Non-singular Cocycles and Piecewise Linear Time Changes

Cocycles of Zm-actions on compact metric spaces provide a means for constructing Rm-actions or flows, called suspension flows. It is known that all Rm flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. In this paper we investigate when the space for a suspension flow depends only on the given Zm-action and not on the actual cocycle. The identity map I of Rm determines perhaps the simplest cocycle for any Zm-action. We introduce invertible cocycles, and show that they produce the same space as the cocycle determined by the identity map I. The main result, Theorem 5.2, establishes an integration test for invertibility using piecewise linear maps and related topological ideas. Finally, it applies to the known methods for modeling Rm flows as suspensions and leads to refinements of these results.