RESTRICTED ORBIT EQUIVALENCE FOR ERGODIC Z(D) ACTIONS .1.

In [R1] a notion of restricted orbit equivalence for ergodic transform ations was developed. Here we modify that structure in order to genera lize it to actions of higher-dimensional groups, in particular Z(d)-ac tions. The concept of a 'size' is developed first from an axiomatized notion of the size of a permutation of a finite block in Z(d). This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit equi valence. This is shown to be an equivalence relation. Associated to ea ch size is an entropy which is an equivalence invariant. As in the one -dimensional case this entropy is either the classical entropy or is z ero. Several examples are discussed.