  We study the Benjamin-Ono equation, posed on the torus. We prove that an
infinite sequence of weighted gaussian measures, constructed in our previous
work, are invariant by the flow of the equation. These measures are supported
by Sobolev spaces of increasing regularities. As a by product we deduce
informations on the long time behaviour of regular solutions. To our knowledge
this is the first result which gives an evidence about recurrence properties of
the Benjamin-Ono equation flow.
