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 the 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.
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