Three results on mixing shapes

Let a be a Zd-action (d ³ 2) by automorphisms of a compact metric abelian group. For any non-linear shape I Ì Zd, there is an a with the property that I is a minimal mixing shape for a. The only implications of the form ``I is a mixing shape for a Þ J is a mixing shape for a'' are trivial ones for which I contains a translate of J. If all shapes are mixing for a, then a is mixing of all orders. In contrast to the algebraic case, if b is a Zd-action by measure-preserving transformations, then all shapes mixing for b does not preclude rigidity. Finally, we show that mixing of all orders in cones - a property that coincides with mixing of all orders for Z-actions - holds for algebraic mixing Z2-actions.