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Abstract. We consider the following Hamiltonian equation on a special manifold of rational functions,where Π denotes the Szegő projector on the Hardy space of the circle S 1 . The equation with α = 0 was first introduced by Gérard and Grellier in [6] as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. For α < 0, any compact subset of initial data leads to a relatively compact subset of trajectories. For α > 0, there exist trajectories on which high Sobolev norms exponentially grow in time.

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