Contemporary Mathematics Two Examples of Resurgence C. Oliv´ e, D. Sauzin, and T. M. Seara Abstract. This paper gives an account of two articles which illustrate the use of Resurgence theory for estimating the difference between two complex invariant manifolds associated with the area-preserving H´ enon map for the first one, and with a Hamiltonian system which stems from the rapidly forced pendulum for the second one. 1. Introduction 1.1. Motivation. It is usual that exponentially small phenomena appear in the study of near-integrable Hamiltonian systems with a weakly hyperbolic singular point. One of our goals is to estimate asymptotically the splitting of the separatrices associated with such a point, which is exponentially small with respect to the perturbation parameter, by using Resurgence theory. A common approach in separatrix splitting problems is to look for good aprox- imations of the stable and unstable invariant manifolds. But in the cases we are interested in, which are singular in the sense that the hyperbolicity disappears when the perturbation parameter goes to zero (the fixed point is said to be weakly hyperbolic), these manifolds have different approximations in different regions of the complex plane. Using “matching techniques”, one can obtain a so-called inner equation, which retains the dominant part of the invariant manifolds and of their splitting. The papers [GS01] and [OSS03] are devoted to the resurgent study of the inner equations (1.6) and (1.7) below, obtained in two different settings: an area- preserving map and a Hamiltonian system. Here, we shall try to give an account of these two papers and to explain in a systematic way how Resurgence theory can be a used to estimate the exponentially small difference between special solutions of these inner equations. The passage from results for the inner equations (1.6) and (1.7) to the initial problems of singular separatrix splitting which have motivated them 2000 Mathematics Subject Classification. 34M25, 34M30, 34M37, 34M40, 37G20. Key words and phrases. Resurgence, Divergent series, Asymptotic calculus, Splitting of sep- aratrices, Hyperbolic singular point, Homoclinic trajectories. This work has been partially supported by the Catalan grants DRAC 2002-PDI and 2001SGR- 70, the Spanish grants DPI2002-00706 and DGICYT n ◦ BFM2000-0805-C02, the INTAS grant 2002-221 and the Franco-Spanish grant Picasso n ◦ 05241UA. c 2003 American Mathematical Society 1