Dynamical systems
On the resurgent approach to Écalle–Voronin's invariants
[Approche résurgente des invariants d'Écalle–Voronin]

Présenté par : Jean-Pierre Ramis

Dudko, Artem1 ; Sauzin, David2
1 Institute for Mathematical Sciences, University of Stony Brook, NY, USA
2 CNRS UMI 3483 – Laboratoire Fibonacci, Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore di Pisa, Italy
Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 265-271.

Un germe parabolique simple admet une paire de coordonnées de Fatou qui ont la même série asymptotique résurgente. Nous montrons comment utiliser les opérateurs étrangers d'Écalle pour étudier les singularités dans le plan de Borel et les relier aux applications de corne, de façon à obtenir chaque invariant d'Écalle–Voronin comme une série numérique géométriquement convergente.

Given a holomorphic germ at the origin of C with a simple parabolic fixed point, the Fatou coordinates have a common asymptotic expansion whose formal Borel transform is resurgent. We show how to use Écalle's alien operators to study the singularities in the Borel plane and relate them to the horn maps, providing each of Écalle–Voronin's invariants in the form of a convergent numerical series. The proofs are original and self-contained, with ordinary Borel summability as the only prerequisite.

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Accepté le :
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DOI : 10.1016/j.crma.2014.11.003
Dudko, Artem 1 ; Sauzin, David 2

1 Institute for Mathematical Sciences, University of Stony Brook, NY, USA
2 CNRS UMI 3483 – Laboratoire Fibonacci, Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore di Pisa, Italy 
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Dudko, Artem; Sauzin, David. On the resurgent approach to Écalle–Voronin's invariants. Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 265-271. doi : 10.1016/j.crma.2014.11.003. http://archive.numdam.org/articles/10.1016/j.crma.2014.11.003/

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[2] Écalle, J. Les fonctions résurgentes, vols. 1 & 2, Publ. Math. d'Orsay, vols. 81-05 & 81-06, 1981

[3] Malgrange, B. Sommation des séries divergentes, Exp. Math., Volume 13 (1995), pp. 163-222

[4] Ramis, J.-P. Séries divergentes et théories asymptotiques, Panoramas et Synthèses, vol. 121, Société mathématique de France, Paris, 1993 Bull. Soc. Math. Fr. (suppl.) 74 p

[5] Sauzin, D. On the stability under convolution of resurgent functions, Funkc. Ekvacioj, Volume 56 (2013) no. 3, pp. 397-413

[6] D. Sauzin, Nonlinear analysis with resurgent functions, Preprint oai:hal.archives-ouvertes.fr:hal-00766749, 35 p., 2014. To appear in Ann. Sci. Éc. Norm. Supér.

[7] D. Sauzin, Introduction to 1-summability and resurgence, Preprint oai:hal.archives-ouvertes.fr:hal-00860032, 2014, 125 p.

[8] Voronin, S.M. Analytic classification of germs of conformal mappings (C,0)→(C,0) with identity linear part, Funct. Anal. Appl., Volume 15 (1981) no. 1, pp. 1-13 (transl. from Funkts. Anal. Prilozh. 15 (1) (1981) 1–17. https://zbmath.org/journals/?q=se:00000424) | DOI

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