Spectrum of positive entropy multidimensional dynamical systems with a mixed time

Abstract:

It is shown that if an abelian countable group $G = G_{1}\oplus G_{2}$ is such that $G_{2}$ is a finite group and every aperiodic positive entropy action $\Phi$ of $G_{1}$ on a Lebesgue probability space $(X,\mathcal {B}, \mu )$ has a countable Haar spectrum in the subspace $L^{2}_{0}(X,\mu )\ominus L^{2}_{0}(X,\Pi (\Phi ),\mu )$, where $\Pi (\Phi )$ denotes the Pinsker $\sigma$- algebra of $\Phi$, then every aperiodic positive entropy action of $G$ on $(X, \mathcal {B}, \mu )$ has the same property. A positive answer to the question of J.P. Thouvenot is obtained as a corollary.