THE STRUCTURE OF LOCAL HOMEOMORPHISMS, III*1

In this paper, we continue the study of local homeomorphisms in Euclidean n-space begun in [13] and [14]. The main result to be proved here is that any C- volume preserving transformation defined in some neighborhood of the origin, keeping the origin fixed, whose Jacobian matrix at the origin satisfies a formal condition (**) to be given below, can be brought by a volume preserving change of coordinates to a certain normal form. The set of germs of these normal forms fall into a finite number of classes, each of which constitutes a maximal commutative subgroup of the group of local Cvolume preserving maps. Furthermore, the condition (**) will be interpreted as a regularity condition. Thus, we will establish for the group of local C- volume preserving maps a theorem analogous to the one which asserts