Algebraic $\mathbb {Z}^d$-actions of entropy rank one

Abstract

We investigate algebraic $\mathbb Z^d$-actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic $\mathbb Z^d$-action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.