On an entropy of ℤ + k -actions

In this paper, a definition of entropy for ℤ+k(k ≥ 2)-actions due to 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 ℤ+k-actions on circles generated by expanding endomorphisms is given. Moreover, an upper bound of this entropy for the 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.