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Conjugacy classes are dense in the space of mixing \(\mathbb{Z}^d\)-actions. (English) Zbl 1376.37009
The main result of the paper is the extension of Halmos’ conjugacy lemma on the space of mixing \({\mathbb{Z}}^d\)-actions, namely, it proves the density of the conjugacy class of any mixing \({\mathbb{Z}}^d\)-action.
This implies that the set of \({\mathbb{Z}}^d\)-actions of rank 1 is massive, and the rank 1 property is generic for mixing \({\mathbb{Z}}^d\)-actions. This ensures the genericity for mixing of consequences of rank 1, such as triviality of the centralizer and the absence of factors.
The proof is based on an auxiliary result: “For any mixing \({\mathbb{Z}}^d\)-action \(T\) and any numbers \(\alpha \in (0,1)\) and \(\epsilon >0\), there exists a set \(A\) of measure \(\alpha\) whose images \(\{T^g A\}_{g\in {\mathbb{Z}}^d}\) intersect almost independently up to \(\epsilon\)”, which is formalized as a separate theorem (in a somewhat more general form).

MSC:
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37A25 Ergodicity, mixing, rates of mixing
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
22D40 Ergodic theory on groups
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