March  2011, 31(1): 165-207. doi: 10.3934/dcds.2011.31.165

Resurgence of inner solutions for perturbations of the McMillan map

1. 

Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Ed-C3, Jordi Girona, 1-3, 08034 Barcelona

2. 

IMCCE, Observatoire de Paris, 77 Avenue Denfert-Rochereau, 75014 Paris, France

3. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain

Received  January 2010 Revised  January 2011 Published  June 2011

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.
Citation: Pau Martín, David Sauzin, Tere M. Seara. Resurgence of inner solutions for perturbations of the McMillan map. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 165-207. doi: 10.3934/dcds.2011.31.165
References:
[1]

B. Candelpergher, J.-C. Nosmas and F. Pham, "Approche de la Résurgence,", (French) [Approach to resurgence] Actualités Mathématiques, (1993). Google Scholar

[2]

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], (1981). Google Scholar

[3]

V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map,, Ann. Inst. Fourier, 51 (2001), 513. Google Scholar

[4]

B. Malgrange, Resommation des séries divergentes,, (French) [Summation of divergent series], 13 (1995), 163. Google Scholar

[5]

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

[6]

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, 53 (2003), 1185. Google Scholar

[7]

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

show all references

References:
[1]

B. Candelpergher, J.-C. Nosmas and F. Pham, "Approche de la Résurgence,", (French) [Approach to resurgence] Actualités Mathématiques, (1993). Google Scholar

[2]

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], (1981). Google Scholar

[3]

V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map,, Ann. Inst. Fourier, 51 (2001), 513. Google Scholar

[4]

B. Malgrange, Resommation des séries divergentes,, (French) [Summation of divergent series], 13 (1995), 163. Google Scholar

[5]

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

[6]

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, 53 (2003), 1185. Google Scholar

[7]

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

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