3.10 gives such a deformation.

-When ζ ∈ ℓ e for e ∈ E Ω , we can take a neighborhood

U ζ ⊂ U e of ζ so that [ω ′ n ′ ,+ (ζ ′ ), p Ω (ζ ′ )] ⊂ U e for all ζ ′ ∈ U ζ , where ω ′ n ′ ,+ (ζ ′ ) is the intersection point of [ω ′ n ′ , p Ω (ζ ′ )] and C ε ′ ω ′ n ′ . Define a deformation Q ε ′ ζ,ζ ′ (ζ ′ ∈ U e ) of P ε ′ ζ by continuously varying the arc of C ε ′ ω ′ n ′ from ω ′ n ′ ,-to ω ′ n ′ ,+ (ζ ′ ) and the line segment [ω ′ n ′ ,+ (ζ ′ ), p Ω (ζ ′ )] and fixing the other part of P ε ′ ζ . Then, shrinking U ζ if necessary, we find that Q ε ′ ζ,ζ ′ satisfies Q ε ′ ζ,ζ ′ ∈ Π Ω and L(Q ε ′ ζ,ζ ′ ) < L(ζ ′ ) + 2ε for each ζ ′ ∈ U ζ .
Beware that, when the edge (v, v ↑ ) is the type i),

Q ε ′ ζ,ζ ′ is different from P ε ′ ζ ′ for ζ ∈ ℓ v→v ↑ and ζ ′ ∈ U ζ ∩ U v ↑ . On the other hand, Q ε ′ ζ,ζ ′ = P ε ′ ζ ′ holds for ζ ′ ∈ U ζ ∩ U v . When the edge (v, v ↑ ) is the type ii) or iii), Q ε ′ ζ,ζ ′ = P ε ′ ζ ′ holds for ζ ∈ ℓ v→v ↑ and ζ ′ ∈ U ζ .
Let (X, p, 0) be an Ω-endless Riemann surface. For each ζ ∈ X Ω , take γ ∈ Π Ω such that γ(1) = ζ and let γ X be its lift on X. Then, define a map q : X Ω → X by q(ζ) = γ X (1). We now show the well-definedness of q. For that purpose, it suffices to prove the following Proposition 3.12. Let γ 0 , γ 1 ∈ Π Ω such that γ 0 (1) = γ 1 (1). Then, there exists a continuous family (H s ) s∈[0,1] of Ω-allowed paths satisfying the conditions 1. H s (0) = 0 and H s (1) = γ 0 (1) for all s ∈ [0, 1],

2. H j = γ j for j = 0, 1.

The proof of Proposition 3.12 is reduced to the following Lemma 3.13. For each γ ∈ Π Ω and ε ′ > 0 sufficiently small, there exists a continuous family ( Hs ) s∈[0,1] of Ω-allowed paths satisfying the following conditions:

1. L Hs ≤ L(γ |s ) and Hs (1) = γ(s) for all s ∈ [0, 1],

2. Hs = P ε ′ γ(s) for s = 0, 1.

Notice that, since γ(0

) = 0 Ω , P ε ′ γ(0) is the constant map P ε ′ γ(0) = 0.
Reduction of Proposition 3.12 to Lemma 3.13. For each γ ∈ Π Ω and s ∈ (0, 1], define H s using Hs constructed in Lemma 3.13 as follows:

H s (t) := Hs (t/s) when t ∈ [0, s], γ(t) when t ∈ [s, 1].
It extends continuously to s = 0 and gives a continuous family (H s ) s∈[0,1] of Ω-allowed paths satisfying the assumption in Proposition 3.12 with γ 0 = γ and γ 1 = P ε ′ γ(1) . Now, let γ 0 and γ 1 be the Ω-allowed paths satisfying the assumption in Proposition 3.12. Applying the above discussion to each of γ 0 and γ 1 , we obtain two families of Ω-allowed paths connecting them to P ε ′ γ 0 (1) and, concatenating the deformations at P ε ′ γ 0 (1) , we obtain a deformation (H s ) s∈[0,1] satisfying the conditions in Proposition 3.12.

Proof of Lemma 3.13. Take δ, L > 0 so that γ ∈ Π δ,L Ω . We first show the following:

(3.3) When γ(t 0 ) ∈ U v→(0,0) for t 0 ∈ (0, 1] and v = ((0, 0), (ω 2 , σ 2 )), the following estimate holds for t ∈ [t 0 , 1]:

L(γ(t)) + |ω 2 | 2 + δ 2 -|ω 2 | ≤ L(γ |t ).
Notice that, since γ ∈ Π δ,L Ω , the length L(γ |t 0 ) of γ |t 0 must be longer than that of the polygonal curve C obtained by concatenating the line segments [0, ω 2 + δe iθ ] and [ω 2 + δe iθ , γ(t 0 )], where θ = arg(ω 2 )σ 2 π/2. Then, we find that, for an arbitrary ε > 0, taking ε ′ > 0 sufficiently small, the path γε ′ obtained by concatenating the paths P ε ′ γ(t 0 ) and γ|

[t 0 ,1] satisfies γε ′ ∈ Π Ω , γε ′ (t) = γ(t) and L(γ ε ′ |t ) ≤ L(γ 0 |t ) + ε for t ∈ [t 0 , 1]. Therefore, we have L(γ(t)) ≤ L(γ 0 |t ) for t ∈ [t 0 , 1] Since L(C) ≥ |ω 2 | 2 + δ 2 + |γ(t 0 ) -ω 2 |, we find L(γ |t ) = L(γ 0 |t ) + L(γ |t 0 ) -L([0, γ(t 0 )]) ≥ L(γ(t)) + |ω 2 | 2 + δ 2 -|ω 2 |
holds for t ∈ [t 0 , 1], and hence, we obtain (3.3). Now, we shall construct (H s ) s∈[0,1] . Let ε > 0 be given. We assign the path P

ε ′ t γ(t) (ε ′ t > 0) to each t ∈ [0, 1]
and take a neighborhood U γ(t) of γ(t) and the deformation Q

ε ′ t γ(t),ζ ′ (ζ ′ ∈ U γ(t) ) of P ε ′ t γ(t)
constructed in Lemma 3.11. Then, we can cover [0, 1] by a finite number of intervals

I j = [a j , b j ] (j = 1, 2, • • • , k) satisfying the following conditions: -The interior I • j of I j satisfies I • j 1 ∩ I • j 2 = Ø when |j 1 -j 2 | ≤ 1 and I j 1 ∩ I j 2 = Ø otherwise.
-There exists t j ∈ I j such that t j < t j+1 for j = 1,

• • • , k -1 and γ(I j ) ⊂ U γ(t j ) .
Notice that, since U γ(t) is taken for each t ∈ [0, 1] so that it is contained in one of the charts U v (v ∈ V Ω ) or U e (e ∈ E Ω ), one of the followings holds:

-

γ(t j ) ∈ U v and γ(I j ) ⊂ U v (v ∈ V Ω ).
-γ(t j ) ∈ ℓ e and γ(I j ) ⊂ U e (e ∈ E Ω ).

We set ε ′ = min j {ε ′ t j | γ(t j ) / ∈ U (0,0) }. Then, P ε ′ γ(t j ) and its deformation

Q ε ′ γ(t j ),ζ ′ (ζ ′ ∈ U γ(t j )
) also satisfy the conditions in Lemma 3.10 and Lemma 3.11. Let J E ⊂ {1, • • • , k} denote the set of suffixes satisfying the condition that there exists e ∈ E Ω such that γ(t j ) ∈ ℓ e and let j 0 be the minimum of J E . Shrinking the neighborhood U γ(t) for each t ∈ [0, 1] at the first, we may assume without loss of generality that,

-|γ(t) -γ(t j )| ≤ ε for t ∈ I j and j = 1, • • • , k, -if j, j + 1 ∈ J E , there exists an edge e ∈ E Ω such that γ(t j ), γ(t j+1 ) ∈ ℓ e .
Recall that, from the construction of

Q ε ′ ζ,ζ ′ , Q ε ′ γ(t j ),γ(t) = Q ε ′ γ(t j+1 ),γ(t) for t ∈ I j ∩ I j+1
except for the cases where there exists an edge e

= (v, v ↑ ) ∈ E Ω of the type i) such that -γ(t j ) ∈ U e and γ(t j+1 ) ∈ U v ↑ , -γ(t j ) ∈ U v ↑ and γ(t j+1 ) ∈ U e .
In the first case, the difference between

Q ε ′ γ(t j ),γ(t) and Q ε ′ γ(t j+1 ),γ(t) is the part from ω t (v ↑ ) to γ(t), where ω t (v ↑ ) is the intersection point of C ε ′ ω(v ↑ ) and [ω(v ↑ ), γ(t)]:
Let ω e,i (i = 0, • • • , m + 1) be the points on the line segment [ω(v ↑ ), ω(v)] satisfying the conditions (L e,i , ω e,i ) ∈ S Ω and L e,i < L e,i+1 , where L e,i :=

L(v ↑ ) + |ω e,i -ω(v ↑ )|. Then, the part of Q ε ′ γ(t j ),γ(t) from ω t (v ↑ ) to γ(t) is given by concatenating the arcs of C ε ′ ω e,i (i = 0, • • • , m + 1), the intervals of the line segment [ω(v ↑ ), ω(v)] and [ω t (v), γ(t)], where ω t (v) is the intersection point of C ε ′ ω(v) and [ω(v), γ(t)]. (See Figure 4 (a).) On the other hand, Q ε ′ γ(t j+1 ),γ(t) goes directly from ω t (v ↑ ) to γ(t). (See Figure 4 (d).) Now, let ω t i,+ (resp. ω t i,-) be the intersection point of C ε ′ ω e,i and [ω t (v ↑ ), ω t (v)] that is the closer to ω t (v) (resp. ω t (v ↑ )). While t moves on I j ∩ I j+1 , we first deform the part of Q ε ′ γ(t j ),γ(t) from ω t (v ↑ ) to ω t (v) to the line segment [ω t (v ↑ ), ω t (v)] by shrinking the part of Q ε ′ γ(t j ),γ(t) from ω t i,-
to ω t i,+ (resp. from ω t i,+ to ω t i+1,-) to the line segment [ω t i,-, ω t i,+ ] (resp. [ω t i,+ , ω t i+1,-]) for each i. (See Figure 4 (b) and (c).) Then, further shrinking the polygonal line given by concatenating [ω t (v ↑ ), ω t (v)] and [ω t (v), γ(t)] to the line segment [ω t (v ↑ ), γ(t)], we obtain a continuous family of Ω-allowed paths Hs s∈[t j ,t j+1 ] satisfying the following conditions:

-Hs = Q ε ′ γ(t j ),γ(s) when s ∈ [t j , t j+1 ] \ I j+1 , -Hs = Q ε ′ γ(t j+1 ),γ(s) when s ∈ [t j , t j+1 ] \ I j , -L Hs ≤ L Q ε ′ γ(t j ),γ(s) and Hs (1) = γ(s) when s ∈ I j ∩ I j+1 . ω ,0 ω ,1 ω ,2 γ( ) (a) (b) (c) (d)
Figure 4:

For the second case, we can also construct a continuous family of Ω-allowed paths Hs s∈[t j ,t j+1 ] satisfying the first and the second conditions above and -L Hs ≤ L Q ε ′ γ(t j+1 ),γ(s) and Hs (1) = γ(s) when s ∈ I j ∩ I j+1 .

Then, we can continuously extend Hs to [0, 1] by interpolating it by Q ε ′ γ(t j ),γ(s) so that it satisfies

(3.4) L Hs ≤ max j L Q ε ′ γ(t j ),γ(s) | s ∈ I j and Hs (1) = γ(s) for all s ∈ [0, 1].
Since I j 0 is taken so that |γ(t)γ(t j 0 )| ≤ ε holds on I j 0 , applying (3.3) with t 0 = t j 0 , we have the following estimates:

L(γ(t)) + |ω 2 | 2 + δ 2 -|ω 2 | -ε ≤ L(γ |t ) for t ∈ [a j 0 , 1].
On the other hand, since γ(t) ∈ U (0,0) for t ∈ [0, a j 0 ], we find L Q ε ′ γ(t j ),γ(t) = L(γ(t)) holds for t ∈ I j and j < j 0 from the construction of

Q ε ′ ζ,ζ ′ . Therefore, taking ε > 0 sufficiently small so that 3ε ≤ |ω 2 | 2 + δ 2 -|ω 2 |,
we obtain the following estimates from Lemma 3.10 and (3.4):

L Hs ≤ L(γ |s ) for s ∈ [0, 1].
Finally, from the construction of Hs , we find that Hs satisfies Hs = P ε ′ γ(s) for s = 0, 1.

Since p Ω = p • q and p is isomorphic near 0, all the maps q : X Ω → X must coincide near 0 Ω , and hence, uniqueness of q follows from the uniqueness of the analytical continuation of q. Finally, X Ω is unique up to isomorphism because X Ω is an initial object in the category of Ω-endless Riemann surfaces.

Supplement to the properties of X Ω

Let O X denote the sheaf of holomorphic functions on a Riemann surface X and consider the natural morphism p * Ω :

p -1 Ω O C → O X Ω induced by p Ω : X Ω → C .
Since X Ω is simply connected, we obtain the following: Proposition 3.14. Let φ ∈ O C,0 . Then the followings are equivalent:

i) φ ∈ O C,0 is Ω-continuable, ii) p * Ω φ ∈ O X Ω ,0 Ω can be analytically continued along any path on X Ω , iii) p * Ω φ ∈ O X Ω ,0 Ω can be extended to Γ(X Ω , O X Ω ).
Therefore, we find

p * Ω : RΩ ∼ -→ Γ(X Ω , O X Ω ).
Notation 3.15. For L, δ > 0, using Π δ,L Ω of (2.7), we define a compact subset

K δ,L Ω of X Ω by (3.5) K δ,L Ω := ζ ∈ X Ω | ∃γ ∈ Π δ,L Ω such that ζ = γ(1) .
Notice that X Ω is exhausted by (K δ,L Ω ) δ,L>0 . Therefore, the family of seminorms

• δ,L Ω (δ, L > 0) defined by f δ,L Ω := sup ζ∈K δ,L Ω | f (ζ)| for f ∈ Γ(X Ω , O X Ω )
induces a structure of Fréchet space on Γ(X Ω , O X Ω ).

Definition 3.16. We introduce a structure of Fréchet space on RΩ by a family of seminorms

• δ,L Ω (δ, L > 0) defined by φ δ,L Ω := |ϕ 0 | + p * Ω φ δ,L Ω for φ ∈ RΩ , where B( φ) = ϕ 0 δ + φ ∈ C δ ⊕ RΩ . Let Ω ′ be a d.f.s. such that Ω ⊂ Ω ′ . Since Π Ω ′ ⊂ Π Ω , X Ω is Ω ′ -endless. Therefore, Theorem 3.2 yields a morphism q : (X Ω ′ , p Ω ′ , 0 Ω ′ ) → (X Ω , p Ω , 0 Ω ), which induces a morphism q * : q -1 O X Ω → O X Ω ′ . Since q(K δ,L Ω ′ ) ⊂ K δ,L Ω , we have q * f δ,L Ω ′ ≤ f δ,L Ω for f ∈ Γ(X Ω , O X Ω ),
and hence,

φ δ,L Ω ′ ≤ φ δ,L Ω for φ ∈ RΩ .
In view of Theorem 4.8 below, the product map RΩ × RΩ ′ → RΩ * Ω ′ is continuous and hence, when Ω * Ω = Ω, RΩ is a Fréchet algebra.

3.4

The endless Riemann surface associated with a d.d.f.s.

In this section, we discuss the construction of the endless Riemann surfaces associated with an arbitrary d.d.f.s. Ω. Let us first define the skeleton of Ω:

Definition 3.17. Let V Ω ⊂ ∞ n=1 (C × Z) n be the set of vertices v := ((ω 1 , σ 1 ), • • • , (ω n , σ n )) ∈ (C × Z) n
that satisfy the conditions 1) and 2) in Definition 3.3 and

3') M j (v), L j (v), ω j ∈ S Ω for j = 2, • • • , n, with L j (v) := j-1 i=1 |ω i+1 -ω i | (j = 2, • • • , n), M j (v) :=        0 (j = 2), j-1 i=2 A i (v) + 2π(|σ i | -1) (j = 3, • • • , n),
and

A i (v) := |θ i | if θ i σ i ≥ 0, 2π -|θ i | if θ i σ i < 0,
where

θ i := arg ω i+1 -ω i ω i -ω i-1 is taken so that θ i ∈ (-π, π]. Let E Ω ⊂ V Ω × V Ω
be the set of edges e = (v ′ , v) that satisfy one of the conditions i) ∼ iii) in Definition 3.3. We denote the directed tree diagram (V Ω , E Ω ) by Sk Ω and call it skeleton of Ω. Now, assigning a cut plane U v (resp. an open set U e ) to each v ∈ V Ω (resp. each e ∈ E Ω of type i)) defined by totally the same way with Section 3.1 and patching them as in Section 3.1, we obtain an initial object (X Ω , p Ω , 0 Ω ) in the category of Ω-endless Riemann surfaces associated with a d.d.f.s. Ω. We denote the lift of γ ∈ Π dv Ω on X Ω by γ.

Estimates for the analytic continuation of iterated convolutions

In this section, our aim is to prove the following theorem, which is the analytical core of our study of the convolution product of endlessly continuable functions.

Theorem 4.1. Let δ, L > 0 be real numbers. Then there exist c, δ ′ > 0 such that, for every d.f.s. Ω such that Ω 4δ = Ø, for every integer n ≥ 1 and for every f1 , . . . , fn ∈ RΩ , the function 1 * f1 * • • • * fn (which is known to belong to RΩ * n ) satisfies

(4.1) p * Ω * n 1 * f1 * • • • * fn (ζ) ≤ c n n! sup L 1 +•••+Ln=L p * Ω f1 δ ′ ,L 1 Ω • • • p * Ω fn δ ′ ,Ln Ω for ζ ∈ K δ,L Ω * n
(with notation (3.5)).

Using the Cauchy inequality, the identity d dζ (1 * f1 * • • • * fn ) = f1 * • • • * fn and the inverse Borel transform, one easily deduces the following Corollary 4.2. Let δ, L > 0 be real numbers. Then there exist c, δ ′ , L ′ > 0 such that, for every d.f.s. Ω such that Ω 4δ = Ø, for every integer n ≥ 1 and for every f1 , . . . , fn ∈ RΩ without constant term, the formal series f1 • • • fn (which is known to belong to RΩ * n ) satisfies

f1 • • • fn δ,L Ω * n ≤ c n+1 n! f1 δ ′ ,L ′ Ω • • • fn δ ′ ,L ′ Ω .
In fact, one can cover the case f1 ∈ RΩ 1 , . . . , fn ∈ RΩn with different d.f.s.'s Ω 1 , . . . , Ω n as well-see Theorem 4.8-, but we only give details for the case of one d.f.s. so as to lighten the presentation.

Notations and preliminaries

We fix an integer n ≥ 1 and a d.f.s. Ω. In view of Remark 2.12, without loss of generality, we can suppose that Ω coincides with its upper closure:

(4.2) Ω = Ω.
Let ρ > 0 be such that Ω 3ρ = Ø. We set (See [START_REF]Nonlinear analysis with resurgent functions[END_REF] for the notations and notions related to integration currents.)

U := { ζ ∈ C | |ζ| < 3ρ }.
As in [START_REF]Nonlinear analysis with resurgent functions[END_REF], our starting point will be Lemma 4.3. Let f1 , . . . , fn ∈ RΩ and β := (p * Ω f1 )

ζ 1 • • • (p * Ω fn ) ζ n dζ 1 ∧ • • • ∧ dζ n ,,
where we denote by dζ 1 ∧ • • • ∧ dζ n the pullback by p ⊗n Ω :

X n Ω → C n of the n-form dζ 1 ∧ • • • ∧ dζ n . Then 1 * f1 * • • • * fn (ζ) = D(ζ) # [∆ n ](β) for ζ ∈ U .
Proof. This is just another way of writing the formula

(4.4) 1 * f1 * • • • * fn (ζ) = ζ n ∆n f1 (ζs 1 ) • • • fn (ζs n ) ds 1 • • • ds n .
See [START_REF]Nonlinear analysis with resurgent functions[END_REF] for the details.

Notation 4.4. We set

N (ζ) := ζ 1 , . . . , ζ n ∈ X n Ω | p Ω ζ 1 + • • • + p Ω ζ n = ζ for ζ ∈ C, (4.5) N j := ζ 1 , . . . , ζ n ∈ X n Ω | ζ j = 0 Ω for 1 ≤ j ≤ n. (4.6)

γ-adapted deformations of the identity

Let us consider a path γ : [0, 1] → C in Π Ω * n for which there exists a ∈ (0, 1) such that We now introduce the notion of γ-adapted deformation of the identity, which is a slight generalization of the γ-adapted origin-fixing isotopies which appear in [Sau15, Def. 5.1]. Definition 4.5. A γ-adapted deformation of the identity is a family (Ψ t ) t∈[a,1] of maps

Ψ t : V → X n Ω , for t ∈ [a, 1],
where

V := D γ(a) (∆ n ) ⊂ X n Ω , such that Ψ a = Id, the map t, ζ ∈ [a, 1] × V → Ψ t ζ ∈ X n Ω
is locally Lipschitz, and for any t ∈ [a, 1] and j = 1, . . . , n,

(4.8) Ψ t V ∩ N γ(a) ⊂ N γ(t) , Ψ t V ∩ N j ⊂ N j
(with the notations (4.5)-(4.6)).

Let γ denote the lift of γ in X Ω starting at 0 Ω . The analytical continuation along γ of a convolution product can be obtained as follows: The following is the key estimate: Theorem 4.7. Let δ ∈ (0, ρ) and L > 0. Let γ ∈ Π δ,L Ω * n satisfy (4.7) and let (4.12)

Proposition 4.6 ([Sau15]). If (Ψ t ) t∈[a,1] is a γ-adapted deformation of the identity, then (4.9) p * Ω * n 1 * f1 * • • • * fn γ(t) = Ψ t • D γ(a) # [∆ n ](β) for t ∈ [a,
δ ′ (t) := ρ e -2 √ 2δ -1 L(γ| [a,t] ) , c(t) := ρ e 3δ -1 L(γ| [a,t] ) for t ∈ [a, 1].
Then there exists a γ-adapted deformation of the identity Proof that Theorem 4.7 implies Theorem 4.1. Let δ, L > 0. We will show that (4.1) holds with δ ′ := min δ, ρ e -4 √ 2(1+δ -1 L) , c := max 2ρ, ρ e 6(1+δ -1 L) , where ρ := 4 3 δ. Let Ω be a d.f.s. such that Ω 4δ = Ø. Without loss of generality we may suppose that Ω = Ω.

(Ψ t ) t∈[a,1] such that (4.13) Ψ t • D γ(a) (∆ n ) ⊂ L 1 +•••+Ln=L(γ |t ) K δ ′ (t),L 1 Ω × • • • × K δ ′ (t)
In view of formula (4.4), the inequality (4.1) holds for ζ ∈ K δ,L Ω * n ∩ U , where U is defined by (4.3), because the Lebesgue measure of ∆ n is 1/n!.

Let ζ ∈ K δ,L Ω * n \ U . We can write ζ = γ(1) with γ ∈ Π δ,L Ω * n ,
is not C 1 , then we use a sequence of paths γ k ∈ Π δ/2,L+δ Ω * n such that γ k | [0,a] = γ| [0,a] , γ k (1) = γ(1), γ k | [a,1] is C 1 and sup t∈[a,1] |γ(t) -γ k (t)| → 0 as k → ∞;
(4.15) p * Ω 1 * f1 * • • • * fn (ζ) ≤ c n n! sup L 1 +•••+Ln=L p * Ω 1 f1 δ ′ ,L 1 Ω 1 • • • p * Ωn fn δ ′ ,Ln Ωn for ζ ∈ K δ,L Ω . Proposition 4.10. Let ζ = L (ζ 1 ), . . . , L (ζ n ) ∈ V , i.e. ζ j = s j γ(a) with (s 1 , . . . , s n ) ∈ ∆ n . We define v := (|ζ 1 |, ζ 1 ), . . . , (|ζ n |, ζ n ) ∈ (R × C) n and Γ = (γ 1 , . . . , γn ) : [0, 1] → (R × C) n by t ∈ [0, a] ⇒ Γ(t) := t a (|ζ 1 |, ζ 1 ), . . . , t a (|ζ n |, ζ n ) , t ∈ [a, 1] ⇒ Γ(t) := Φ a,t ( v ).
Then, for each j ∈ {1, . . . , n}, γj is a path [0, 1] → R × C whose C-projection γ j belongs to Π Ω , and the formula

(4.21) Ψ t ζ := γ 1 (t), . . . , γ n (t) ∈ X n Ω for t ∈ [a, 1].
defines a γ-adapted deformation of the identity.

Proof. We first prove that γ 1 , . . . , γ n ∈ Π Ω . In view of (2.5), we just need to check that, for each j ∈ {1, . . . , n}, the path γj = (λ j , γ j ) satisfies

v ∈ (R × C) n | v j = v * j } is invariant by the maps Φ t 1 ,t 2 (because η(v j ) = 0 implies that X j = 0 on this submanifold), in particular Φ t,a (R × C) n \ M n Ω ⊂ (R × C) n \ M n Ω
, whence (4.23) follows because Φ a,t and Φ t,a are mutually inverse bijections.

Therefore the paths γ 1 , . . . , γ n are Ω-allowed and have lifts in X Ω starting at 0 Ω , which allow us to define the maps Ψ t by (4.21) on V .

We now prove that (Ψ t ) t∈[a,1] is a γ-adapted deformation of the identity. The map (t, v ) → Ψ t ( v ) is locally Lipschitz because the flow map (4.20) is locally Lipschitz, and Ψ a = Id because Φ a,a is the identity map of (R × C) n ; hence, we just need to prove (4.8).

We set which can be itself checked as follows: consider first an arbitrary initial condition v ∈ (R × C) n and the corresponding solution v(t) := Φ a,t ( v ), and let v 0 (t) := v 1 (t) + We now show that the γ-adapted deformation of the identity that we have constructed in Proposition 4.10 meets the requirements of Theorem 4.7.

Ñ (w) := (v 1 , . . . , v n ) ∈ (R × C) n | v 1 + • • • + v n = w for w ∈ R × C, Ñj := (v 1 , . . . , v n ) ∈ (R × C) n | v j = (0, 0) for 1 ≤ j ≤ n.
In view of (2.6)-(2.7) and (3.5), the inclusion (4.13) follows from Φ a,t Ṽ ⊂

L 1 +•••+Ln=L(γ |t ) M L 1 ,δ ′ (t) Ω × • • • × M Ln,δ ′ (t) Ω
for all t ∈ [a, 1], with δ ′ (t) as in (4.12).

Proof of Lemma 4.11. Let us consider an initial condition v ∈ Ṽ and the corresponding solution v(t) := Φ a,t ( v ), whose components we write as v j (t) = λ j (t), ζ j (t) for j = 1, . . . , n. We also have v j (a) = s j γ(a) for some (s 1 , . . . We first notice that . . .

X n := η n (v n ) D(t, v ) γ′ (t),
where η j (v) := dist v, {(0, 0)} ∪ S Ω j , D t, v := η 1 (v 1 ) +

• • • + η n (v n ) + |γ(t) -(v 1 + • • • + v n )|.

The case of endless continuability w.r.t. bounded direction variation

In this subsection, we extend the estimates of Theorem 4.1 to the case of a d.d.f.s.. Let us fix an arbitrary d.d.f.s. Ω. We fix ρ > 0 such that Ω 3ρ,M = Ø for every M ≥ 0. We consider a path γ : [0, 1] → C in Π δ,M,L Ω * n , with arbitrary δ ∈ (0, ρ) and L > 0, satisfying the following condition: (see proof of Theorem 4 in [START_REF]Nonlinear analysis with resurgent functions[END_REF] for the detail). Since F (z, w) ∈ RΩ {w}, we obtain from Corollary 4.2 the following estimates: For every δ, L > 0, there exist δ ′ , L ′ , C > 0 such that

F k δ ′ ,L ′ Ω ≤ C k+1 and Hm δ,L Ω * ∞ ≤ k≥1 (m + k -1)! m!k! n 1 +•••+n k =m+k-1 n 1 ,••• ,n k ≥1 C k+1 k! Fn 1 δ ′ ,L ′ Ω • • • Fn k δ ′ ,L ′ Ω ≤ k≥1 2 m+k n 1 +•••+n k =m+k-1 n 1 ,••• ,n k ≥1 C m+3k k! ≤ k≥1 2 2m+3k-2 C m+3k k! ≤ e 8C 3 (4C) m .
This yields H(z, w) ∈ RΩ * ∞ {w}, whence, H(z, F0 (z)) ∈ RΩ * ∞ .

  Definition 1.2. A formal series φ(z) = ∞ j=0 ϕ j z -j ∈ C[[z -1 ]] is said to be resurgent if φ(ζ) = ∞ j=1 ϕ j ζ j-1

  ) ψ(ζξ) dξ for ζ in the intersection of the discs of convergence of φ and ψ.

  Let us illustrate this point on two examples. -The equation dϕ dz λϕ = b(z), where b(z) is given in z -1 C{z -1 } and λ ∈ C * , has a unique formal solution in C[[z -1 ]], namely φ(z) := -λ -1 Id -λ -1 d dz -1 b, whose Borel transform is φ(ζ) = -(λ + ζ) -1 b(ζ); here, the Borel transform b(ζ) of b(z) is entire, hence φ is meromorphic in C, with at worse a pole at ζ = -λ and no singularity elsewhere. Therefore, heuristically, for a nonlinear equation dϕ dz λϕ = b 0

  n+1ω n+1 | gives the minimum of |ω ′ n+1ω n+1 | for such vertices and assign an open line segment ℓ v ′ →v := {ω n+1 + s(ω

  we obtain a lift of γ| [0,a] by concatenating γ |b and γ| [b,a] in the coordinate. It contradicts the maximality of I, and hence, I = [0, 1].

  Figure 3:

.

  by totally the same discussion. Notice that, since the sequence v ′ in the proof of Lemma 3.10 is uniquely determined by ζ ∈ X Ω , the choice of the path P ε ′ ζ depends only on the radius ε ′ of the circles C ε ′ ω ′ j Further, from the construction of the path P ε ′ ζ , we can extend Lemma 3.10 as follows: Lemma 3.11. For all ε > 0 and ζ ∈ X Ω , there exist a neighborhood U ζ of ζ and, for ε ′ small enough, a continuous deformation

For

  each ζ ∈ U , the path γ ζ : t ∈ [0, 1] → tζ is Ω-allowed and hence has a lift γ ζ on X Ω starting at 0 Ω . Then L (ζ) := γ ζ (1) defines a holomorphic function on U and induces an isomorphism(4.3) L : U ∼ -→ U , where U := L (U ) ⊂ X Ω , such that p Ω • L = Id.Let us denote by ∆ n the n-dimensional simplex∆ n := { (s 1 , . . . , s n ) ∈ R n ≥0 | s 1 + • • • + s n ≤ 1 }with the standard orientation, and by [∆ n ] ∈ E n (R n ) the corresponding integration current. For ζ ∈ U , we define a map D(ζ) on a neighbourhood of ∆ n in R n by D(ζ) : s = (s 1 , . . . , s n ) → D(ζ, s ) := L (s 1 ζ), . . . , L (s n ζ) ∈ U n ⊂ X n Ω and denote by D(ζ) # [∆ n ] ∈ E n (X n Ω ) the push-forward of [∆ n ] by D(ζ).

  (4.7) γ(t) = t a γ(a) for t ∈ [0, a], |γ(a)| = ρ, γ| [a,1] is C 1 .

  for k large enough one has γ k (1) = ζ, thus one then can replace γ by γ k . Hence we can assume that (4.7) holds. Let (Ψ t ) [t∈[a,1]] denote the γ-adapted deformation of the identity provided by Theorem 4.7, possibly with (δ, L) replaced by (δ/2, L + δ). Proposition 4.6 shows that, for f1 , . . . , fn ∈ RΩ , p * Ω * n 1 * f1 * • • • * fn (ζ) can be written as (4.10) with t = 1, and (4.13)-(4.14) then show that (4.1) holds because δ ′ (t) ≥ δ ′ and c(1) ≤ c. Therefore, (4.1) holds on K δ,L Ω * n \ U too. In fact, in view of the proof of Theorem 4.7 given below, one can give the following generalization of Theorem 4.1: Theorem 4.8. Let δ, L be positive real numbers. Then there exist positive constants c and δ ′ such that, for every integer n ≥ 1 and for all d.f.s. Ω 1 , . . . , Ω n with Ω j,4δ = Ø (j = 1, • • • , n) and f1 ∈ RΩ 1 , . . . , fn ∈ RΩn , the function 1 * f1 * • • • * fn belongs to RΩ , where Ω := Ω 1 * • • • * Ω n , and

(

  Let j ∈ {1, . . . , n}. The second part of (4.8) follows from the inclusionΦ a,t Ñj ⊂ Ñj for t ∈ [a, 1],which stems from the fact that the jth component of the vector field (4.19) vanishes on Ñj (because η (0, 0) = 0).Sinceζ 1 + • • • + ζ n = γ(a) ⇒ |ζ 1 | + • • • + |ζ n | = |γ(a)| for any (ζ 1 , . . . , ζ n ) ∈ V, the first part of (4.8) follows from the inclusion Φ a,t Ñ γ(a) ⊂ Ñ γ(t) for t ∈ [a, 1],

  ), whence γ(t)v 0 (t) ≤ γ(a)v 0 (a) exp δ -1 √ 2 L(γ| [a,t] )for all t; now, if v ∈ Ñ γ(a) , we find v 0 (a) = γ(a), whence v 0 (t) = γ(t) for all t.

  Lemma 4.11. Let Ṽ := s 1 γ(a), . . . , s n γ(a) | (s 1 , . . . , s n ) ∈ ∆ n ∈ (R × C) n . Then (4.24)

  , s n ) ∈ ∆ n , whence λ 1 (a) + • • • + λ n (a)≤ |γ(a)| = ρ and |v j (a)| = ρ for j = 1, . . . , n.

Notation 4. 14 .

 14 Given δ, M, L > 0, we denote by Π δ,M,L Ω the set of all paths γ ∈ Π dv Ω such that V (γ) ≤ M , L(γ) ≤ L and inf t∈[0,t * ]

  dist 1 γ dv (t), S Ω ≥ δ, where γ dv is as in (2.13) and dist 1 is the distance associated with the norm• 1 defined on R 2 × C by (µ, λ, ζ) 1 := |µ| + |λ| 2 + |ζ| 2 .

  (4.29)There exists a ∈ (0, 1) such that γ(t) = t a γ(a) for t ∈ [0, a] and |γ(a)| = ρ.Then, for t ∈ [0, 1] and v ∈ R × C, we setη(t, v) := dist 1 (V (γ |t ), v), R ×{(0, 0)} ∪ S Ω . and, for v = (v 1 , • • • , v n ) ∈ (R × C) n , D t, v := η(t, v 1 ) + • • • + η(t, v n ) + |γ(t) -(v 1 + • • • + v n )|.

  See the proof of [Sau15, Prop. 5.2].

	Note that the right-hand side of (4.9) must be interpreted as
	(4.10)	∆n	(p * Ω f1 ) ζ t 1 • • • (p * Ω fn ) ζ t n det	∂ζ t i ∂s j 1≤i,j≤n	ds 1 • • • ds n
	with the notation
	(4.11) (each function ζ t i is Lipschitz on ∆ n and Rademacher's theorem ensures that it is differentiable ζ t 1 , . . . , ζ t n := Ψ t • D γ(a) , ζ t i := p Ω • ζ t i for 1 ≤ i ≤ n
	almost everywhere on ∆ n , with bounded partial derivatives).

1]

for any f1 , . . . , fn ∈ RΩ , with β as in Lemma 4.3.

Proof.

  assuming without loss of generality that the first two conditions in (4.7) hold. If the third condition in (4.7) does not hold, i.e. if γ|[a,1]

  4.22)t ∈ [0, 1] ⇒ γj (t) ∈ M Ω and dλ j /dt = |dγ j /dt|. Since ζ j ∈ U and γ j (t) = t a ζ j for t ∈ [0, a], the property (4.22) holds for t ∈ [0, a]. For t ∈ [a, 1], the second property in (4.22) follows from the fact that the R-projection of X j (t, v ) ∈ R × C coincides with the modulus of its C-projection.Since γ1 (t), . . . , γn (t) = Φ a,t γ1 (a), . . . , γn (a) and the first property in (4.22) holds at t = a, the first property in (4.22) for t ∈ [a, 1] is a consequence of the inclusion

	(4.23)	Φ a,t M n Ω ⊂ M n

Ω , which can itself be checked as follows: suppose

v * ∈ (R × C) n \ M n Ω ,

then it has at least one component v * j in S Ω and, in view of the form of the vector field (4.19), the submanifold {

  • • • + v n (t); then (4.19) shows that d dt γ(t)v 0 (t) = γ(t)v 0 (t) D(t, v(t)) γ′ (t), hence the Lipschitz function h(t) := γ(t)v 0 (t) has an almost everywhere defined derivative which satisfies |h ′ (t)| ≤ d dt γ(t)v 0 (t) ≤ 1 D(t, v(t)) |γ ′ (t)| h(t), which is ≤ δ -1 √ 2 |γ ′ (t)| h(t) by (4.17

  Let j ∈ {1, . . . , n}. Since η is 1-Lipschitz, we can define a Lipschitz function on [a, 1] by the formula h j (t) := η v j (t) , and its almost everywhere defined derivative satisfies |γ ′ (t)| ≤ g(t)h j (t), where g(t):= δ -1 √ 2 |γ ′ (t)|. (a) e -δ -1 √ 2 L(γ| [a,t] ) ≤ η v j (t) ≤ η v j (a) e δ -1 √ 2 L(γ| [a,t] ) for all t ∈ [a, 1].Let us now fix t ∈ [a, 1]. We conclude by distinguishing two cases. , Gronwall's lemma yields V (t) ≤ V (a) e 3δ -1 L(γ| [a,t] ) , and hence, sinceV (a) = ρ n j=1 |s js ′ j |, we have (4.28) V (t) ≤ ρ e 3δ -1 L(γ| [a,t] )Then, (4.28) entails via Rademacher's theorem that the following estimate holds a.e. on ∆ n : ≤ ρ e 3δ -1 L(γ| [a,t] ) .Remark 4.13. Theorem 4.8 is verified by replacing the vector field (4.19) by

	n j=1 |γ |h ′ |λ ′ j (t)| = n j=1 η v j (t) D(t, v(t)) j (t)| ≤ |v ′ j (t)| = h j (t) D(t, v(t)) a g(τ ) dτ = δ -1 √ t 2 L(γ| [a,t] ), we deduce that j=1 n i=1 ∂ζ t i ∂s j Finally, (4.14) follows from the inequality Since (4.26) det ∂ζ t i ∂s j 1≤i,j≤n ≤ n j=1 X(t, v ) = η v j Thereforen X 1 := η 1 (v 1 ) |s j -s ′ j |. n i=1 ∂ζ t i ∂s j D(t, v ) γ′ (t)	.

′ (t)| ≤ |γ ′ (t)|, hence λ 1 (t) + • • • + λ n (t) ≤ λ 1 (a) + • • • + λ n (a) + t a |γ ′ | ≤ L(γ |t ). Therefore, we just need to show that (4.25) dist v j (t), S Ω ≥ δ ′ (t) for j = 1, . . . , n. Suppose first that η(v j (a)) ≥ ρ e - √ 2 δ -1 L(γ| [a,t]

) . Then the first inequality in (4.26) yields η(v j (t)) ≥ δ ′ (t), and since dist v j (t), S Ω ≥ η(v j (t)) we get (4.25).

Graduate School of Sciences, Hiroshima University. 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan.

2 IMCCE, CNRS-Observatoire de Paris, France.

Acknowledgements. This work has been supported by Grant-in-Aid for JSPS Fellows Grant Number 15J06019, French National Research Agency reference ANR-12-BS01-0017 and Laboratoire Hypathie A*Midex. The authors thank Fibonacci Laboratory (CNRS UMI 3483), the Centro Di Ricerca Matematica Ennio De Giorgi and the Scuola Normale Superiore di Pisa for their kind hospitality.

Proof of Theorem 4.7

We suppose that we are given n ≥ 1, ρ > 0, a d.f.s. Ω such that Ω = Ω and Ω 3ρ = Ø, and γ ∈ Π δ,L Ω * n satisfying (4.7) with δ ∈ (0, ρ) and L > 0. We set γ(t) := L(γ |t ), γ(t) and define functions

by the formulas (4.16) η(v) := dist v, {(0, 0)} ∪ S Ω , D t, v := η(v 1 ) +

where | • | is the Euclidean norm in R × C ≃ R 3 . The assumptions Ω = Ω and γ ∈ Π δ,L Ω * n yield Lemma 4.9. The function D satisfies (4.17)

Proof. Let (t, v ) ∈ [a, 1] × (R × C) n . For each j ∈ {1, . . . , n}, pick u j ∈ {(0, 0)} ∪ S Ω so that η(v j ) = |v ju j |, and let

Either all of the u j 's equal (0, 0), in which case u = (0, 0) too, or u = (λ, ω) is a non-trivial sum of at most n points of the form u j = (λ j , ω j ) ∈ S Ω , in which case we have in fact ω j ∈ Ω λ j because of Lemma 2.4 and the assuption Ω = Ω, hence (2.12) then yields ω ∈ Ω * n λ . We thus find

Otherwise, u ∈ S Ω * n and (4.18) shows that D t, v ≥ δ because γ ∈ Π δ,L Ω * n .

Since D never vanishes, we can define a non-autonomous vector field

. . .

The functions X j : [a, 1] × (R × C) n → R × C are locally Lipschitz, thus we can apply the Cauchy-Lipschitz theorem on the existence and uniqueness of solutions to differential equations and get a locally Lipschitz flow map

(value at time t of the unique maximal solution to d v/dt = X(t, v ) whose value at time t * is v ). We construct a γ-adapted deformation of the identity out of the flow map as follows:

) . Then the second inequality in (4.26) yields

and we are done.

Only the inequality (4.14) remains to be proved. We first show the following:

Lemma 4.12. For any t ∈ [a, 1] and u, v ∈ (R × C) n , the vector field (4.19) satisfies

Proof of Lemma 4.12. We rewrite X j (t, u ) -X j (t, v ) as follows:

Then, summing up |X j (t, u ) -X j (t, v )| in j, we obtain (4.27) from the inequality n j=1 η(v j ) ≤ D(t, v ).

We conclude by deriving the inequality (4.14) from Lemma 4.12. We use the notation (4.11) to define ζ t 1 , . . . , ζ t n : ∆ n → C, and we now define v t j : ∆ n → R × C for t ∈ [a, 1] by the formulas v a j ( s ) := s j γ(a) and

We obtain from (4.17) and (4.27) the following estimate:

Choosing (µ j , u j ) ∈ R ×{(0, 0)} ∪ S Ω so that η(t, v j ) = (V (γ |t ), v j ) -(µ j , u j ) 1 for each j and using (µ

We can thus define a map (t

. . .

) be the flow of (4.30) with the initial condition v a j := (|γ(a)|s j , γ(a)s j ) with s ∈ ∆ n . Since γ′ (t), η(t, v j ) and D(t, v ) are Lipschitz continuous on [a, 1] × (R × C) n , we find by Rademacher's theorem that dζ t j /dt is differentiable a.e. on [a, 1] and satisfies

when s j = 0. Since η(v t j ) and D(t, v t ) are real valued functions, we have

Therefore, the following holds for every t ∈ [a, 1]:

Arguing as for Theorem 4.1, we obtain Theorem 4.15. Let δ, L, M > 0 be real numbers. Then there exist c, δ ′ > 0 such that, for every d.d.f.s. Ω such that Ω 4δ,M = Ø (M ≥ 0), for every integer n ≥ 1 and for every f1 , . . . , fn ∈ R dv Ω , the function 1 * f1 * • • • * fn belongs to R dv Ω * n and satisfies

where the seminorm

Applications

In this section, we display some applications of our results of Section 4. We first introduce convergent power series with coefficients in RΩ :

Definition 5.1. Given Ω a d.f.s. and r ≥ 1, we define RΩ {w 1 , • • • , w r } as the space of all

such that, for every δ, L > 0, there exists a positive constant C satisfying Fk

where |k| := k 1 + • • • + k r (with the notation of Definition 3.16 for • δ,L Ω ).

We can now deal with the substitution of resurgent formal series in a context more general than in Theorem 1.3. Theorem 5.2. Let r ≥ 1 be an integer and let Ω 0 , . . . , Ω r be d.f.s. Then for any F (w 1 , . . . , w r ) ∈ RΩ 0 {w 1 , • • • , w r } and for any φ1 , . . . , φr ∈ C[[z -1 ]] without constant term, one has

where

Proof. Since the family

f.s. satisfies the conditions in Theorem 4.8 for sufficiently small δ > 0, for every L > 0, there exist δ

Therefore, since F (w 1 , . . . , w r ) ∈ RΩ 0 {w 1 , • • • , w r }, we find that F ( φ1 , . . . , φr ) converges in RΩ 0 * Ω * ∞ and defines an Ω 0 * Ω * ∞ -resurgent formal series.

Notice that, in view of Theorem 2.13, Theorem 1.3 is a direct consequence of Theorem 5.2.

Next, we show the following implicit function theorem for resurgent formal series: