Endless continuability and convolution product

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https://hal.archives-ouvertes.fr/hal-01071005v2

Preprint submitted on 26 Mar 2015

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Eric Delabaere, Yafei Ou

To cite this version:

Eric Delabaere, Yafei Ou. Endless continuability and convolution product. 2014. �hal-01071005v2�

PRODUCT

by

Yafei Ou & Eric Delabaere

Abstract. — We provide a rigorous analysis for the so-called endlessly continuable germs of holomorphic functions or in other words, the Ecalle’s resurgent functions. We follow and complete an approach due to Pham, based on the notion of discrete filtered set Ω⋆ and the associated Riemann surface defined as the space of Ω⋆-homotopy classes of paths. Our main contribution consists in a complete though simple proof of the stability under convolution product of the space of endlessly continuable germs.

Contents

1. Introduction. . . 1

2. Discrete filtered set and associated Riemann surface. . . 2

3. Endless continuability. . . 12

4. Endless continuability and convolution product. . . 15

5. Conclusion. . . 19

References. . . 20

1. Introduction

Today, the resurgence theory of Ecalle has demonstrated its efficiency in many instances for dealing with divergent series arising from differential and difference equations, PDEs, semiclassical analysis, etc.. see for instance [11, 9, 15, 16, 4, 5, 24] and references therein. In particular, this theory provides the necessary tools for understanding the nonlinear Stokes phenomena in asymptotic analysis and leads up to both theoretical results and even numerical methods, see e.g. [8].

In its simplest form, the Ecalle’s theory generalizes the Borel resummation theory, whose main objects are 1-Gevrey formal seriesϕ(z) =e X

n≥1

an

zⁿ ∈C[[z⁻¹]], that is the formal Borel transformϕ(ζ) =b X

n≥1

an

(n−1)!ζⁿ∈C{ζ}defines a germ of holomorphic functions at the origin. The formal seriesϕeis said to be resurgent ifϕbis “endlessly

2000 Mathematics Subject Classification. — 34M37, 30Hxx, 30D05, 37F99.

Key words and phrases. — Asymptotics, Resurgent algebras, Convolution product.

continuable”. Roughly speaking, ϕb is endlessly continuable ifϕb can be analytically continued along any path on C avoiding a set of possibly singular points, each of them being locally isolated. However, this set may be everywhere dense and is so for many important applications, in particular those stemming from high energy physics, string theory and related models [6, 7, 1, 19, 10]

It turns out that the Cauchy product(ϕ.eψ)(z)e of 1-Gevrey formal series becomes the convolution product ϕb∗ψ(ζb ) =

Z ζ

0 ϕ(η)b ψ(ζb −η)dη of germs of holomorphic functions by Borel transformation. Since the Ecalle’s theory aims at analyzing non- linear problems, it is an essential demand that the convolution product preserves the notion of endless continuability. This ends up with the definition of various subalge- bras of resurgent functions in the so-called Borelζ-plane and their counterpart in the initial z-plane, as well as alien operators designed for encoding the singularities by microlocalisation and describing the nonlinear Stokes phenomena [12, 13, 2, 24, 5].

As a matter of fact, there exist various definitions of “endless continuability” in the literature. The more general one is that of Ecalle [12, 13]. Another one is due to Phamet al. [3, 2], is easier to handle with and is based on the construction of a Riemann surface governed by the datum of a discrete filtered set. This provides the notion of endless Riemann surface. A germ of holomorphic functions that can be an- alytically continued to such a Riemann surface is by definition endlessly continuable.

This is this second approach that we use in this paper.

As already said, the endless continuability and its stability under convolution product are key-properties at the very root of the resurgence theory. Unfortunately, the existing proofs for the stability in its full generality [12, 2] are difficult and subject to controversy. Our main goal in this article is to show in a simple and rigorous way that the convolution product of any two endlessly continuable functions is endlessly continuable. Though inspired by [2], our methods differ from these authors for key-arguments. The method that we present in this paper can be seen as an extension of ideas detailed in [20, 22, 24] for the case of holomorphic functions that can be analytically continued along any path avoiding closed discrete subsets of C. Consequently, we thought that our results were worth the attention of the specialists in this field of research.

The paper is organized as follows. We introduce the notion of discrete filtered sets and their associated Riemann surfaces that we study properly (Sect. 2). We define endless Riemann surfaces and endlessly continuable germs of holomorphic functions and we make a link with the endlessly continuable functions of Ecalle (Sect. 3). The main result of the paper concerns the stability under convolution product and this is detailed in Sect. 4. We end the paper with some open problems.

2. Discrete filtered set and associated Riemann surface

2.1. Discrete filtered sets. — The following definitions are adapted from [2].

Definition 2.1. — Adiscrete filtered set Ω_⋆ centred at ω ∈C is an increasing sequence of finite setsΩ_L⊂C,L >0, such that :

– for any L >0,Ω_Lbelongs to the open disc centred at ω with radius L;

– if L₁ ≤L₂ thenΩ_L1 jΩ_L2;

– for L >0 small enough,ΩL={ω}.

For L >0, we denoteΩ^⋆_L = Ω_L\ {ω}. The number ρ_Ω⋆(ω) = sup{L >0|Ω^⋆_L=∅}

is called the distance of ω to Ω_⋆.

Definition 2.2. — Let Ω_⋆ and Ω^′_⋆ be two discrete filtered sets centred at ω ∈ C. Their unionΩ_⋆∪Ω^′_⋆ is the discrete filtered set centred at ω defined by : for every L > 0, (Ω_⋆ ∪Ω^′_⋆)_L = Ω_L∪Ω^′_L. Their sum Ω_⋆ + Ω^′_⋆ is the filtered set centred at ω defined by : for every L > 0, (Ω_⋆ + Ω^′_⋆)_L = {−ω + Ω_L+ Ω^′_L} ∩D(ω, L).

Their fine sumΩ_⋆∗Ω^′_⋆ is the filtered set centred atω given by : for every L > 0, (Ω_⋆∗Ω^′_⋆)_L={ζ =−ω+ω₁+ω₂|ω₁∈Ω_L1, ω₂∈Ω^′_L2, L₁+L₂=L}.

IfΩ_⋆ is a discrete filtered set, we remark that S

L>0Ω_L can be dense inCas it is shown in the following example.

Example 2.3. — Assume that ω₁ ∈C^⋆ and define – for any L∈]0,|ω₁|],Ω_L={0},

– for any n∈N^⋆ and anyL∈]n|ω1|,(n+ 1)|ω1|],Ω_L={0,±ω1,· · · ,±nω1}.

This define a discrete filtered setΩ1⋆ centred at0.

Assume now thatω₁, ω₂, ω₃ ∈C^⋆ are rationally independent, that is linearly inde- pendent overZ. We consider the three discrete filtered setsΩ_1⋆,Ω_2⋆ andΩ_3⋆centred at 0 defined as above. We note Ω_⋆ = Ω_1⋆+ Ω_2⋆+ Ω_3⋆ their sum. Then S

L>0Ω_L is everywhere dense in C. The conclusion is the same when Ω_⋆ = Ω_1⋆∗Ω_2⋆∗Ω_3⋆ is defined by fine sums.

For a given discrete filtered set Ω_⋆ centred at ω, its iterated fine sums P

n∗Ω_⋆= Ω_⋆∗ · · · ∗Ω_⋆

| {z }

n times

makes a direct system (for the injections (P

n∗Ω_⋆)_L֒→P

(n+1)∗Ω_⋆

L, for every L > 0). The fine sums enjoyes the following property (the proof is left to the reader):

Proposition 2.4. — Let Ω⋆ be a discrete filtered set Ω⋆ centred at ω. Then the direct limitΩ^∞_⋆ = lim

→

X

n∗

Ω_⋆ is a discrete filtered set at ω.

Definition 2.5. — The discrete filtered setΩ^∞_⋆ is called thesaturated ofΩ_⋆. 2.2. Reminder about paths. — In what follows, a pathλin a topological space X is any continuous function λ: [a, a+l]→X, where[a, a+l]⊂R is a (compact) interval possibly reduced to {a}. We often work with standard paths, that is paths defined on [0,1]. The path λ:t∈[0,1]7→λ(a+tl) is the standardized path of λ.

For two paths λ₁ : [a, a+l]→X,λ₂: [b, b+k]→X so that λ₁(a+l) =λ₂(b), one defines their product (or concatenation) by

λ₁λ₂:t∈[a, a+l+k]7→

λ₁(t), t∈[a, a+l]

λ₂(t−a−l+b), t∈[a+l, a+l+k]

When the two paths λ₁, λ₂ have same extremities, they are homotopic when there exists a continuous mapH: [0,1]×[0,1]→X that realizes a homotopy between the standardized pathsλ₁ andλ₂. We recall that any pathλ:I →Ccan be uniformaly approached by C^∞-paths. Whenλ:I →C is piecewiseC¹, we denote its length by Lλ.

2.3. Ω_⋆-allowed path, Ω_⋆-homotopy. — The following definitions are inspired from [2] but for slight modifications⁽¹⁾.

Definition 2.6. — ForΩ⋆ a discrete filtered set centred at ω∈C, one denotes by R_Ω⋆(L) the set of paths λ:I →Cstarting fromω and such that :

– λisC¹ piecewise and its length satisfiesL_λ< L;

– λis the constant path or there exists t₀ ∈[0,1[ such thatλ([0, t₀]) ={ω} and λ(]t₀,1])⊂D(ω, L)\Ω_L.

A path λ is said to be Ω_⋆-allowed if λ ∈ R_Ω⋆(L) for some L > 0. We denote by R_Ω⋆ =S

L>0R_Ω⋆(L) the set ofΩ_⋆-allowed paths.

Definition 2.7. — LetΩ_⋆ be a discrete filtered set centred atω ∈C. A continuous mapH : (s, t)∈[0,1]² 7→H_t(s)∈Cis a Ω_⋆-homotopy if H has a continuous partial derivative ∂H

∂s and, for every t∈[0,1], the path H_t isΩ_⋆-allowed.

Two Ω_⋆-allowed paths λ₀ and λ₁ with same extremities are Ω_⋆-homotopic when there exists aΩ_⋆-homotopy that realises a homotopy between the standardized paths λ₀ and λ₁.

ForλaΩ⋆-allowed path, we denote bycl(λ)its equivalence class for the relation∼Ω⋆

of Ω_⋆-homotopy of paths inR_Ω⋆ with fixed extremities.

It is quite important to understand what is the Ω_⋆-homotopy and we make the following remark that we formulate as a lemma:

Lemma 2.8. — Assume that Ω_⋆ be a discrete filtered set centred at ω ∈C and let H be a Ω_⋆-homotopy. Then there exists a good⁽²⁾ open covering (I_i)_0≤i≤n of [0,1]

and real positive numbers L0, L1,· · · , Ln such that,

– for every i= 0,· · · , nand every t∈I_i, H_t belongs to R_Ω⋆(L_i).

– for everyi= 0,· · ·, n−1and for everyt∈Ii∩Ii+1,Ht∈RΩ⋆(Li)∩RΩ⋆(Li+1).

Proof. — From the very definition of a discrete filtered set, one can define an in- creasing sequence of real numbers 0 =l₋₁< l₀ < l₁ < l₂ <· · · with the properties:

– Ω_l0 ={ω} and for any integer i≥1,Ω_li−1 $Ω_li;

1. These definitions are less general than those of [2] but sufficient in practice as far as we know.

2. By “good”, we mean that the covering has finite elements, that each of these elementIiis a connected interval and that there are no 3-by-3 intersections.

– Ω_L= Ω_li for everyL∈]li−1, l_i],i∈N.

Pick any t_⋆ ∈ [0,1] and assume that LH_t⋆ ∈ [l_i−1, l_i[ for some i ∈ N. One has Ht⋆ ∈ R_Ω⋆(li) necessarily since Ht⋆ is Ω⋆-allowed. We say that Ht ∈ R_Ω⋆(li) for any t ∈[0,1] close enough tot_⋆. Indeed, we remark that the map t∈ [0,1]7→ LHt

is continuous because of the existence and the continuity of the partial derivative

∂H

∂s. Thus, if LH_t⋆ ∈]li−1, l_i[, then LHt ∈]li−1, l_i[ for t close enough to t_⋆. Now if LH_t⋆ = l_i−1 (i ≥ 1), then LHt ∈]li−2, l_i[ for t close enough to t_⋆. However, LHt

belongs also to R_Ω⋆(l_i) for tclose enough to t_⋆ because of the continuity of H (the euclidean distanced(H_t⋆,Ω_li)of the pathH_t to the setΩ_li is>0, so doesd(H_t,Ω_li) for t close enough to t_⋆). This way one gets an open covering of [0,1] from which one deduces a finite open covering by compacity. One easily concludes.

ω

ω

λ

λ

λ

ω

ω

0

Figure 1. We assume that Ω_⋆ is a discrete filtered set centred at 0. For 0 < L1 < L2, Ω_L1 = {0, ω1, ω2}, Ω_L2 = Ω_L1 ∪ {ω3, ω4}. The paths λ1, λ2∈RΩ_⋆(L1) are Ω_⋆-homotopic, the paths λ2, λ3 ∈ RΩ_⋆(L2) are Ω_⋆- homotopic, thus λ1 and λ3 are Ω_⋆-homotopic despite the fact that λ1∈/ RΩ_⋆(L2).

From lemma 2.8, observe that if L₂ > L₁, a path λ₁ ∈ R_Ω⋆(L₁) can be Ω_⋆- homotopic to another path λ₂ ∈R_Ω⋆(L₂) and at the same time λ₁ not being homo- topic toλ₂ in the usual way, when both are seen as paths inR_Ω⋆(L₂). Even, we may have λ₁ ∈/R_Ω⋆(L₂), see Fig. 1.

2.4. Riemann surface associated with a discrete filtered set. —

Definition 2.9. — LetΩ⋆ be a discrete filtered set centred atω^• ∈C. We set:

R_Ω

⋆ ={ζ = cl(λ) | λ∈RΩ⋆} and p:ζ = cl(λ)7→ζ^•=λ(1)∈C.

Remark that p⁻¹(ω)^• is reduced to a single point ω = cl(constant path). This is why one usually considers R_Ω

⋆ as a pointed space (R, ω).

Let Ω⋆ be a discrete filtered set centred at ω^• ∈ C, and set ω = p⁻¹(ω). One^• can endow R_Ω

⋆ with a separated topology, a basis B ={U} of open sets defining

this topology being given as follows⁽³⁾. (We adapt the classical construction of a universal covering [17]). Let us consider a point ζ ∈R_Ω

⋆.

– Assume that ζ = ω. For some L > 0 we consider U^•⊂ D(ω, L)^• \Ω^⋆_L a star- shaped domain with respect to ω. Let^• U ⊂ R_Ω

⋆ be the set of all ξ = cl(λ) where λ ∈ R_Ω⋆(L) is any path ending at ξ^• ∈U^• and whose image is the line segment [ω,ξ]. (For a given^• ξ, the length of these paths is^• |ξ|^• < L and all these paths belong to the sameΩ_⋆-homotopy class).

– Suppose thatζ 6=ω. We choose a path λ₁ ∈R_Ω⋆(L) such thatcl(λ₁) =ζ. For someL₂ >0such thatLλ1+L₂< L, we considerU^•⊂D(ζ, L^• ₂)\Ω_L⊂D(ω, L)^• \Ω_L such that U^• is a star-shaped domain with respect to ζ^• = λ1(1). For ξ^• ∈U^•, consider a path λ₂ starting from ζ^•, ending at ξ^• and whose image is the line segment [ζ,^• ξ]. Then the product^• λ₁λ₂ belongs to R_Ω⋆(L) and we consider its Ω_⋆-homotopy classξ = cl(λ₁λ₂). We noteU the set of such pointsξ.

We show that the system B = {U} made of these sets provides a basis for a topology onR_Ω

⋆. Obviously, every elementξ ∈R_Ω

⋆ belongs to at least oneU ∈B. Now assume that ξ∈U ∩V,U,V ∈B.

– Ifξ =ω, then necessarilyU^• and V^• are two star-shaped domains with respect to ω,^• U^• is a subset of D(ω, L^• 1)\Ω^⋆_L1 and V^• is a subset of D(ω, L^• 2)\Ω^⋆_L2. Set L= max{L₁, L₂}. Then W^•=U^• ∩V^•⊂D(ω, L)^• \Ω^⋆_L is also a star-shaped domain with respect to ω^• to which is associated a W ∈ B that satisfies: ξ ∈ W ⊂U ∩V.

– Otherwise ξ 6= ω and ξ ∈ U ∩V. There is no loss of generality in assuming also that ω /∈U ∩V. Thus :

• for someL >0,U^•⊂D(ζ^•₁, L₂)\Ω_Lis a star-shaped domain with respect to ζ^•₁ = λ₁(1) where λ₁ ∈ R_Ω⋆(L) satisfies Lλ1 +L₂ < L. Also we have ξ = cl(λ₁λ₂) where λ₂ starts from ζ^•₁, ends at ξ^• and is such that λ₂([0,1]) = [ζ^•₁,ξ].^•

• for someL^′ >0,V^•⊂D(ζ^•′₁, L^′₂)\Ω_L^′ is a star-shaped domain with respect to ζ^•′₁ = λ^′₁(1) where λ^′₁ ∈ R_Ω⋆(L^′) satisfies L_λ^′₁ +L^′₂ < L^′. Also ξ = cl(λ^′₁λ^′₂) where λ^′₂ starts fromζ^•′₁, ends at ξ^• and is such that λ^′₂([0,1]) = [ζ₁^′, ξ].

Now choose L^′′₂ >0such that Lλ1λ2 +L^′′₂ < Land L_λ^′_1λ^′₂+L^′′₂ < L^′. Consider

•

W⊂U^• ∩ V^• ∩D(ξ, L^{• ′′}₂) a star-shaped domain with respect to ξ. To^• W^• is associated W ∈B and ξ∈W ⊂U ∩V.

We show that the topology thus defined byBis Hausdorff. We consider two pointsζ and ζ^′ inR_Ω

⋆. Clearly ifζ^•6=ζ^•′, then ζ andζ^′ have disjoints neighbourhoods. Thus

3. This topoloy is not detailed in [2].

assume thatζ^•=ζ^•′ (6=ω, say) but^• ζ 6=ζ^′. Suppose the existence of a neighbourhood U ∈B ofζ, a neighbourhoodU^′∈Bofζ^′such thatU ∩U^′ 6=∅. This means that there exists ξ∈U ∩U^′ that satisfies:

– ξ = cl(λ₁λ₂) with ζ = cl(λ₁), λ₂ starting from ζ^•, ending at ξ^• with image the line segment [ζ,^• ξ]^• ⊂U^•;

– ξ = cl(λ^′₁λ^′₂) with ζ^′ = cl(λ^′₁), λ^′₂ starting from ζ^•′ = ζ^•, endings at ξ^• the line segment [ζ,^• ξ]^• ⊂U^•′ for its range.

This implies that λ₁ and λ^′₁ are in the same class, that is ζ = ζ^′ and we get a contradiction.

The topological space R_Ω

⋆ is obviously (arc)connected. Also, by the very con- struction of the topology, for every U ∈ B, the restriction p|^U : ζ 7→ ζ^• ∈U^• is a homeomorphism. This means that R_Ω

⋆ is an étalé space on C(but of course not a covering space). We have thus shown the following proposition.

Proposition 2.10. — Let Ω_⋆ be is a discrete filtered set. The (pointed) space R_Ω

⋆

is a topologically (arc)connected separated space. With the projection p, the space R_Ω

⋆ is an étalé space on C.

When pulling back by p the complex structure of C, R_Ω

⋆ becomes a Riemann surface. This Riemann surface is the smallest in the following sense:

Lemma 2.11. — Let (R^′,p^′, ω^′) be a Riemann surface over C, with p^′(ω^′) = ω.^• We suppose that any Ω_⋆-allowed path can be lifted on R^′ from ω^′ with respect to p^′. Then (R_Ω

⋆,p, ω) is contained in (R^′,p^′, ω^′), that is there exists a continuous map τ :R_Ω

⋆ →R^′ such that p^′◦τ =p and τ(ω) =ω^′.

Proof. — From the very definition of the Riemann surface(R_Ω

⋆,p) associated with a discrete filtered setΩ_⋆ centred atω^• ∈C, every Ω_⋆-allowed path can be lifted with respect to p into a path on R_Ω

⋆ with initial point ω = p⁻¹(ω). Indeed, assume^• that λ ∈ R_Ω⋆(L) and define λ_t : s ∈ [0,1] 7→ λ_t(s) = λ(ts) for t ∈ [0,1]. Then λ_t∈ R_Ω⋆(L) and the mapping Λ : t∈ [0,1]7→ Λ(t) = cl(λ_t) is continuous and is a lifting ofλfromω. This lifting is unique thanks to the uniqueness of lifting [17]. Pick anyΩ⋆-allowed path λ. Its lifting with respect to pfromω ends atζ while its lifting with respect to p^′ from ω^′ ends at ξ. This gives a mapping τ :ζ ∈R_Ω

⋆ 7→ ξ ∈R^′ which is well-defined and injective (uniqueness of lifting), continuous (we work with étalé spaces) and preserves fibers.

Definition 2.12. — Let Ω_⋆ be a discrete filtered set centred at ω. Any Riemann^• surface (R^′,p^′, ω^′) over C which is isomorphic to (R_Ω

⋆,p, ω), is called a Riemann surface associated withΩ_⋆.

In this definition, isomorphic means the existence of a fiber preserving homeomor- phismτ :R_Ω

⋆ →R^′.

We would like to point out a consequence of the topology considered on these Riemann surfaces. On Fig. 2, we consider a discrete filtered set Ω_⋆ centred at 0

such that, for 0 < L₁ < L₂, Ω_L1 = {0, ω1, ω₂}, Ω_L2 = Ω_L1 ∪ {ω3, ω₄}. We have drawn two paths λ₁, λ₂ ∈ R_Ω⋆(L₁) ending at the same point ζ and Ω_⋆-homotopic, cl(λ₁) = cl(λ₂). Also we have drawn a path λ₃ starting at ζ and ending at ξ such that the product pathλ₂λ₃ belongs toR_Ω⋆(L₂). We remark that λ₁λ₃ ∈/R_Ω⋆(L₂).

The path λ₁, resp. λ₂, can be lifted with respect to p to a path Λ₁, resp. Λ₂, on R_Ω

⋆ with initial point 0 = p⁻¹(0) and common end point cl(λ₁) = cl(λ₂). From that point, λ₃ can be lifted with respect to p to a path Λ₃ ending atcl(λ₂λ₃). We thus see that the path Λ₁Λ₃ is well defined and p(Λ₁Λ₃) =λ₁λ₃. This means that λ₁λ₃ can be lifted onR_Ω

⋆ with respect top from0despite the fact thatλ₁λ₃ is not Ω_⋆-allowed. We say thatω₃ is a “removable Ω_⋆-point” forλ₁λ₃.

λ

λ

ω

ω

ω

ω

λ

ζ ξ

0

Figure 2

Definition 2.13. — LetΩ_⋆be a discrete filtered set centred atω^•₀∈Cand(R_Ω

⋆,p) its associated Riemann surface. Let Λ be a piecewiseC¹ path on R_Ω

⋆ starting from ω₀ = p⁻¹(ω^•₀) and λ = p◦Λ. Let L > L_λ. If λ meets a point ω ∈ Ω_L, then ω is called a removable Ω_⋆-point for λ.

2.5. Distance of a path to Ω_⋆. — LetΩ_⋆ be a discrete filtered set. We consider a point ζ ∈R_Ω

⋆. For r >0 small enough and since a disc is a star-shaped domain with respect to its origin, there exists a connected neighbourhoodU ⊂R_Ω

⋆ of ζ so thatp|^U :U →D(ζ, r)^• is a homeomorphism.

Definition 2.14. — For any ζ ∈R_Ω

⋆ and forr >0small enough, theballD(ζ, r) centred at ζ with radius r is the connected neighbourhood of ζ so that p|_D(ζ,r) : D(ζ, r) →D(ζ, r)^• ⊂C is a homeomorphism. The distance ρΩ⋆(ζ) of ζ to Ω⋆ is the supremum of the r >0 such that ζ has a neighbourhood of the form D(ζ, r).

In other words, ρΩ⋆(ζ) is the distance of ζ (for the norm |.|) to the boundary of R_Ω

⋆. IfΩ_⋆is centred atω^• =p(ω), then of courseρ_Ω⋆(ω) =^• ρ_Ω⋆(ω)and there is no risk of misunderstanding. Notice that the mappingζ ∈R_Ω

⋆7→ρΩ⋆(ζ)is continuous since p is continuous. This implies that t ∈ [0,1] 7→ ρ_Ω⋆ Λ(t)

is a continuous mapping when Λon R_Ω

⋆ starting fromω, thusinf_t∈[0,1]ρ_Ω⋆ Λ(t)

>0by compactness.

Definition 2.15. — Let Ω_⋆ be a discrete filtered set centred at ω^• =p(ω)∈C. For any path λ issued from ω^• that can be lifted to R_Ω

⋆ with respect to p from ω into the path Λ, one defines the distance d(λ,Ω_⋆) of λ to Ω_⋆ by d(λ,Ω_⋆) = inf_t∈[0,1]ρ_Ω⋆ Λ(t)

>0.

2.6. Some properties of the Riemann surface R_Ω

⋆. — Proposition 2.16. — The Riemann surface R_Ω

⋆ associated with the discrete fil- tered set Ω_⋆ is simply connected.

ThusR_Ω

⋆ is conformally equivalent to the open unit disc as a consequence of the uniformization theorem.

Proof. — LetΩ_⋆ be a discrete filtered setΩ_⋆ centred atω^• ∈Cand let(R_Ω

⋆,p)be its associated pointed Riemann surface, ω =p⁻¹(ω). Pick a non-constant closed curve^• Λ on R_Ω

⋆. We want to show thatΛ is null-homotopic. Since R_Ω

⋆ is arcconnected, one can suppose that Λ starts and ends at ω and there is no loss of generality in assuming that Λ does not meet ω apart from its extremities. Also, up to making a slight deformation of Λ in its homotopy class, one can assume thatλ=p◦Λ isC¹, with lengthLλ< Lfor someL >0and thatλ|_]0,1[avoidsΩ_L. One can writeΛunder the form⁽⁴⁾ Λ = Λ₁Λ⁻¹₂ where bothΛ₁,Λ₂ are paths starting from ω and ending at a point ζ ∈R_Ω

⋆ with ζ 6=ω. We set λ₁ =p◦Λ₁ and λ₂ =p◦Λ₂. Bothλ₁,λ₂ are Ω_⋆-allowed paths, precisely their belong to R_Ω⋆(L). The pathλ₁, resp. λ₂, can be lifted with respect topfromωand, by uniqueness of lifting, corresponds toΛ₁, resp.

Λ2. This implies that λ1 and λ2 are Ω⋆-homotopic with ζ = cl(λ1) = cl(λ2). The Ω_⋆-homotopy between λ₁ and λ₂ can be lifted with respect to pand this provides a homotopy between Λ₁ andΛ₂. Therefore, Λ = Λ₁Λ⁻¹₂ is null-homotopic.

Proposition 2.17. — Let (R_Ω

⋆,p) be the Riemann surface associated with a dis- crete filtered set Ω⋆ = Ω⋆(ω) centred at ω. Then, for every^• ζ ∈ R_Ω

⋆, there exists a discrete filtered set Ω_⋆(ζ) centred at ζ^• = p(ζ) such that every Ω_⋆(ζ)-allowed path starting from ζ^• can be lifted on R_Ω

⋆ fromζ with respect top.

Proof. — We consider the Riemann surface (R_Ω

⋆,p) associated with a discrete fil- tered setΩ_⋆ centred atω^• ∈Cand setω =p⁻¹(ω). Pick a point^• ζ ∈R_Ω

⋆ withζ 6=ω and assume that ζ = cl(λ₀), λ₀ ∈ R_ΩL

0 for some L₀ > 0. We consider a path λ (C¹ piecewise) starting formζ^•and of length L_λ. IfL_λ< ρΩ⋆(ζ), thenλcan be lifted from ζ with respect to p : this is just a consequence of the topology considered on R_Ω

⋆.

Assume now that λ satisfies the properties : λ(0) = ζ^•, λ(]0,1]) ⊂ C\Ω_L0+L and L_λ < L for some L >0. For ε >0 small enough, one can construct a Ω_⋆-homotopy H :t∈[0,1]7→H_t∈R_Ω⋆ such that

– H₀=λ₀;

– for every t∈[0, ε],Ht∈RΩ⋆(L0) and Ht(1) =λ(t);

– H_ε belongs toR_Ω⋆(L₀)∩R_Ω⋆(L₀+L);

– for every t∈[ε,1],H_t∈R_Ω⋆(L₀+L)and H_t(1) =λ(t).

4. Λ⁻₂¹ is the inverse path,Λ⁻₂¹(s) = Λ⁻₂¹(1−s).

Indeed, for t∈[0, ε],H_trealizes a small deformation of λ₀ so as to avoid the points of Ω_L0+L while for t ∈ [ε,1], H_t is for instance the standardized product of H_ε with λ|_[ε,t]. This Ω_⋆-homotopy can be lifted with respect to p into a homotopy H:t∈[0,1]7→ Ht where, for everyt∈[0,1],Ht: [0,1]→R_Ω

⋆ is a path starting at ω. Therefore, the pathΛ :t∈[0,1]7→ Ht(1)∈R_Ω

⋆ is a lifting ofλfromζ. This has the following consequences. There exists a discrete filtered setΩ_⋆(ζ) centred atζ,^•

– for L >0 small enough,ΩL(ζ) ={ζ},

– for L >0 large enough,Ω_L(ζ) = Ω_L0+L∩D(ζ, L)^•

∪ {ζ}

such that everyΩ_⋆(ζ)-allowed path can be lifted onR_Ω

⋆with respect topfromζ.

2.7. Seen and glimpsed points. — We denote byS¹ ⊂Cthe circle of directions about 0of half-lines on C. We usually identify S¹ withR/2πZ.

Definition 2.18. — Let I ⊂S¹ be an open arc, L >0 and ω ∈C. We denote by s^L

ω(I) the following open sector adherent toω:

s^L

ω(I) ={ζ =ω+ξe^iθ ∈C | θ∈I,0< ξ < L}.

Assume thatΩ_⋆ is a discrete filtered set centred atω ∈C,θ∈S¹ is a given direc- tion andL >0. SinceΩ_Lis a finite set, observe thatΩ_L∩s^L

ω(I) = Ω_L∩]ω, ω+e^iθL[

when I =]−α+θ, θ+α[with α >0chosen small enough.

Definition 2.19. — Let Ω_⋆ be a discrete filtered set centred at ω ∈ C, θ ∈ S¹ and L > 0. We denote Ω^⋆_L(θ) = ΩL∩]ω, ω +e^iθL[. One says that α ∈]0,^π₂[ is a Ω_⋆(θ, L)-angle if Ω_L∩s^L

ω(I) = Ω^⋆_L(θ) withI =]−α+θ, θ+α[.

Definition 2.20. — Let Ω_⋆ be a discrete filtered set centred at ω ∈C,θ∈S¹ and L > 0. We denote by R(Ω_⋆, θ, L) the set of piecewise C¹ paths λ that satisfy the conditions:

– λ(0) =ω andLλ< L;

– for every t∈[0,1], the right and left derivatives λ^′(t) do not vanish;

– there exists a Ω_⋆(θ, L)-angle α ∈]0,^π₂[ such that for every t ∈ [0,1], argλ^′(t)∈]−α+θ, θ+α[.

Remark that, apart from its origin, a pathλ∈R(Ω_⋆, θ, L)stays in an open sector of the forms^L

ω(I),I =]−α+θ, θ+α[, with aαaΩ_⋆(θ, L)-angle. Moreoverλalways moves forward in that sector.

Proposition 2.21. — Let Ω⋆ be a discrete filtered set centred atω0∈Candθ∈S¹ a direction. There exists a uniquely defined discrete and closed setGLIMP^⋆_Ω⋆(θ)⊂C that satisfies the following conditions for any L >0:

– GLIMP^⋆_Ω⋆(θ, L)⊂Ω^⋆_L(θ), whereGLIMP^⋆_Ω⋆(θ, L) = GLIMP^⋆_Ω⋆(θ)∩D(0, L);

– any path belonging to R(Ω_⋆, θ, L) that circumvents to the right or the left the set GLIMP^⋆_Ω⋆(θ, L), can be lifted on the Riemann surface (R_Ω

⋆,p) with respect to p fromp⁻¹(ω₀).

– when at least one point is removed from GLIMP^⋆_Ω⋆(θ), then the above property is no more satisfied.

Proof. — We show proposition 2.21 by constructingGLIMP^⋆_Ω⋆(θ).

If S

L>0Ω^⋆_L(θ) = ∅, then GLIMP^⋆_Ω⋆(θ) = ∅. Otherwise, from the very definition of Ω_⋆, one can define an increasing sequence 0 =L₋₁< L₀< L₁< L₂<· · · such that

– Ω^⋆_L0(θ) =∅,

– for everyi∈N^⋆,Ω^⋆_Li(θ) = Ω^⋆_Li−1(θ)∪ {ωi1,· · ·, ωil} with{ωi1,· · · , ωil}a finite subset of ]ω₀, ω₀+e^iθL_i−1];

– Ω^⋆_L(θ) = Ω^⋆_Li(θ) for every L∈]Li−1, Li]and every i∈N. We construct GLIMP^⋆_Ω⋆(θ)by induction on i∈N.

Case i = 0. Since Ω^⋆_L0(θ) = ∅, then for every L ≤ L0, every λ ∈ R(Ω⋆, L, θ) is Ω_⋆-allowed and thus can be lifted onR_Ω

⋆ with respect topfromp⁻¹(ω). Therefore, we setGLIMP^⋆_Ω⋆(θ, L) =∅ for L≤L0.

Case i = 1. From the above property, if ω∈Ω^⋆_L1(θ) satisfied |ω| < L₀, then it is a removable Ω_⋆-point for every path λ ∈ R(Ω_⋆, θ, L) with L ≤ L₁. Let us take L∈]L0, L1], so that Ω^⋆_L(θ) = Ω^⋆_L1(θ):

– either there is ω ∈ Ω^⋆_L1(θ) of the form ω =L₀e^iθ. In this case the condition GLIMP^⋆_Ω⋆(θ, L) = GLIMP^⋆_Ω⋆(θ, L₀)∪ {ω} is both needed and sufficient so as to ensure that for every L ≤ L1, any path λ ∈ R(Ω⋆, θ, L) that circumvents GLIMP^⋆_Ω⋆(θ, L)to the right or the left can be lifted on R_Ω

⋆; – or we setGLIMP^⋆_Ω⋆(θ, L) = GLIMP^⋆_Ω⋆(θ, L₀) =∅.

Induction. Pick some i ∈ N^⋆ and suppose that the following properties are valid for every integer j∈[1, i]:

– for every L∈]L_j−1, L_j], GLIMP^⋆_Ω⋆(θ, L) = GLIMP^⋆_Ω⋆(θ, L_j)⊂Ω^⋆_Lj(θ);

– GLIMP^⋆_Ω⋆(θ, L_j)∩]ω0, ω₀+e^iθL_j−1[= GLIMP^⋆_Ω⋆(θ, L_j−1);

– for every L ≤L_i, any path λ∈ R(Ω_⋆, L, θ) that circumvents GLIMP^⋆_Ω⋆(θ, L) to the right or the left, can be lifted on R_Ω

⋆ with respect top fromp⁻¹(ω0).

From these properties, every ω∈Ω^⋆_Li+1(θ)\GLIMP^⋆_Ω⋆(θ, L_i) such that |ω|< L_i is a removable Ω_⋆-point for every path λ ∈ R(Ω_⋆, θ, L) with L ≤ L_i+1. From the fact thatΩ^⋆_L,θ = Ω^⋆_Li+1,θ for every L∈]Li, L_i+1]:

– either there is ω ∈ Ω^⋆_Li+1(θ) of the form ω = L_ie^iθ. In that case one sets GLIMP^⋆_Ω⋆(θ, L) = GLIMP^⋆_Ω⋆(θ, L_i)∪ {ω}for L∈]L_i, L_i+1]and this provides a necessary and sufficient condition to ensure that any pathλ∈R(Ω_⋆, θ, L)that circumvents GLIMP^⋆_Ω⋆(θ, L) to the right or the left, can be lifted on R_Ω

⋆, for every L≤L_i+1;

– or we simply setGLIMP^⋆_Ω⋆(θ, L) = GLIMP^⋆_Ω⋆(θ, L_i) for L∈]Li, L_i+1].

This ends the proof.

Definition 2.22. — LetΩ⋆ be a discrete filtered set centred atω0∈C,θ∈S¹. The discrete and closed set⁽⁵⁾ GLIMP^⋆_Ω⋆(θ) ={ωi∈]ω0, ω₀+e^iθ∞[, ω0 ≺ω₁≺ω₂· · · } given by proposition 2.21 is the set of glimpsed Ω_⋆-points in the direction

5. The symbol≺stands for the total order on[ω0, ω0+e^iθ∞[induced byr∈[0,∞]7→ω0+re^iθ∈ [ω0, ω0+e^iθ∞[.

θ. The glimpsed point ω₁ is the seen Ω_⋆-point in the direction θ. The com- pleted set of glimpsed Ω_⋆-points GLIMP_Ω⋆(θ) in the direction θ is defined by GLIMP_Ω⋆(θ) = GLIMP^⋆_Ω⋆(θ)∪ {ω₀}.

Remark 2.23. — The notion of glimpsed point can be defined in a simpler way but the presentation we have made here is fitted to the methods that we develop in the paper.

3. Endless continuability 3.1. Endless Riemann surface. —

Definition 3.1 (Endless Riemann surface). — A Riemann surface (R,p), given as an étalé space onC, is said to beendless if for everyζ ∈R, there exists a discrete filtered set Ω_⋆(ζ) centred at ζ^• =p(ζ) so that every Ω_⋆(ζ)-allowed path can be lifted on R with respect to p fromζ.

Example 3.2. — LetΩbe a closed discrete subset ofC. Then the universal covering

^C\Ω ofC\Ω is an endless Riemann surface.

The following result is a direct consequence of proposition 2.17.

Proposition 3.3. — Let Ω_⋆ be a discrete filtered set. Then the associated Riemann surface (R_Ω

⋆,p) is endless.

3.2. Endless continuability. —

Definition 3.4 (Endless continuability). — A germ of holomorphic functions b

ϕ∈ Oω at ω∈Cis endlessly continuable on Cifϕbcan be analytically continued to an endless Riemann surface. One denotes byRˆ_ω,endlthe space of germ of holomorphic functions atωthat are endlessly continuable onC. When ω= 0we use the abridged notation Rˆ_endl = ˆR_0,endl.

Proposition 3.5. — A germ of holomorphic functionsϕb∈ O_ω atω∈Cis endlessly continuable onCif and only if there exists a discrete filtered setΩ_⋆ centred atω such that ϕb can be analytically continued along any Ω_⋆-allowed path.

Proof. — We suppose that ϕb∈ O_ω^• is endlessly continuable, thus ϕbcan be analyti- cally continued to an endless Riemann surface (R,p). This means that there exist a neighbourhood U^•⊂Cofω^• =p(ω) and a neighbourhood U ⊂R of ω such that the restrictionp|U :U →U^• is a homeomorphism, and there is a functionΦholomorphic onRso thatφ= Φ◦p|⁻¹U represents the germϕ. By the very definition of an endlessb Riemann surface, one can find a discrete filtered set Ω_⋆ centred atω^• so that every Ω_⋆-allowed pathλcan be lifted with respect topinto a pathΛstarting atω. SinceΦ can be analytically continued alongΛ, one gets thatϕbcan be analytically continued alongλas an upshot.

We now suppose thatϕb∈ O_ω^• can be analytically continued along anyΩ_⋆-allowed path, where Ω⋆ is a discrete filtered set Ω⋆ centred at ω. By proposition 2.17, the^• Riemann surface(R_Ω

⋆,p)associated with this discrete filtered set is endless. Toϕbis associated a germ of holomorphic functionsΦatω,ω^• =p(ω)that can be analytically continued along the path Λ starting fromω and deduced from any Ω_⋆-allowed path λ. SinceR_Ω

⋆ is simply connected, this implies that Φcan be analytically continued to a function holomorphic on R_Ω

⋆.

Definition 3.6. — IfΩ⋆ is a discrete filtered set centred atω, one denotes byRˆ_Ω

⋆

the space of germs of holomorphic functions atω that can be analytically continued to the Riemann surface R_Ω

⋆.

3.3. Seen and glimpsed points. — We have introduced the notion of glimpsed Ω_⋆-points (definition 2.22) associated with a discrete filtered setΩ_⋆centred atω₀. Ifϕb is an endless continuable germ atω0that belongs toRˆ_Ω

⋆ then, by the very definition ofGLIMP^⋆_Ω⋆(θ),ϕbcan be analytically continued along any path that closely follows the half-line[ω0, ω0+e^iθ∞[in the forward direction, while circumventing to the right or to the left each point from the set GLIMP^⋆_Ω⋆(θ). However, this set is not always the smaller one and one easily gets the following proposition.

Proposition 3.7. — Let ϕb∈Rˆ_ω,endl be an endlessly continuable germ of holomor- phic functions atω∈Cand letθ∈S¹ be a direction. There exists a uniquely defined discrete and closed set GLIMP^⋆_ϕb(θ) ={ωi ∈]ω0, ω₀+e^iθ∞[, ω0 ≺ω₁ ≺ω₂· · · } such that:

– ϕb can be analytically continued along any path that closely follows the half-line [ω₀, ω₀+e^iθ∞[in the forward direction, while circumventing (to the right or to the left) each point of the set GLIMP^⋆_ϕb(θ).

– this property is no more valid if at least one point is removed fromGLIMP^⋆_ϕb(θ).

If Ω_⋆ a discrete filtered set centred at ω₀, then GLIMP^⋆_ϕb(θ)⊆GLIMP^⋆_Ω⋆(θ) for any b

ϕ belonging to Rˆ_Ω

⋆.

Definition 3.8. — The elements of GLIMP^⋆_ϕb(θ) are called the glimpsed singu- lar points in the direction θ ∈ S¹ for the endlessly continuable germ ϕb∈Rˆ_ω,endl. Specifically, ω₁ is theseen singular point in the directionθ for ϕ.b

3.4. Continuability without cut. — We complete this Sect. with a brief com- parison to Ecalle’s endless continuability.

Definition 3.9 (Riemann surface without cut[12]). — Let (R,p) be a Rie- mann surface given as an étalé space on C. This surface is said to be without cut if for every ω ∈R, there exists a closed and discrete set sing(ω) ⊂C that satisfies the following properties. Introducing ω^• =p(ω):

1. if the line segment[ω,^• ω^•′]⊂C does not meetsing(ω), then[ω,^• ω^•′]can be lifted homeomorphically on R with respect to pfrom ω.

2. for every line segment [ω,^• ω^•′]⊂Cthat meets the points ω^•₁,· · · ,ω^•_r of sing(ω), there exists an open rectangleW neighbourhood of[ω,^• ω^•′]such that each of the 2^r simply connected open sets W_j deduced from W by making lateral cuts at ω•₁,· · · ,ω^•_r (see Fig. 3) can be lifted homeomorphically onR with respect top to an open setW_j ⊂R containing ω.

3. if at least one point is removed from sing(ω), then properties 1-2 are no more satisfied.

One says that the point ω^•₁ ∈ sing(ω) is seen from ω if [ω,^• ω^•₁]∩sing(ω) ={ω^•₁}.

Otherwise the points ω^•_j ∈sing(ω) areglimpsedfrom ω.

ω₁ ω₂ ω₃ ω₄ ω ω

Figure 3

We note that in Definition 3.9, condition 3 is added so as to define the seen and glimpsed singular points.

Definition 3.10 (Continuability without cut). — A germ of holomorphic functions ϕb ∈ O0 at 0 ∈ C is said to be analytically continuable without cut on Cif its Riemann surface is without cut.

There is the following relationship between endless continuability in the sense of definition 3.1 and continuability without cut.

Proposition 3.11. — Let (R,p) be a Riemann surface. IfR is endless, thenR is without cut.

Proof. — We assume that the Riemann surface (R,p) is endless. We consider a pointω∈R,ω^• =p(ω): there exists a discrete filtered setΩ_⋆ centred atω^• such that every Ω_⋆-allowed path λ ∈ R^⋆_Ω⋆ can be lifted on R from ω with respect to p. We consider the line segment [ω,^• ω^•′]⊂Cand L > l >0where l=|ω^•′−ω|.^•

1. Assume that the line segment[ω,^• ω^•′]does not meetΩLapart fromω. Then the^• path λ:t∈[0,1]7→ω^•+t(ω^•′−ω)^• belongs toR_Ω⋆(L)and thus can be lifted by p fromω.

2. Assume that the line segment [ω,^• ω^•′] meets the points ω,^• ω^•₁,· · · ,ω^•_r of Ω_L. Consider an open rectangleW centred on (and thus neighbourhood of) [ω,^• ω^•′], of lengthl+2l^′and width2l^′wherel^′ >0satisfiesl+2l^′ > L. Forl^′small enough one has W ∩Ω_L ={ω,^• ω^•₁,· · · ,ω^•_r} and for every ζ^•∈W \ {ω^•₁,· · ·,ω^•_r}, there exists a path λ ∈ RΩ⋆(L)^⋆ such that λ([0,1]) ⊂ W \ {ω^•₁,· · · ,ω^•_r}, λ(0) =ω^• and λ(1) =ζ.^•