Main publications :

  1. Théorie des invariants holomorphes.
    Thèse d'Etat, Orsay, March 1974.

  2. Théorie iterative: introduction à la théorie des invariants holomorphes.
    J. Math. pures et appl., 54, 1975, p. 183-258.

  3. The main results about holomorphic invariants were announced in:
    C.R.A.S., t. 272, A 1971, p 225-228
    C.R.A.S., t. 272, A 1971, p 308-311
    C.R.A.S., t. 272, A 1971, p 372-375
    C.R.A.S., t. 276, A 1973, p 179-182
    C.R.A.S., t. 276, A 1973, p 261-264
    C.R.A.S., t. 276, A 1973, p 375-378
    C.R.A.S., t. 276, A 1973, p 471-474
    Vestnik L.G.U., 7 1973, p 69-71
    Vestnik L.G.U., 13 1973, p 166-169

  4. Some new criteria for the Riemann hypothesis:
    C.R.A.S., t. 277, A 1973, p 23-25

  5. The seminal ideas behind resurgence theory were set forth in:
    C.R.A.S., t. 282, A 1976, p 203-206
    C.R.A.S., t. 282, A 1976, p 861-864
    C.R.A.S., t. 282, A 1976, p976-982
    Un analogue distant des fonctions automorphes: les fonctions résurgentes,
    Séminaire Choquet,1977-78.

  6. Les fonctions résurgentes, Vol. 1:
    Algèbres de fonctions résurgentes.
    Publ. Math. Orsay 81.05 (1981), # 248 pp.

  7. Les fonctions résurgentes, Vol. 2:
    Les fonctions résurgentes appliquées à l'itération.
    Publ. Math. Orsay 81.06 (1981), # 283 pp.

  8. Les fonctions résurgentes, Vol. 3:
    L'équation du pont et la classification analytique des objets locaux.
    Publ. Math. Orsay 85.05 (1985), # 585 pp.

  9. Iteration and analytic classification of local diffeomorphisms of C^n.
    in Iteration theory and its functional equations. Lecture Notes 1163, Springer, 1985, p 41-48.

  10. Cinq applications des fonctions résurgentes.
    Prepub. Math. d'Orsay, 1984, 84T62, # 110 pp.
    One of these five articles is available here: Singularités irrégulieres et résurgence multiple.

  11. Classification analytique des champs hamiltoniens. Potentiels de résurgence et hamiltoniens étrangers.
    Proc. of the Dijon 1985 Conference on Differential Equations in the Complex Field, Asterisque.

  12. L'accélération des fonctions résurgentes,
    Unpublished typescript, Orsay 1985, # 54 pp.
    A scan of the original typescript (-unpublished because unsubmitted, and unsubmitted
    because widely circulated and then half-forgotten-) can be accessed here.
    We didn't attempt to bring the text in line with our later notations and nomenclature on accelaration theory.

  13. (with J. Martinet, R. Moussu, J.-P. Ramis)
    Non-accumulation des cycles-limites.
    C.R.A.S., t. 304, série I,no 14, 1987, p 375-378
    C.R.A.S., t. 304, série I,no 14, 1987, p 431-434

  14. Finitude des cycles limite et accéléro-sommation de l'application de retour.
    Bifurcations of Planar Vector Fields, Proceedings, Luminy 1989, Lecture Notes 1455, Springer, p 74-159.

  15. The acceleration operators and their applications. Proc. Internat. Cong. Math., Kyoto, 1990, vol.2, Springer, Tokyo, 1991, p1249-1258.

  16. The Bridge Equation and its Applications to Local Geometry.
    in Proc. of the Intern. Confer. on Dynamical Systems and Related Topics, K. Shiraiwa ed.,
    Advanced Series in Dynamical Systems, Vol.9, 1991, p 100-122.

  17. Singularités non abordables par la géométrie.
    Ann. Inst. Fourier,Grenoble, 42, 1992, p 73-164.

  18. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac.
    (book), Actual. Math., Hermann, Paris, 1992, # 337 pp.
    The contents of the book, along with an update, are described in Dulac: constructive proof .
    The book itself can be downloaded from this portal .

  19. Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac's Conjecture .
    Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk ed., p.75-184, 1993, Kluwer

  20. Discretised Resurgence.
    Annexe C, p 161-218,
    Included in F. Menous' PhD Thesis:
    Les bonnes moyennes uniformisantes et leurs applications à la resommation réelle
    PhD Thesis, 26.5.1999, Laboratoir Emile Picard, Université P. Sabatier, Toulouse, France.

  21. Cohesive functions and weak accelerations.
    Journal d'Analyse Mathématique, Vol. 60 (1993), p 71-97.

  22. (with D. Schlomiuk)
    The nilpotent part and distinguished form of resonant vector fields or diffeomorphisms.
    Annales de Institut Fourier, 1993, p 1407-1483.

  23. Weighted products and parametric resurgence
    in Méthodes résurgentes, Travaux en Cours, 47, L. Boutet de Monvel ed., 1994, p 7-49.

  24. Compensation of small denominators and ramified linearisation of local objects.
    in Complex Analytic Methods in Dynamical Systems, IMPA, Asterisque, 1994, p 135-199.

  25. (with F. Menous)
    Well-behaved convolution averages and the non-accumulation theorem for limit-cycles.
    in: The Stokes Phenomenon and Hilbert's 16th Problem, eds B.L.J. Braaksma, G.K. Immink, M. van der Put, p 71-101, World Scient. Publ.

  26. (with B. Vallet)
    Passive/active resonance. Non-linear resurgence and isoresurgent deformations.
    in: The Stokes Phenomenon and Hilbert's 16th Problem, eds B.L.J. Braaksma, G.K. Immink, M. van der Put, p ...-..., World Scient. Publ.

  27. Prenormalisazation, correction, and linearization of resonant vector fields or diffeomorphisms.
    Prepub. Orsay 95-32 (1995), # 90 pp.

  28. (with B. Vallet)
    Correction, and linearization of resonant vector fields or diffeomorphisms.
    Math. Zeitschrift 229, p 249-318 (1998)

  29. A Tale of Three Structures: the Arithmetics of Multizetas, the Analysis of Singularities, the Lie Algebra ARI.
    Diff. Eq. and the Stokes Phenomenon, BLJ Braaksma, GK Immink, M van der Put, J Top Eds 2002, World Scient. Publ.,p 89-146.

  30. Recent Advances in the Analysis of Divergence and Singularities.
    Proceedings of the July 2002 Montreal Seminar on Bifurcations, Normal forms and Finiteness Problems in Differential Equations,
    C. Rousseau,Yu. Ilyashenko Publ., 2004 Kluwer Acad. Publ., p 87-187

  31. ARI/GARI. la dimorphie et l'arithmétique des multizetas: un premier bilan.
    Journal de Théorie des Nombres de Bordeaux, 15 (2003), p 411-478.

  32. Twisted Resurgence Monomials and canonical-spherical synthesis of Local Objects.
    Proc. of the June 2002 Edinburgh conference on Asymptotics and Analysable Functions, O. Costin ed, World Scient. Publ., # 105 pp

  33. (with B. Vallet)
    Intertwined mappings.
    prepub. Orsay 2003-73, # 92 pp.
    Ann. Fac. Sc. Toulouse,  Tome XIII, no 3 (2004),  p. 291-376.

  34. (with B. Vallet)
    The arborification-coarborification transform: analytic, combinatorial. and algebraic aspects
    prepub. Orsay 2004-30, # 80 pp
    Ann. Fac. Sc. Toulouse, Ser.6, 13, no 4 (2004), p. 575-657.

  35. Multizetas, perinomal numbers, arithmetical dimorphy, and ARI/GARI.
    prepub. Orsay 2004-37, # 26 pp (a survey)
    Ann. Fac. Sc. Toulouse, Tome XIII, no. 4 (2004), p. 683-708.

  36. (with Sh. Sharma)
    Power series with sum-product Taylor coefficients and their resurgence algebra.
    prepub. Orsay 2010-04, # 146 pp
    Ann.Scuo.Norm.Pisa , 2011, Vol.1, Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation; ed. O.Costin, F.Fauvet, F.Menous, D.Sauzin.

  37. The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles.
    prepub. Orsay 2010-05, # 163 pp.
    Ann.Scuo.Norm.Pisa , 2011, Vol.2, Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation; ed. O.Costin, F.Fauvet, F.Menous, D.Sauzin.
    An enlarged version, March 2011, # 185 pp, can be found here and the penultimate galley proof is here.
    Tables for lomi/lami/lumi and their singulands are available here . They illustrate the dramatic "compression of information" made possible by the flexion potentials.

  38. (X. Buff, J. Ecalle, A. Epstein)
    Limits of Degenerate Parabolic Quadratic Rational Maps.
    Final version 2012-08, # 46 pp
    To appear in Geometric and Functional Analysis.

  39. (with O. Bouillot)
    Invariants of identity-tangent diffeomorphisms: explicit formulae and effective computation.
    Orsay 2012-07, # 42 pp

  40. Eupolars and their bialternality grid.  Appeared in A.M.V., 2015.
    A preliminary version (# 85 pp) was posted in February 2014 and slightly expanded in March 2014.
    The complete version (# 114 pp) was posted in April 2014.
    Numerous illustrative Tables can be accessed here or here.

  41. Singulators vs Bisingulators. (# 21 pp, 7 June 2014)
    This is a preview of the last chapter of a forthcoming monograph:
    Finitary Flexion Algebras. (to be posted in July 2014)

  42. Singularly Perturbed Systems, Coequational Resurgence, and Flexion Operations. (# 21 pp, 7 June 2014)
    This is a preview of the first part of a larger monograph:
    The Three Bridge Equations (forthcoming).
  43. Singular ODEs and Resurgence. (June 2015)
    (in Russian)

  44. The Natural Growth Scale. (January 2016,145 pp)
    Updated in March 2018.
    45. From the abstract: The paper starts with the group of all germs of analytic self-mappings of R+ at infinity, and concerns itself with its successive closures under (i) fractional iteration (ii) conjugation (iii) the solving of general composition equations.
    Rather than attempting a systematic treatment, we focus on the typical difficulties attendant upon these extensions. On the formal side, power series make way first for transseries, then for ultraseries, involving finite resp. transfinite iterates of the exponential mapping. On the analysis side, the first casualty are convergence and analyticity: from the start, we have to face generic resurgence (multicritical but of a weakly polarising type) and, further down the road, generic cohesiveness (a natural and very inclusive extension of Denjoy quasi-analyticity).
    Fortunately, none of these complications destroys the bi-constructive correspondence between the formal objects (series, transseries, ultraseries) and the geometric germs. We describe, and illustrate on numerous examples, the apparatus required for upholding this correspondence: mainly accelero-summation, which uses convolution-respecting integral transforms to ascend from one Borel plane to the next, and the so-called display, a semi-algebric construct that supplements the genuine variable with a host of pseudo-variables and encapsulates in highly convenient form all the information about the resurgence pattern and Stokes constants of a given germ.
    We also devote three sections to the (non-linear) iso-differential operators which, on top of their surprising algebraic properties, are uniquely adapted to germ composition, the analysis of deep convexity, and the description of the universal asymptotics of very slow- or fast-growing germs.
    Lastly, we reflect on the seemingly unsurmountable indeterminacy inherent in the choice of transfinite exponential iterates, and on what that implies for the natural growth scale (-- by which we mean, roughly speaking, the ultimate extension of our groups of non-oscillating germs --): far from being the quintessential continuum that one would expect, the natural growth scale -- on the formal as on the analysis side, in the large as well as locally -- displays a granular, almost fractal-like structure.

  45. Combinatorial tidbits from resurgence theory and mould calculus.
    46. (June 2016, 36 pp)
    Based on a series of talks delivered at a meeting of combinatoricists.
    1. From moulds to bimoulds, and back.
    2. Mould extensions of classical functions.
    3. Natural projectors.
    4. Minimal convolution domains.
    5. Iso-differential operators and their natural basis.

  46. Invariants of identity-tangent diffeomorphisms expanded as series of multitangents and multizetas. (November 2016, 120 pp)

  47. Taming the coloured multizetas. (forthcoming).
    Here are the slides of a talk given on 27/06/2017 at CIRM (Marseilles-Luminy).
    The comments on the black-numbered, star-marked slides were added after the event.
    A full exposition can be found in the chapters 1 and 4 of this long paper.
  48. The scrambling operators applied to multizeta algebra and singular perturbation analysis. (October 2018, 156 pp).
    The scrambling operators applied to multizeta algebra and singular perturbation analysis. (expanded version, March 2019, 203 pp).
    (i) Gives a synopsis of the three scrambling operators (scram, viscram, discram) and applies them:
    (ii) to the theory of singular pertubations, to derive the Second and Third Bridge Equations in the most general situation,
    (iii) to multizeta algebra, to show (among other results) how the complete set of coloured multizetas can be recovered from any of three small subsets of boundary data (known as "satellites").
    For a survey of the chapters on bicoloured multizetas (ch.3-4 of the 2018 version; ch. 5-7 of the 2019 version), see these slides .
    For a survey of the chapters on singular perturbations (ch. 2 of the 2018 version; ch. 3-4 of the 2019 version) see these slides .
    A compact exposition of the question is also available in sections 2-8 of here , while sections 9-11 contain new material.

  49. Resurgent analysis of singularly perturbed systems: exit Stokes, enter Tes. (25 pp).
    To appear in the Proceedings of the Nov. 2018 Moscow Conference in Memory of Prof. B. Yu. Sternin.
    For the related slides, go there .


  50. Flexion algebra meets tree algebra: a tale of asymmetric cross-ferlilisation. (January 2023, 117 pp).
    From the abstract: As mathematical objects, finite trees would seem to be nearly as basic and ubiquitous as the integers, were it not for their apparent 'chemical inertness', by which we mean the paucity of natural operations acting on them. As an attempt to redress this state of affairs, we bring trees into close relation with Flex(E) -- the flexion polyalgebra generated by a flexion unit E -- and upload the rich structure of that polyalgebra onto trees. The rapprochement also benefits Flex(E), leading in particular to a neat filtration by alternality codegree, to exact formulae for the corresponding dimensions, and to gratifyingly explicit expansions for all main elements of Flex(E). We conclude by introducing a notion of pre-associative algebra, parallel to that of pre-Lie algebra and potentially capable of rendering roughly the same services.


  51. Guided tour through resurgence theory. (January 2023, 24 pp).
    A fast-paced introduction to resurgence and a host of related topics (alien calculus; acceleration operators; display; the Bridge equations I,II,III; transseries and analyzable functions; the dichotomies cohesive/loose and autark/non-autark) in chronological perspective and in relation to each other.


  52. Multizeta algebra and rational associators in the light of the mirror transform. (May 2025, 130 pp)
    From the abstract: This long paper (more a progress report than a regular treatise) is about the mirror reflection, a recently discovered, degree reversing transform of great relevance to flexion theory and multizeta algebra. It sharply differs from -- while in some contexts oddly overlapping with -- the familiar length-degree exchange. It suffuses the various ari-algebras and gari-groups with additional structure: stratification, quadratic forms, natural bases etc. In multizeta algebra, it associates mirror images to the generating series zag=gari(zag1,zag2,zag3) and its three factors, most notably zag1, which encodes a rational associator. While information equivalent to their sources, these mirror images are incomparably richer in explicit structure, and for that reason of much help in unravelling the nearly inexhaustible wealth of multizeta properties. We also touch on related topics, like perinomal analysis (the search for meromorphic zag1, zag2 with the study of their multipoles) and the uniqueness of perinomal potentials.

    Some conference related slides

    Singular and singularly perturbed systems and multiple resurgence.
    (Nov. 2018, Moscow. In memory of Prof. B. Yu. Sternin)

    Resurgence's two main types and their signature complications: tessellation, isography, autarchy.
    (June 2019, IHES)

    A leisurely walk through the resurgence landscape.
    (May 2025, Orsay)

    FORTHCOMING:

    A --- One or two books on resummation theory and the underlying structures.

    B --- One or two books on numerical dimorphy, multizeta arithmetic, ARI-GARI and the flexion structure.


    SOME RELATED PUBLICATIONS: