The joinings within a class of \({\mathbb{Z}{}}^ 2\)-actions (Q1803630)

The general theory of minimal self joinings for actions of locally compact groups was established by \textit{A. del Junco} and \textit{D. Rudolph} [Ergodic Theory Dyn. Syst. 7, 531-558 (1987; Zbl 0646.60010)] but until recently the only concrete cases which have been studied have been \(\mathbb{Z}\) and \(\mathbb{R}\) actions. The aim of the authors is to generalize the construction of Chacon type automorphisms to obtain what they call Chacon \(\mathbb{Z}^ 2\)-actions and to obtain a fairly complete theory for joinings within this class. Using the fact that \(\mathbb{Z}^ 2\) actions arise from a pair of commuting automorphisms, these \(\mathbb{Z}^ 2\)-actions are constructed using a 2- dimensional analogue of the usual rank 1 cutting and stacking construction. The main rsults include the following: 1. Chacon \(\mathbb{Z}^ 2\)-actions are weakly mixing but not mixing. 2. Chacon \(\mathbb{Z}^ 2\)-actions have minimal self joinings. This implies that they are totally ergodic, have a trivial commutant and are prime (i.e., have no non-trivial factors).