**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 66, 1 (1997)

pp. 1-20

Non-singular Cocycles and Piecewise Linear Time Changes

K. M. MADDEN and N. G. MARKLEY

**Abstract**.
Cocycles of \zm -actions on compact metric spaces provide a means for constructing m -actions or flows, called suspension flows. It is known that all m 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 m 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 m flows as suspensions and leads to refinements of these results.

**AMS subject classification**.
58F25; Secondary 28D10, 54H20

**Keywords**.

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