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**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.

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.

Reviewer: Geoffrey R. Goodson (Towson)

### MSC:

37A25 | Ergodicity, mixing, rates of mixing |

28D15 | General groups of measure-preserving transformations |

22D40 | Ergodic theory on groups |