Abstract. 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 resur- gence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Ecalle's alien derivations to measure the discrepancy between di erent Borel-Laplace sums.

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