Topological and metric product $$\mathbb{Z}^d $$ -actions and their applications-actions and their applications

Abstract

We show that for a\(\mathbb{Z}^2 \)-action Ψ being the Kronecker sum of a symbolic strictly ergodic\(\mathbb{Z}\)-actionT and a Chacon\(\mathbb{Z}\)-actionS, the rank (covering number) of Ψ is the same as that forT. Using this result we construct, for a given natural numberr≥2 and a real numberb∈(0,1) withr\b≥1, a\(\mathbb{Z}^d \)-action with rankr, covering numberb and a simple spectrum. On the other hand, for any positive integersr, m with 1≤m≤r≤∞ we construct a\(\mathbb{Z}^d \)-action with rankr and spectral multiplicitym.