Abstract:The isometries of a hyperbolic space are classified into three classes - elliptic, parabolic, and loxodromic; this classification plays the major role in homogeneous dynamics of hyperbolic manifolds. Since the work of Serge Cantat in early 2000-ies it is known that a similar classification exists for complex surfaces, that is, compact complex manifolds of dimension 2. These results were recently generalized to holomorphically symplectic manifolds of arbitrary dimension. I would explain the ergodic properties of the parabolic automorphisms, and prove the ergodicity of the automorphism group action for an appropriate deformation of any compact holomorphically symplectic manifold. This is a joint work with Ekaterina Amerik.