New York Journal of Mathematics
Volume 3A (1997-1998) 1-10


T. Ward

Three Results on Mixing Shapes

Published: November 24, 1997
Keywords: Mixing, mixing shapes, algebraic dynamical systems
Subject: 28D15, 22D40

Let α be a Zd-action (d≧ 2) by automorphisms of a compact metric abelian group. For any non-linear shape I⊂Zd, there is an α with the property that I is a minimal mixing shape for α. The only implications of the form "I is a mixing shape for α ⇒ J is a mixing shape for α'' are trivial ones for which I contains a translate of J.

If all shapes are mixing for α, then α is mixing of all orders. In contrast to the algebraic case, if β is a Zd-action by measure-preserving transformations, then all shapes mixing for β 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.


The author gratefully acknowledges support from NSF grant DMS-94-01093.

Author information

School of Mathematics, University of East Anglia, Norwich NR4 7TJ, U.K.