Mixing and descriptive set theory (Q2505480)

This interesting paper displays a thought-provoking interplay between the ergodic theory and descriptive set theory. The author presents an example of a complete analytic set using the concept of invariant measure mixing. Let \(X\) be a compact metric space and \(H(X)\) be the group of all homeomorphisms of \(X\). Further let \(H_{m}(X)=\{T\in H(X)\): there exists a \(T\)-invariant probability measure \(\mu\) such that \(T\) defines a mixing operator in \(L^{2}(\mu)\}\). There are two theorems and one example. In the first theorem, the author shows that \(H_{m}(X)\) is always an analytic subset of \(H(X)\). But \textit{S. Siboni} [Nonlinearity 7, No.~4, 1133--1141 (1994; Zbl 0809.58039)] has given an example of a compact metric space \(X_1\) such that \(H_{m}(X_1)\) is not closed in \(H(X_1)\). By using this example, the author constructs in the second theorem a compact metric space \(X_2\) of infinite topological dimension such that \(H_{m}(X_2)\) becomes a complete analytic subset of \(H(X_2)\). But in order to remove the ``stylistic defect'' of \(X_2\) having infinite topological dimension, the author re-works Siboni's example with a 0-dimensional compact metric space \(X_3\) and consequently for his second theorem, the author produces a compact metric space \(X_2\) of topological dimension 1.