Let α be a Zd-action(d≧ 2) byautomorphisms of a compact metric abelian group.For any non-linear shape I⊂Zd, there is anα with the property that I isa minimal mixing shape for α. The onlyimplications of the form "I is a mixingshape for α ⇒J is a mixing shape for α'' aretrivial ones for which I containsa translate of J.If all shapes are mixing for α, thenα is mixing of all orders. In contrast to thealgebraic case, if β isa Zd-action by measure-preserving transformations,then all shapes mixing for β does not precluderigidity.Finally, we show that mixing of all orders incones -- a property that coincides with mixing of all ordersfor Z-actions -- holds for algebraic mixing Z2-actions.
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