  Let $f\colon X\dashrightarrow X$ be a bimeromorphic transformation of a
complex irreducible symplectic manifold $X$. Some important dynamical
properties of $f$ are encoded by the induced linear automorphism $f^*$ of
$H^2(X,\mathbb Z)$. Our main result is that a bimeromorphic transformation such
that $f^*$ has at least one eigenvalue with modulus $>1$ doesn't admit any
invariant fibration (in particular its generic orbit is Zariski-dense).
