Abstract
Dynamical properties of actions of groups of automorphisms on Heisenberg nilmanifolds H/Γ are studied. It is proved that such a group G has only finite or dense orbits if the induced action on the associated torus has the same property. This gives a partial answer to a question of Margulis. Moreover, the G-invariant (or even stationary) measures on H/Γ are determined.
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