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