  Near an arbitrary finite gap potential we construct real analytic, canonical
coordinates for the Benjamin-Ono equation on the torus having the following two
main properties: (1) up to a remainder term, which is smoothing to any given
order, the coordinate transformation is a pseudo-differential operator of order
0 with principal part given by a modified Fourier transform (modification by a
phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in
normal form up to order three and the corresponding Hamiltonian vector field
admits an expansion in terms of para-differential operators. Such coordinates
are a key ingredient for studying the stability of finite gap solutions of the
Benjamin-Ono equation under small, quasi-linear perturbations.
