Conjugacy classes are dense in the space of mixing \(\mathbb{Z}^d\)-actions (Q295953)

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).