In the study of the exponentially small splitting that occurs in certain perturbations of the McMillan map a sequence of "inner equations" has to be considered. An essential step in the measure of the splitting is to know some special solutions of these equations and to be able to give an asymptotic value of their difference.

The present work relies on ideas from resurgence theory: we obtain the desired solutions as Borel-Laplace sums of the formal solutions, studying the analyticity of their Borel transforms. Moreover, using 'Ecalle's alien derivations we are able to measure the discrepancy between different Borel-Laplace sums.

(Joint work with P. Martin and D. Sauzin)