On an entropy of . k+-actions

In this paper, a definition of entropy for . 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 . k+-actions on circles generated by expanding endomorphisms is given. Moreover, an upper bound of this entropy for the . 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.