On the primitivity of birational transformations of irreducible symplectic manifolds

Let $f\text{\hspace{0.17em}⁣}:X⇢X$ be a bimeromorphic transformation of a complex irreducible symplectic manifold $X$. Some important dynamical properties of $f$ are encoded by the induced linear automorphism $f^{\ast }$ of $H^2(X,\mathbb{Z})$. Our main result is that a bimeromorphic transformation such that $f^{\ast }$ has at least one eigenvalue with modulus $>1$ doesn’t admit any invariant fibration (in particular its generic orbit is Zariski-dense).