The well-behaved Catalan and Brownian averages and their applications to real resummation

Publicacions Matem`atiques, Vol 41 (1997), 209–222. THE WELL-BEHAVED CATALAN AND BROWNIAN AVERAGES AND THEIR APPLICATIONS TO REAL RESUMMATION Fr´ ed´ eric Menous Abstract The aim of this expository paper is to introduce the well-behaved uniformizing averages, which are useful in resummation theory. These averages associate three essential, but often antithetic, properties: respecting convolution; preserving realness; reproducing lateral growth. These new objects are serviceable in real resummation and we sketch two typical applications: the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields. 1. Introduction This paper introduces the well-behaved uniformizing averages, which answer to the problem of real resummation: how to assign a real sum to a real divergent series of “natural origin”. We first give some heuristics which point out the difficulties that relate to the real resummation. The well-behaved averages solve these problems and we will explain the action of these objects. We will also present some well-behaved averages. Once these objects are given, we sketch two simple examples of their use (among their large range of application): the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields. Most of these results are due to J. Ecalle and a complete exposition can be found in [1]. 210 F. Menous 2. Some heuristics. The need for well-behaved averages 2.1. The resummation scheme. All the series and functions in zintroduced here are considered at infinity. Let ˜ϕ(z) be a real (with real coefficients) divergent series of “natural origin”: for instance the formal solution of a local analytic equation or system: (1) E(˜ϕ)=0. The most simple resummation scheme for resumming ˜ϕ(z) goes like this: (2) ϕ(z)−→ ϕ(z)  ˆϕ(ζ) We begin by subjecting ϕ(z) to the formal Borel transform (to obtain ˆϕ(ζ)) which, for instance, turns each monomial z−σinto ζσ−1/Γ(σ) (σ>0). Then we carry out a Laplace transform: (3) ˆϕ(ζ)−→ ϕ(z)=+∞ 0 e−zζ ˆϕ(ζ)dζ. This procedure for turning the formal object ϕ(z) into a geometric one ϕ(z) is the most simple one, but it is already representative of the difficulties arising from the need for a real resummation. Although the move ˆϕ(ζ)→ ϕ(z) seems to be one single step, it actually involves three distinct substeps. 2.2. Three steps in one. (i) First substep: calculating a germ. The function ˆϕ(ζ) is obtained, by the Borel transform, as a germ near ζ= +0 and, generally speaking, it converges only for small enough values of ζ. (ii) Second substep: getting a global function. We must continue this germ from +0 to +∞so as to get a global function, which could be Laplace transformed. This will be generally possible by analytic continuation, owing to the “natural origin” of ˜ϕ(z). Catalan and Brownian averages 211 (iii) Third substep: uniformizing the global function. Although there are no obstacles to analytic continuation, there may be analytic singularities. Indeed, the existence of singularities in the ζ-plane is precisely what causes the divergence of ˜ϕ. There are often compelling reasons for them to be located on R+. Then ˆϕis multivalued (many-branched) over R+ (its determination depends on the choice of a path of analytic continuation and on the way this one dodges the singularities.). If so, we must turn ˆϕ(ζ), “in some suitable way”, into a univalued (uniform) function (mˆϕ)(ζ), so as to be able to carry out the Laplace transform. 2.3. What does “suitable way” mean?. If we assume that ˆϕ(ζ) is multivalued, we will turn it into a uniform function by making an average of the different analytic continuations of ˆϕand then, the difficulty relates to the choice of a “suitable” (or wellbehaved) uniformizing average m. Here, suitable means three things: P1: mmust respect convolution: it is indispensable in all non-linear situations that mturns convolution into convolution. The Borel and Laplace transforms are algebra homomorphisms. Thus our average m must be an algebra homomorphism (for the convolution of “ramified” functions and the convolution of “uniform” functions) so as to assign to ˜ϕ(z) a sum ϕ(z) which is also a solution of the original equation. P2: mmust respect realness: this is rather necessary if ˜ϕ(z) has real coefficients and if we want to assign a real sum for some compelling reason: for example, if it represents a physical or real-geometric object. P3: mmust respect the lateral growth: for a series ˜ϕof natural origin, ˆϕdisplays, generally speaking, the good growth rate (that is to say exponential growth) which allows to carry out the Laplace transform. But that statement must be restricted. In fact, this exponential growth is obtained only: – on singularity-free axes Γθ(from 0 to infinity in the direction θ). – on both sides (right and left) of a singularity-carrying axis. If ˆϕ(ζ) has singularities on R+, this exponential growth is, generally speaking, also ensured on paths Γ which are close to the positive axis and cross it only a finite number of times. But, if R+carries infinitely many singularities, on paths Γ following R+, but with infinitely many crossings, the function ˆϕ(ζ) often has faster-than-exponential growth: (4) |ˆϕ(ζ)|≤ c0exp(c1(|ζ|+|ζ|log |ζ|)). Unfortunately, the uniformizing averages which are P1 and P2 will involve the analytic continuations of ˆϕ(ζ) on such “often-crossing” paths. 212 F. Menous Thus we must carefully choose the average msuch that (mˆϕ)(ζ) has a no-faster-than-exponential growth (|(mˆϕ)(ζ)|≤ c0exp(c1|ζ|)). Now an average mwill be called a “well-behaved” uniformizing average if the three properties P1, P2, P3 hold. In order to define and present such well-behaved averages, we need to introduce the convolution algebras of resurgent functions. But, for the sake of simplicity, we will restrict ourselves to the definition of the convolution algebra RESUR(R+//N, int.) of resurgent functions, with singularities over N∗and which are locally integrable. Nonetheless, the following statements can be extended to more general convolution algebras (with some different set of singularities and without the condition of local integrability). 2.4. The algebra RESUR(R+//N, int.). Definition. The algebra RESUR(R+//N, int.) is defined as follows. Let ˆϕ(ζ) be an element of this algebra, then: •ˆϕ(ζ) is defined and holomorphic at the root of R+(on ]0,[). •ˆϕ(ζ) is analytically continuable along any path that follows R+ and dodges each point of N∗to the left or to the right, but without ever going back. •All the determinations of ˆϕ(ζ) are locally integrable on R+. Moreover, the convolution is defined by: ˆϕ3(ζ)=(ˆϕ1∗ˆϕ2)(ζ)=ζ 0 ˆϕ1(ζ1)ˆϕ2(ζ−ζ1)dζ1(0 <ζ1)(5) (ˆϕ1,ˆϕ2∈RESUR(R+//N, int.)). This expression is purely local (at ζ= 0) and the germ ˆϕ3(ζ) must then be extended, by analytic continuation, to a global function. For details see [1]. Now, for a function of RESUR(R+//N, int.), we can give the following notation: Let ˆϕ(ζ) be a function of RESUR(R+//N, int.) and (ε1,... ,ε n)be a sequence of nplus or minus signs, then, for ζin ]n, n + 1[, we will note ˆϕε1,... ,εn(ζ) the analytic continuation of ˆϕfrom 0 to ζon the path that follows R+and dodges each singularity k(1 ≤k≤n) to the right (resp. to the left)ifεk= + (resp. εk=−). Catalan and Brownian averages 213 Example. If ζ∈]4,5[, then ˆϕ+,−,−,+(ζ) is the analytic continuation of ˆϕalong the following path: •✲ ••••• ζ 0 Of course, ˆϕ∅(ζ)(O<ζ<1) is the unique determination of ˆϕon ]0,1[. Once this notation is given, for any fixed integer n, a function ˆϕ of RESUR(R+//N, int.) has 2npossibly different determinations ˆϕε1,... ,εn(ζ)over the interval ]n, n + 1[ and a uniformizing average m will return an actual average of these 2ndeterminations. 2.5. The uniformizing averages. A uniformizing average mis a uniformizing projection of the space RESUR(R+//N, int.) into the space UNIF(R+, int.) of uniform, locally integrable functions on R+. It can be define as a collection of “weights”: (6) m={mε1,... ,εn;n∈N;εi=±;mε1,... ,εn∈C} subject to the self-consistency relations: (7)  εn=± mε1,... ,εn=mε1,... ,εn−1(resp. m∅=1)ifn>1 (resp. n=1) and the action of the average mon a function ˆϕof RESUR(R+//N, int.) is defined as follows: (8) ∀n∈N;∀ζ∈]n, n +1[ (mˆϕ)(ζ) = ε1=±···±εn=± mε1,... ,εnˆϕε1,... ,εn(ζ). Thus an average mturns a multivalued function into a uniform one by averaging its different determinations and it is important to point out that the self-consistency relations are a necessity: for instance, whenever a function ˆϕof RESUR(R+//N, int.) has only fictive singularities, that is to say ˆϕis uniform, then we would like to obtain mˆϕ=ˆϕ, which is ensured by the self-consistency relations. Once these definitions are given, there exists more precise statements for the properties P1, P2, P3: P1: An average mrespects convolution if and only if, for any two functions ˆϕand ˆ ψin RESUR(R+//N, int.): (9) m(ˆϕ∗ˆ ψ)=(mˆϕ)∗(mˆ ψ) 214 F. Menous where the first star ∗(resp. the second) denotes the convolution on RESUR(R+//N, int.) (resp. on UNIF(R+, int.)). This condition is ensured if and only if the weights of mverify a universal multiplication table which reads, for example: (10)                m+m+=m+,+−m−,+ m+m−=m+,−+m−,+ m−m−=m−,−−m+,− m+m+,+=m+,+,+−m+,−,+−m−,+,+ . . . For proofs and complements, see [1], [4]. P2: The fact that an average mrespects realness can easily be read on its weights. Let ˆϕbe a function of RESUR(R+//N, int.) and let us assume that ˆϕis the formal Borel transform of a real divergent series. Then ˆϕ(ζ)isreal for small enough real values of ζand assumes complex conjugate values on complex conjugate paths of analytic continuation: (11) ∀n∈N;∀ζ∈]n, n + 1[; ∀εi∈{+,−} ˆϕε1,... ,εn(ζ)= ˆϕ¯ε1,... ,¯εn(ζ) where ¯εiis the opposite sign to ε. Therefore, a uniformizing average mrespects realness ((mˆϕ) is real on R+) if and only if: (12) mε1,... ,εn=m¯ε1,... ,¯εn(∀n≥0; ∀εi∈{+,−}). P3: Although this appears to be the main demand, we won’t go into details in this introductory paper (see [1]). This condition does not reduce to growth conditions on the weights. It actually involves some compensation phenomena. This faster-than-exponential growth on “oftencrossing” paths is a precise mechanism, which has to do with the nature of the “acting alien algebra”. But, whenever a function ˆϕis of “natural origin”, the analysis of this nuisance (with the “Bridge equation”) shows how to construct, independently of ˆϕitself, “well-behaved” averages m which produce mean values mˆϕwith the requisite exponential growth. These three properties tend to be mutually exclusive but the main fact is that such well-behaved averages exist. We shall give now some examples of uniformizing averages (well-behaved or not). Catalan and Brownian averages 215 3. Examples of uniformizing averages 3.1. The right-lateral average mur and the left-lateral average mul. It can be proved that they are the only convolution-preserving averages that involve only one determination over each interval. In term of weights: murε1,... ,εn= 1 (resp. 0) if ε1=ε2=···= + (resp. otherwise)(13) mulε1,... ,εn= 1 (resp. 0) if ε1=ε2=···=−(resp. otherwise)(14) mur and mul are P1 (convolution) and P3 (lateral growth) but clearly fail to preserve realness. 3.2. The median average mun. The weights of mun depend only on the number p(resp. q) of + signs (resp. −signs) in the address (ε1,... ,ε n): (15) munε1,... ,εn≡Γ(p+1/2) Γ(q+1/2) Γ(1/2)Γ(p+q+ 1)Γ(1/2) ≡(2p)! (2q)! 4p+q(p+q)! p!q! and mul,mur and mun can be embedded in an interval of averages (depending also on the number of + and −signs) muα,β with weights: muε1,... ,εn α,β ≡Γ(p+α)Γ(q+β) Γ(α)Γ(p+q+ 1)Γ(β)(α, β ∈R+;α+β=1)(16) mu1,0=mur;mu1/2,1/2=mun;mu0,1=mul.(17) The averages muα,β respect convolution (P1). Only mun respects realness but only mur and mul respect lateral growth: no one of these averages is “well-behaved”. 3.3. The Catalan average man. The weights of man assume rational values, and can be obtained by the following formula: (18) manε1,... ,εn≡4−ncan1can2...ca ns(1 + ns) with the classical Catalan numbers: (19) can def =(2n)! n!(n+ 1)! (can∈N) 216 F. Menous which in this case are indexed by the integers n1,n 2,... ,n swhich denote the numbers of identical consecutive signs within the address (ε1,... ,ε n): (20) (ε1,... ,ε n)=(±)n1(∓)n2...(εn)ns(of course n1+···+ns=n). Like mun, the Catalan average man may be embedded in an interval of averages maα,β (as usual α+β= 1) with weights: (21) maε1,... ,εn α,β def =(αβ)n(can1can2...ca ns−1)Cans((α/β)εn). Once again nidenotes the cardinality of the ith cluster of identical signs. The new formula, however, alongside with the Catalan numbers can, also involves the Catalan polynomials Can, which are distinguished by a capital Cand inductively definable by: Ca0(x)=1(22) Ca1+n(x)=−(1 + x−1)can+(x+2+x−1)Can(x).(23) All negative powers of xcancel out, and it may be noted that: Can(0) = can;Can(1) = (1 + n)can (24) lim x→−1(x+1) −1Can(x)=can−1.(25) The Catalan average man is interesting because it is a well-behaved uniformizing average (P1, P2, P3). Moreover, under a “rescaling” and a “passage to the limit”, the Catalan average gives rise to the so-called Brownian average mown. For details, see [5], [1], [4]. 3.4. The Brownian average mown. If we define, for any positive integers mi: (26) manm1,ε1,... ,mn,εn def = ηi k=± manη1 1...η1 m1−1ε1η2 1...η2 m2−1ε2...εn−1ηn 1...ηn mn−1εn then the following limits exist for any address (ε1,... ,ε n) and define the weights of an average: (27) mownε1,... ,εn= lim m→+∞manm,ε1,... ,m,εn. These weights determine the Brownian average mown, which is also a well-behaved uniformizing average. For man and mown this point follows from their direct study (by using the family maα,β [5]). But this also follows from the fact that these averages belong to a larger set of well-behaved convolution averages, introduced and studied by J. Ecalle: “The averages induced by a diffusion”. Catalan and Brownian averages 217 3.5. Averages induced by a diffusion. We fix an integrable function fon Rsuch that: (28) +∞ −∞ f(x)dx =1. The function fmay be viewed as representing the probability distribution at the time t= 1, on the vertical axis 1+iR, of a particle starting from the origin at t= 0, moving along R+with unit speed, and diffusing randomly in the vertical direction. To any such “diffusion”, we may associate a uniformizing average mwith weights defined as follows: Definition (J. Ecalle). mε1,... ,εnis the probability for the particle to hit the half-axis n+iεnR+at the time nafter successively crossing each half-axis j+iεjR+(1 ≤j<n) at the time j. Analytically, this translates into the following formula: (29) mε1,... ,εn =f(x1)...f(xn)σε1(x1)σε2(x1+x2)...σ εn(x1+···+xn)dx1...dx n with integration over Rnand with the classical step functions σ+and σ−: (30) σ±(x)≡1 (resp. 0) if ±x>0 (resp. ±x≤0). J. Ecalle proved that any average induced by a diffusion respects both convolution and lateral growth. Moreover, as soon as the function fis even (f(x)=f(−x)), the average also respects realness and thus is a well-behaved uniformizing average (for details see [1]). Now it can be checked that both man and mown are induced by a diffusion: man induced by f(x)=1 2exp(−|x|)(31) mown induced by f(x)= 1 2√πexp −x2 4.(32) The construction and the study of the averages induced by a diffusion are due to J. Ecalle. Moreover, this method for building well-behaved averages allows to generalize them, so that they can act on resurgent functions having their singularities in any discrete additive semi-group of R+. J. Ecalle also proved that the condition of local integrability, for the resurgent functions, is not a necessity. Thus the well-behaved averages have a great range of application and we will present, in the two following sections, some illustrations of the use of such averages (for details, see [1],[4]).