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We consider a perturbation of the Benjamin Ono equation with periodicboundary conditions on a segment. We consider the case where the perturbationis Hamiltonian and the corresponding Hamiltonian vector field is analytic as amap form energy space to itself. Let $\epsilon$ be the size of theperturbation. We prove that for initial data close in energy norm to an $N$-gapstate of the unperturbed equation all the actions of the Benjamin Ono equationremain $\cO(\epsilon^{\frac{1}{2(N+1)}})$ close to their initial value fortimes exponentially long with $\epsilon^{-\frac{1}{2(N+1)}}$.

A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori