  We consider a rational map f:S->S of a complex projective surface together
with an invariant meromorphic two form. Under a mild topological assumption on
the map, we show that the zeroes of the invariant form can be eliminated by
birational change of coordinate. In this context, when the form has no zeroes,
we investigate the notion of algebraic stability for f. We show in particular
that algebraic stability is equivalent to a more tractable condition involving
the behavior of f on the poles of the form. Finally, we illustrate our results
in the particular case where S is the projective plane and the invariant form
is dx dy / xy, showing that our criterion for stability translates to whether
or not the rotation number for a certain circle homeomorphism is rational.
