Proof of Some Auxiliary Results
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The goal of this Chapter is to prove that the non-resonance condition required for the normal form construction is fulfilled for all the values of the gravity-capillary parameters except a zero measure set. A key property of the proof is that the function which describes the linear dispersive relation of the gravity-capillary water waves equations is sub-analytic. In addition in the last section we prove an expansion of the Dirichlet–Neumann operator that has been used in the preceding chapters.