Given a holomorphic germ at the origin of 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.
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.
Accepted:
Published online:
Artem Dudko 1; David Sauzin 2
@article{CRMATH_2015__353_3_265_0, author = {Artem Dudko and David Sauzin}, title = {On the resurgent approach to {\'Ecalle{\textendash}Voronin's} invariants}, journal = {Comptes Rendus. Math\'ematique}, pages = {265--271}, publisher = {Elsevier}, volume = {353}, number = {3}, year = {2015}, doi = {10.1016/j.crma.2014.11.003}, language = {en}, }
Artem Dudko; David Sauzin. On the resurgent approach to Écalle–Voronin's invariants. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 265-271. doi : 10.1016/j.crma.2014.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.11.003/
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