Parabolic automorphisms of hyperk{ä}hler manifolds: Orbits and Betti maps

We study parabolic automorphisms of irreducible holomorphically symplectic manifolds with a lagrangian fibration. Such automorphisms are (possibly up to taking a power) fiberwise translations on smooth fibers, and their orbits in a general fiber are dense ([1]). We provide a simple proof that the associated Betti map is of maximal rank, in particular, the set of fibers where the induced translation is of finite order is dense as well. R{É}SUM{É}. Nous {é}tudions les automorphismes paraboliques des vari{é}t{é}s symplectiques holomorphes qui sont irr{é}ductibles et projectives.