Even and odd primality of dynamical systems with invariant measure (Q1387353)

The paper deals with a dynamical system meant as an action \(\Phi=\{T_g\mid g\in G\}\) of a group \(G\) in a Lebesgue probabilistic space \((X,\mu)\) such that \(\Phi\) preserves the measure \(\mu\). A measure \(\nu\) on the cube \(X^n\) is said to belong to the class \(M(n-1,n)\), \(n>2\), if its projections on the \((n-1)\)-dimensional faces of the cube are equal to \(\mu^{n-1}\). The action \(\Phi\) belongs to the class \(S(n-1,n)\) if \(\mu^{n-1}\) is the only measure of the class \(M(n-1,n)\) that is invariant under the action \(\Phi\otimes\Phi\otimes\cdots\otimes\Phi\). In the paper under review one constructs a noncommutative action \(\Phi\) belonging to the class \(S(2q,2q+1)\), but not belonging to the class \(S(2p-1,2p)\). One says that such an action has odd primality. The study of the \(S(n,n-1)\) properties is related to problems of joining theory [\textit{A. del Junco} and \textit{D. Rudolf}, Ergodic Theory Dyn. Syst. 7, 531-557 (1987; Zbl 0646.60010)].