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Resurgence of inner solutions for perturbations of the McMillan map

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  • A sequence of "inner equations" attached to certain perturbations of the McMillan map was considered in [5], their solutions were used in that article to measure an exponentially small separatrix splitting. We prove here all the results relative to these equations which are necessary to complete the proof of the main result of [5]. The present work relies on ideas from resurgence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Écalle's alien derivations to measure the discrepancy between different Borel-Laplace sums.
    Mathematics Subject Classification: 37J10, 34C37, 34E10, 34M37.


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  • [1]

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    J. Écalle, "Les Fonctions Résurgentes. Tome I," (French) [Resurgent functions. Vol. I] Les algébres de fonctions résurgentes. [The algebras of resurgent functions] With an English foreword. Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 5, Université de Paris-Sud, Département de Mathématique, Orsay, 1981, 247 pp.


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    P. Martín, T. M. Seara and D. Sauzin, Exponentially small splitting of separatrices in perturbations of the McMillan map, preprint, 2009.


    C. Olivé, D. Sauzin and T. Seara, Resurgence in a Hamilton-Jacobi equation, Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), Ann. Inst. Fourier, Grenoble, 53 (2003), 1185-1235.


    D. Sauzin, Resurgent functions and splitting problems, RIMS Kokyuroku, 1493 (2005), 48-117.

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