## Three results on mixing shapes.(English)Zbl 0909.28015

Let $$(X,{\mathcal B},\mu,\alpha)$$ be an algebraic dynamical system, i.e., $$X$$ is a compact metrizable Abelian group, $${\mathcal B}$$ the Borel sets, $$\mu$$ a normalized Haar measure and $$\alpha$$ a $$\mathbb Z^d$$-action for which each $$\alpha_{\mathbf n}$$, $${\mathbf n}\in\mathbb Z^d$$, is a group automorphism. It is assumed that $$d\geq 2$$. The usual notions of rigidity and mixing of order $$r$$ can be defined for such actions [see K. Schmidt, “Dynamical systems of algebraic origins. Progress in Mathematics. 128. Basel: BirkhĂ¤user (1995; Zbl 0833.28001)].
The shape $$F= \{{\mathbf n}_1,\dots,{\mathbf n}_r\}$$ is said to be mixing for $$\alpha$$ if $\forall B_1,\dots, B_r\in{\mathcal B};\quad \lim_{k\to\infty} \mu\Biggl(\bigcap^r_{l= 1}\alpha_{-k{\mathbf n}_l}(B_l)\Biggr)= \prod^r_{i= 1}\mu(B_l).$ $$F$$ is a minimal non-mixing shape for $$\alpha$$ if $$F$$ is non-mixing, but any subset of $$F$$ is mixing, and $$F$$ is admissible if it does not lie on a line in $$\mathbb Z^d$$, contains $$0$$, and for any $$k>0$$ the set $${1\over k}F$$ contains non-integral points. The following results are proved:
Theorem 1. If $$S$$ is any admissible shape, then there is an algebraic $$\mathbb Z^d$$-action for which $$S$$ is a minimal non-mixing shape. If $$S$$ and $$T$$ are admissible shapes, then there is an algebraic $$\mathbb Z^d$$-action that is mixing on $$S$$ and not mixing on $$T$$ unless a translate of $$T$$ is a subset of $$S$$.
Theorem 2. If $$\alpha$$ is an algebraic $$\mathbb Z^d$$-action for which every shape is mixing then $$\alpha$$ is mixing of all orders. (In general, a measure-preserving $$\mathbb Z^d$$-action for which every shape is mixing can be rigid.)
Finally, for algebraic $$\mathbb Z^d$$-actions it is shown that there is a collection $${\mathcal L}= \{l_j\}$$ of half lines in $$\mathbb Z^2$$ such that $$\alpha$$ is mixing of all orders in the oriented cones associated to these lines. Examples are given.

### MSC:

 37A25 Ergodicity, mixing, rates of mixing 28D15 General groups of measure-preserving transformations 22D40 Ergodic theory on groups

Zbl 0833.28001
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