The Acceleration Operators and Their Applications
to Differential Equations, Quasianalytic Functions,
and the Constructive Proof of Dulac's Conjecture
Jean P. Ecalle
Mathématiques, Bâtiment 425, Université de Paris-Sud, Centre d'Orsay
F-91405 Orsay, France
We introduce (§§1,2) a general apparatus (resurgence, alien derivations, acceleration, etc.) that enables one to study and resum most divergent expansions of
natural .origin. We then proceed to give three select applications (§§3-5).
§1. Resurgent Functions, Alien Derivations and Medianization
The Singularity Algebras ^(Se)
and ^ i n t ( 5 ö )
Let Ç be the Riemann surface of log £ and So (resp. Solto2) the semi-axis arg( = 9
(resp. the sector 9\ < arg£ < Ö2). A major cp on So is an analytic germ defined
on So-2n,e close to 0 ("at the root of So-2n,o")' The minor cp of cp is defined at
the root of SQ by cp(Ç) = cp(Q — <p(( • e~2nî). A singularity of direction 0 is a class
cp of majors cp modulo the space of regular (i.e. holomorphic) functions at 0.-A
singularity cp is said to be integrable iff:
' Çcp(Ç) —> 0 as C —• 0
(1.1)
[°mo\\dc\< +00
on So^nß
and
for Co £ SQ close to 0.
Jo
Integrable singularities are fully determined by their minor.
For any two classes $ i 5 $ 2 G ^(Se) and u E So-2n close to 0, the class <p3 of
the major <piiU defined by
(1.2)
h,«(0= [ V(Ci)fa(C-Ci)dC,
Ju
(u E S6-2n J C and ( - W G SG-2n,G-n)
depends neither on u nor on the choice of fa in $,-. The convolution <p1 * cp2 = cp3
thus defined turns £F(SQ) into a commutative algebra. The space SFmX(So) of
integrable singularities is a subalgebra and its convolution reduces to :
(1.3) (pi*fa(0= - / fa(Ci)fo(C-Ci)<iCi
(CCi.C-CionSfl and close to 0).
Jo
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
1250
Jean P. Ecalle
The Resurgence Algebras ffl(Se) and ffiint(SQ)
The subspace &(SQ) C £F(SQ) of all $ whose minor cp can be analytically continued
along any path that keeps close to SQ (without going back) and bypasses to the
right or to the left all intervening singular points cot E SQ, is closed under
convolution. For any sequence e* of + signs and any f in ]cor,œr+i[, denote by
<pSi7.%r(0 t n e determination of q>(Q obtained by starting from 0 and bypassing
each cot to the right (resp. left) if e* = + (resp. —). The space of all cp in
0£(SQ) whose minors cp have all their determinations cp%. integrable on their
segment of definition ]cor9œr+i\j constitutes a subalgebra 0tmi(So). We call 0£(SQ)
(resp. 0tmi(SQ)) the algebra of resurgent functions (resp. integrable resurgent
functions) of direction 6.
Alien Derivations and Medianization
For any finite sequence st = + denote by p (resp. q) the number of + (resp. —)
signs and consider the weights:
(1.4)
S- - = 3M = r - ^ 4 1 T T ; A — = XMM =
(p + 4 + 1 ) ! '
{2P) !
™'
4P+iplq\{p + q)\
For any m € Se the operator Aœ of ${Se) onto itself defined by:
(
1
.
5
)
4
„
:
#
.
—
•
&
,
w
i
t
h
3
U
Q
=
Y
,
«
"
^
W
Ä
i
,
«
+
û
>
)
(for C on SQ and close to 0) is a derivation of the algebra ^(S^) :
(1.6)
Acotâi * $2) = ( 4 ^ ) * q>2 + & * ( ^ # 2 ) .
We call zdû, the alien derivation with index œ. On ^mt(Sû,) it reduces to:
(1.7)
Aœ
with
:q>\—>
<pœ
cpœ(o = £ a«-*-* {MÄ+,(C + œ) - %;^:^(C + ©)}.
Along with the natural derivation d : cp H-> —Ç#, the zU finitely generate all
continuous derivations of M(SQ).
Similarly, the operator med ("medianization") :
(1.8)
med : q>(0 • - > med cp(0 = X ^ Ä ( 0
( if ^ < C < œ r+ i)
e«
mt
is an homomorphism of the algebra $ (So) into the algebra JLmt(S0) of univalued,
locally integrable functions on SQ :
(1.9)
med (cpi * q>2) = (med pi) * (med cp2)
with * as in (1.3).
Medianization has the added advantage of preserving realness: if the germ cp is
real, so is the function med cp.
The Acceleration Operators and Their Applications
1251
The Resurgence Algebras M and MmX
The convolution algebra of all classes q> which, along with their successive alien
derivatives AŒr... A^cp (Vû>/ E Ç) belong to all 0t(So) (resp. all Mint(So)) is known
as the general algebra M (resp. Mint) of resurgent functions (resp. integrable r.f.).
Their majors cp are defined in spiral-like neighbourhoods of 0 on (Ç and their
minors cp can be continued (starting from 0 and bypassing intervening singular
points) along any split line on (Ç.
Resurgent functions of natural origin tend to reproduce themselves at their
singular points. This self-reproduction is exactly described by resurgence equations
linking cp to its alien derivatives Awcp. By a slight abuse, we often extend the label
of "resurgent function" to those power series whose Borei transforms (see §2)
belong to @. See [1-3].
§2. The Acceleration Operators
The Laplace Transform ££ and the Borei Transforms $ and ^
/•+00
(2.1)
JS?: 0(C) •—•?>(*)= /
e**M)dC
Jo
1
(2.2)
rc+ico
» : cp(z).—> HO = ^-. /
e*cp(z) dz
The classical Laplace transform if is a homomorphism of the convolution algebra
lLg"p(IR+) of all univalued, locally integrable functions on R + with (at most)
exponential growth at +oo, into the multiplicative algebra IB of holomorphic
germs bounded in half planes Rez > xo. Its inverse M is known as the Borei
transform. For each formal series cp(z) = J]e„(z) whose general term e„ E IB has
a Borei transform e„, we have a notion of formal (or term-wise) Borei transform :
(2.3) £ : 0(z) = £ e „( Z ) —• HO = £ M 0
( e * Z««^" —• X ^ " " 1 ) •
The Acceleration Operators and Their Kernels
An acceleratrix is a function F holomorphic in a neighbourhood of op E Ç, real
positive on R + = So and such that for z —• oo:
(2.4)
x~1F(x) -+ 0 ; ÖF(z) ~ ÖF(x) ; ö2F(z) ~ ö2F(x)
with 0 < x -> oo ; z = xeiB (9fixedin R) and:
(2.4bis)
ÖF(z) = zF,(z)F(z) ; ö2cp(z) = 1 + zF,,(z)/F/(z) - zF'(z)/F(z).
The co-acceleratrix G of F is the germ G defined on [+0,...] by:
(2.5)
G(f(z)) = F(z) - zf(z) with f(z) = F'(z)
(G(Q -+ +oo as C -+ 0).
1252
Jean P. Ecalle
Borei-Laplace takes the multiplicative endomorphism %? : cp\(z\) \-> ^2(^2) =
cpi (F(z2)) of B into a convolution endomorphism <&F of 14"p(R+) :
/•+00
(2.6)
% = a<ev.se : 9i(Ci)>—fe(C2)= /
cF(f2,Ci)^i(Ci)dCi•
Jo
§> is known as the acceleration z\ -> Z2. Its integral kernel is given by:
i
(2.7)
rc+ico
CF(Ç2,Ci) = T-.
exp(Ç2z2 - CiFfa)) dz2
and it has faster-than-exponential decrease in Ci :
(2.8)
log CF(C2,Ci) ~ -CiG(C2/Ci) for 0 < C2fixedand Ci -• +00.
Therefore, the natural domain of definition of <gF is much larger than !Lj£p(R+);
it is the convolution algbera !Lj?*acc(R+) of all functions #i(Ci) with F-accelerable
growth, i.e. for which there exists 0 0 such that:
(2.9)
|^i(Ci)|<Cst./CF(c,Ci) or
log|<MCi)| < fiG(c/Ci) as Ci - +co.
The largest such c is the acceleration abscissa of <pi.
Strong, Moderate, Weak Accelerations
Strong accelerations (logZ2/logzi -> +00) have kernels with slightly overexponential decrease in (1 and yield germs £2 (£2) which are defined in a
spiral-like neighbourhood of 0 E Ç with infinite aperture. Moderate accelerations (logZ2/logzi —> 1/a with 0 < a < 1) have kernels decreasing like
exp(—caC\ C2 ) w ^ n ß = 1 — & and they yield germs cp2 (Ci) which are defined in a sector of aperture nß/a. Weak accelerations (logZ2/logzi —> 1 but
Z2A1 -> 1) have very fast decreasing kernels but they yield germs fatti) which
are defined only at the root of R + = So and are usually non-analytic, but only
Denjoy-quasianalytic (cf. §4). Of great practical importance are the elementary
accelerations z\ —> Z2 = exp(o-zi) and z\ —> Z2 = z\^ with their respective
kernels :
(2.10)
CF&2, d) = ( k F ' - y r (Ci/cr)
(2.11)
cF(C2,Ci) = C2-1ca(z)
with X = Ci l/^2~a/P
(2.11bis)
(0 < a, ß < 1, a + ß = 1)
C a (r) ~ (C/2TT) 1/2 .X 1/2 . exp(-cX)
when X -» oo with Re X > 0 (c = a a / ^ ) .
The Acceleration Operators and Their Applications
1253
Accelero-summability
A formal series cp(z) — Xifi»(z) *s s a id t 0 be accelero-summable with sum cp(z)
and critical times zi,Z2,...,z,. if it can be subjected to the following operations
(algebra homomorphisms) :
cpi(z\)= cp(z)
fa (Ci) —^
> cp(z) =cpr(zr)
fatti)
—^
fatti)
->•••-•
v
fattr)
multiplicative algebras
convolution algebras
with arrows (i,i + 1) denoting the acceleration z\ —• z, + i. See [4, 5, 7].
§3. Acceleration Applied to Many-Levelled Differential Systems
Resurgent functions are truly ubiquitous. They arise as formal solutions of
differential equations (or difference equations, or general functional equations)
with analytic coefficients, or of systems of such equations. They occur in the
study (normalization, conjugacy, iteration) of local analytic objects, chiefly: local
singular vector fields and local diffeomorphisms. Again, most expansions in a
"singular parameter" (such as the Planck constant in the Schrödinger equation)
turn out to be divergent and resurgent. Indeed, it is no exaggeration to claim
that most divergent expansions met with in actual life are not only resurgent but
also summable by ££0b (one critical time) or by S^^p^ ...^$
(r critical times;
r ^ 2). The former case (r = 1) is by far the more common. Criticity r ^ 2 occurs
only in connection with objects of a certain complexity. Thus it never arises with
vector fields (resp. diffeomorphisms) in less than 3 (resp. 2) dimensions.
We shall describe that phenomenon (namely, r > 2) in the case of a manylevelled but formally separable differential system, because it illustrates all the
relevant analysis while keeping formal complications down to a minimum. So,
consider a local analytic system (3.1) that is formally conjugate to the normal
system (3.2) under transformation (3.3).
(3.1)
— tl+PiXj + ÀjXi = fc/(t,xi,...,xv) E (C{t,x\,...,xv)
(i = l,...,v)
Pi
(3.2)
-tHpiyi
Pi
(3.3)
Xi = hi(t, yi,..., yv) E (C[[t, x i , . . . , xv]]
+ fai = 0
(i = 1,..., v ; k E (C* ; p, E N*)
v
/
(i = 1,..., v) .
Let q\ < q2 < ... < qr be the distinct values taken by the levels p\ and assume
for simplicity's sake that the various X\ attached to any given level display neither
resonance nor quasiresonance (i.e. the combinations £ n\X\ neither vanish nor do
1254
Jean P. Ecalle
they get abnormally close to 0). Under those mild genericity assumptions, the
formal integral x(t,u) of system (3.1):
x(*,tO = Zwn£n(0<Pw(0
(3.4)
(n E W ; un = Y[uini ; En(t) = e x p ^ - r * ; cpn E (€[[*]]) v )
obtained by plugging into (3.3) the elementary solution yt = u\ exp(A,-r~p') of
(3.2), can be shown to be convergent in u and divergent in t, but resurgent
and accelero-summable with critical times z\ = t~qi (i = 1,...,r). This compact
statement translates into the following. Let 9 be a multipolarization, i.e. a choice
of angles 9{,...,9r satisfying the self-compatibility condition:
<-[
2 \qt
(3.5)
qi+ij
(l<i<r~l;
9tETR).
Further, let Qt be the set of all œ E (Ç whose projection co on (C is of the
form Yunjh w ^ h Pj = & a n d nj e ^ ( o r nj = ~~1 f° r o n e J a t most). Then
each component q> (t) = cp\(z\) of x(t,u) has a Borei transform $i((i) with only
isolated singularities and a growth rate not exceeding Gxp(cst.\Ci\q2^q2~~qi). Thus
it has accelerable growth for the acceleration z\ —> Z2 taken along any axis
argCi = 91 that avoids singularities. The corresponding accelerate fatti II 6) can
be analytically continued within a sector S2 containing (at least) all directions 02
linked to 9\ by (3.5). In that sector it possesses only isolated singularities and
has accelerable growth for the acceleration Z2 —> Z3. Thus we get a succession of
accelerates fatti || 9), the last of which (for i = r) has exponential growth and can
be laplaced along any semi-axis 9r compatible with 0r_i, yielding the sought-after
sum cpr(zr\\ 9) = cp(t\\ 9).
Moreover, each i-th Borei transform x(C*,w||0) satisfies the so-called Bridge
Equation, which reads :
(3.6)
(3.6bis)
Aœ xttu 111|fl)= A w , q i t 6 . Stttu 111| 0)
AœiquQ = u<4
X KilqhQ(u).uj^(.Pi^qt
(3.6ter)
{
( X = e-v'Aa, icoEQiH &)
+£
J
Pj«[i
<
q i
M ^ - \
J
)
Ai|l f t i ö €C[[ttfcwithpjk<ft]]
and which describes in compact form the resurgence properties of all the fa-tti || 0).
The Bridge Equation holds, in some form or other, for all local objects. It says in
effect that alien derivations Aœ act on formal integrals like ordinary differential
operators Am, while at the same time enabling one to calculate those Aœ.
Here, the components of the A^^Q relative to each level q\ are formal power
series in all the parameters Uk attached to the lower levels pk < q^ For i — 1,
they are scalar-valued. Lastly, and crucially, these differential operators, taken
together, constitute a complete set of analytic invariants of the system (3.1).
The Acceleration Operators and Their Applications
1255
The proof [6, 8] relies heavily on the study of the operators :
(3.7)
An = e^m{z).d-l.e+m(z)
= (w'(z) + a ) " 1
(d = d/dz ; z = 1/t; m(z) = CûQ + Cû\Z -\
+ coqzq J
and their equivalents in the various £,• planes, for all three cases: precriticai
(qi < q), critical (qj = q) and postcritical (q\ > q).
§4. Acceleration and Quasianalyticity. Cohesive Functions
Transfinite Denjoy Classes of Quasianalytical Functions. Cohesive Functions
Let if be a C00 automorphism of ] . . . , +oo] with if(x) < x. An iterator if* of
^ is any C00 automorphism of ] . . . , +oo] such that i f * o if = — 1 -f if *. For any
transfinite ordinal a = œr.nr H
+ co.ni + no < cDœ (n\ E N) we put:
(4.1) L a = (L)°"° o (L*)oin o (L**)°"2 o • • • o (L*-*)°"' = a-th iterate of L = log .
The function L a is not uniquely determined (unless a < œ) but the algebra a D of
all C00 functions on / = [xi,X2J with derivatives bounded by:
(4.2)
|«p(")(x)| < c". (jT^y
(c = c(q>) = est ; Vx € /)
depends only on a. We call it the Denjoy class of order a. Its elements cp are
quasianalytic, i.e. they vanish if all their derivatives at a given point vanish. The
classes a D increase with a and their union for all a < co03 is the algebra COHES
of cohesive functions.
Weak Accelerates are Cohesive and Cohesive Functions are Weak Accelerates
It can be shown [4, 5] that any cohesive function fatti) on an interval [0,a] is a
weak accelerate (i.e. the result of a weak acceleration z\ —• z2 with logZ2/logzi —>
1) and, under very mild assumptions on the acceleration, the converse holds: each
weak accelerate ^2^2) is cohesive on some interval ]0,cr], i.e. on each [e,cr] with
£>0.
The cohesiveness of weak accelerates (just as the analyticity of moderate or
strong accelerates) is truly providential, for each acceleration z\ —• Z2 is actually a
two-stepped process -.first, we calculate #2 (£2) as a germ (for small (2) by means of
integral (2.6) ; and then we must take its continuation (analytic or quasianalytic)
to get fa as a global, multivalued function on R + . Of course, when singular
points oj{ stand in the way of quasianalytic continuation, their "circumvention"
(right or left) calls for a special construction [4, 5] since fatt2) is not defined
outside R + .
The direct statement (cohesive functions are weak accelerates) is also highly
meaningful, as it leads to a new and fairly elementary procedure for quasianalytic
continuation [4-6].
1256
Jean P. Ecalle
§5. The Finiteness of Limit-Cycles. Analysable Functions
I am indebted to J. Martinet, R. Moussu and J.-P. Ramis for drawing my attention
to a conjecture by Dulac (long known as Dulac's theorem, but unproven by him)
to the effect that the limit-cycles of a vector field on R 2 with polynomial or
real coefficients, cannot possibly accumulate anywhere. Since accumulation could
take place only close to a polycycle (or a point) and since polycyles, under
repeated blowing-ups, can be brought down to a simple form, the problem may
be rephrased as follows. Let ^ be a closed curve on R 2 consisting of r analytic
arcs % intersecting at points Si = %n %+\. Let I b e a real analytic vector field
defined on a neighbourhood of #, with the % as integral curves and with a nonvanishing linear part at each summit St. Next, draw an analytic curve Fi across
each % and endow it with an analytic abscissa x/ = 1/ZJ (x/ ~ + 0 ; z{- ~ +co)
positive towards the "interior" of c€. The integral curve crossing T\ at the point
with inverse abscissa z* crosses I V i at the point zi+\. The germ Gi : Zi i-> Zj+i is
the local map of summit Si and the germ F = Gr o • • • o G\ is the return map of
X. Limit-cycles close to # clearly correspond to large fixed points of F. Thus it
is all a matter of establishing the trichotomy:
(5.1)
F(z)
ES z
or F(z) > z
or F(z) <z
(z > 1).
Due to reduction, the field X has, at each summit Si, either two non-zero
eigenvalues of negative ratio — X\ £ Q (type 1) or —Xi E Q (type II) or only one
non-zero eigenvalue (type III). For all three types, the local map Gi has a formal
counterpart Gi which is an asymptotic or transasymptotic series of the form:
KioPXloHi
with Pxi{z) = zki
(5.2)
Gt =
(5.3)
Gi
(5.4)
Gi =
KioEoU*
with E(z) = expz
(type III+)
(5.5)
Gi =
*VtoLoH{
with L(z) s l o g z
(type HI")
= t i0 u;
(type I)
(type II)
The Acceleration Operators and Their Applications
1257
with the Hj and K/ denoting ordinary real power series of the form az.{l +
£<7„.z~~"} (a > 0, an E R) and with U* and *[// standing for the formal iterators
(direct and inverse) of the "unitary" maps Uj which describe the holonomy of X
at St\
(5.6)
VoU(z)=z;
Ü*o*Ü(z)=z;
Ü* o U(z) = 2ni + U*(z).
The Gi are usually divergent (except for type I and diophantine Xi) but can always
be resummed by ££& with respect to a single critical time z\ — hj(z) of the form:
( h\(z) = logz - clog log z ;
I k(z) ~ log Gi(z) (type III+)
c large (type I or II)
/7,(z) - log z (type UT).
As for the return map F = Gr o • • • o Gy, its formal counterpart F = Gr o • • • oGi
is a transseries with a unique "pulled-down" expansion,
(5.8)
F(z) = z + £ M „ ( z )
(0 < zi < no < û)œ ; fl„ G R)
with finite or transfinite ordinals n and decreasing transmonomials A1± that are
irreducible concatenations of real coefficients and symbols +, x, log, exp.
Like its factors G„ the transseries F is usually divergent, but always accelerosummable with at most r critical times z,- associated with its irregular summits
Sj (actually, the intrinsic notion is that of critical class {z,} regrouping all times
equivalent to zi). For each z,-, only those transseries cp(z) = cpì(zì) that are carried
by F(z) and formally subexponential in z/ possess a Borei transform fatti) in the
d plane. The rest must provisionnally retain their status as symbols and bide their
"times" to be actualized as true functions ! Of course, in order to preserve realness,
it is the median functions med #>/(£/) which are being accelerated or Laplaced.
Accelero-summability is proven by induction on q for Fq = Gq o • • • o G\.
Crucial to the argument is the comparability of non-equivalent critical times z\
and Zj (one is either faster or slower than the other). Since the function Fß taking
Zi to zj and its transseries Fß have the same factorization structure as F and F,
but with a lesser number of factors, one and the same induction takes care of the
accelero-summability of F and Fß. Despite descriptive and notational hurdles,
the proof [4, 5] is amazingly simple *. It reduces to showing, by induction on q,
that accelero-summing the factors Gq and Fq-\ to Gq and Fq-\ and composing
them to get Fq = Gq o Fq-\, yields the same result as composing the transseries
Gq and Fq-\ to get Fq and then accelero-summing it to get Fq. This, in turn,
follows from a permutability of type / ]T = ]T / , due to absolute summability,
with f representing an acceleration or Laplace integral and £ standing for the
infinite sum which translates, in each model Ç/, the operation of composition o.
* Another proof, apparently quite different and non-constructive in nature, has been announced by
Y.S. Ilyashenko.
1258
Jean P. Ecalle
We end up with a formal trichotomy:
J F(z) = z
or
F(z) = z + aoA0(z) + o (AQ(Z))
\ F(z) = z - a0A0(z) + o(A0(z))
(a0
or
>,A0>0)
which, after accelero-summation, translates into the wanted trichotomy (5.1).
Analysable Functions
The return map F is only a special instance of analysable functions. Those are realanalytic germs on ] . . . , +00] that can be represented by an accelero-summable
transseries F with critical times zt which are themselves linked by analysable
functions Fß, e t c . , with a finite critical tree ziu_iir. Unlike analytic functions,
the class of analysable functions enjoys extreme stability (under all common
operations) while retaining the two essential properties of real-analytic functions
(i) being locally comparable (ii) being totally reducible to a formal object, viz. an
infinite set of coefficients. Analysable functions are of very frequent occurence. See
[6, 8]. There, we also introduce an even more comprehensive notion of analysable
function, which subsumes both complex-analytic and cohesive functions and
seems to stretch the Analytic Principle to its farthest possible limit.
Bibliography
[1-3] J. Ecalle: Les fonctions résurgentes (vols. 1, 2, 3). Pub. Math. Orsay (1981, 1981,
1985)
[4] J. Ecalle: Finitude des cycles-limites et accéléro-sommation de l'application de retour.
Dans les Actes du Colloque sur les cycles-limites, Luminy 1989. Lecture Notes in
Mathematics, vol. 1455. Springer, Berlin Heidelberg New York 1990
[5] J. Ecalle: La conjecture de Dulac: une preuve constructive, (à paraître à Travaux
en Cours, Décembre 1990)
[6] J. Ecalle: Calcul accélératoire et applications, (à paraître à Travaux en Cours,
Décembre 1990)
[7] J. Ecalle: Calcul compensatoire et linéarisation quasianalytique des champs de
vecteurs locaux, (à paraître à Travaux en Cours, 1991)
[8] J. Ecalle: Fonctions résurgentes, calcul étranger, calcul accélératoire et applications.
(cours général - en préparation)
[9] J. Ecalle, J. Martinet, R. Moussu, J.P. Ramis: Non-accumulation des cycles-limites.
C. R. Acad. Sci., série I 304, n° 14 (1298), (I) pp. 375-378, (II) pp. 431-434