The Pre-image Entropy Of Z~k-Action

In this paper, four entropy-like invariants of (?)-ctions given by spanning sets and separated sets of a compact metric space whose generators are k commuting continuous maps are introduced and studied. The main results are:(1) the relations between these entropies are established. (2) for positively expansive systems, two types of pointwise pre-image entropies are equal, and the pre-image branch entropy and the pre-image rela-tion entropy are equal too. (3) two classes of systems:(a), a system generated by small C1-erturbations of an expanding map on a closed Riemmanian manifold, and (b). a system generated by equicontinuous maps defined on a finite graph, have zero pre-image branch entropy.