On an entropy of $Z_{+}^k$-actions

In this paper, a definition of entropy for ${\mathbb{Z}}_{+}^k(k\ge 2)$-actions due to S. Friedland \cite{Friedland} is studied. Unlike the traditional definition, it may take a nonzero value for actions whose generators have finite (even zero) entropy as single transformations. Some basic properties are investigated and its value for the ${\mathbb{Z}}_{+}^k$-actions on circles generated by expanding endomorphisms is given. Moreover, an upper bound of this entropy for the ${\mathbb{Z}}_{+}^k$-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies, which are entropy-like invariants depending on the “inverse orbits” structure of the system.