Infinite rank one actions and nonsingular Chacon transformations
An invertible transformation \(T\) of a Lebesgue space \((X,\mu)\) is called conservative if for any subset \(A\subset X\) of positive measure there exists \(n>0\) such that \(\mu(T^nA\cap A)>0\). If all Cartesian powers of \(T\) are ergodic, then \(T\) is said to have infinite ergodic index. If, moreover, for any finite sequence \(n_1,\dots,n_p\) of nonzero integers, the product \(T^{n_1}\times \cdots\times T^{n_p}\) is ergodic, \(T\) is called power weakly mixing. C. Silva proposed the following problem: Is there a nonpower weakly mixing infinite measure-preserving transformation \(T\) with infinite ergodic index such that the Cartesian products \(T^{n_1}\times \cdots \times T^{n_p}\) are all conservative? The author answers, among other things, the problem of C. Silva in a more general setting. In fact, the author first recalls the construction of \((C,F)\)-actions from the author's previous paper [Isr. J. Math. 121, 29-54 (2001; Zbl 1024.28014)]. The author proves the following theorem: Let \(G_\infty\) be the set of elements in \(G\) of infinite order. Then there exists an infinite measure preserving \((C,F)\)-action \(T\) of \(G\) such that (i) \(T_g\) has infinite ergodic index for any \(g\in G_\infty\), (ii) \(T_g\times T_{2g}\) is not ergodic for \(g\in G_\infty\), (iii) \(T_{g_1}\times\cdots \times T_{g_n}\) is conservative for every finite sequence \(g_1,\dots,g_n\) of elements in \(G\) and (iv) \(T_g\) is not conjugate to \(T_q^2\) for \(g\in G_\infty\). The case \(G=\mathbb{Z}\) gives an answer to the problem of C. Silva. The author also treats Chacon transformations in connection with ergodic properties.
scientific article; zbMATH DE number 2140248