Properties of a jointly ergodic action of the direct product of two groups

An ergodic action a of the direct product of ℤ and\(G = \begin{array}{*{20}c} \infty \\ \oplus \\ {n = 1} \\ \end{array} \mathbb{Z}_2 \), not isomorphic to a product of actions of ℤ and G, is constructed, such that the actions of ℤ and G separately are not ergodic. The actions of ℤ on its ergodic components are metrically isomorphic if and only if these components are taken into one another by the action of G. Finally, the centralizerCα(ℤ×G) is such thatCα(ℤ×G)/α(ℤ×G)≈ℤ2.