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

In this paper, a definition of entropy for $\mathbb{Z}_+^k(k\geq2)$-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.