In this paper,we introduce the concept of measure-theoretic r-entropy of a continuous map on a compact metric space,and get the results as follows:1.Measure-theoretic entropy is the limit of measure-theoretic r-entropy and topological entropy is the limit of topological r-entropy(r → 0);2.Topological r-entropy is more than or equal to the supremum of 4r-entropy in the sense of Feldman's definition,where the measure varies among all the ergodic Borel probability measures.
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.
