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A note on mixing properties of invertible extensions. (English) Zbl 0924.28014

Summary: The natural invertible extension \(\widetilde T\) of an \(\mathbb N^d\)-action \(T\) has been studied by Lacroix. He showed that \(\widetilde T\) may fail to be mixing even if \(T\) is mixing for \(d\geq 2\). We extend this observation by showing that if \(T\) is mixing on \((k +1)\) sets then \(\widetilde T\) is in general mixing on no more than \(k\) sets, simply because \(\mathbb N^d\) has a corner. Several examples are constructed when \(d = 2\):
(i) a mixing \(T\) for which \(\widetilde T (n,m)\) has an identity factor whenever \(n \cdot m < 0\);
(ii) a mixing \(T\) for which \(\widetilde T\) is rigid but \(\widetilde T(n,m)\) is mixing for all \((n, m) \neq (0, 0)\);
(iii) a \(T\) mixing on 3 sets for which \(\widetilde T\) is not mixing on three sets.

MSC:

37A25 Ergodicity, mixing, rates of mixing
28D15 General groups of measure-preserving transformations
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