An entropy for ${\displaystyle {\mathbb{Z}}^2}$ -actions with finite entropy generators

We study a definition of entropy for ${\displaystyle {\mathbb{Z}}^{+}\times {\mathbb{Z}}^{+}}$-actions (or ${\displaystyle {\mathbb{Z}}^2}$-actions) due to S. Friedland. Unlike the more traditional definition, this is better suited for actions whose generators have finite entropy as single transformations. We compute its value in several examples. In particular, we settle a conjecture of Friedland [2].