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It is shown, using a result of Rudolph, that any cross-section of an ergodic flow whose return-time function is bounded and bounded away from zero is isomorphic to a cross-section whose return-time function has finite range. A weaker result holds if the boundedness conditions are removed. Introduction-Flows with eigenvalues. In this paper we consider an ergodic flow, { T,: S -Q 2: t E R), on a Lebesgue probability space. A recurring problem in ergodic theory is to describe such flows in the simplest possible way. Such a way involves the construction known as a flow built under a function or special flow, (T, J). Here T: X -* X is an automorphism of a Lebesgue space, f a positive real function on X and s = {(x, r) E X x R: 0 Y and a partition of Y into two sets, A and B, of positive measure such that { T,) is isomorphic to (S, pX4 + qXB)(In fact Rudolph's result says even more, which does not concern us here.) If we have a special flow (T, f) of the type considered by Ambrose then Rudolph's result does not tell us anything about the relationship between S and T except, of course, that they are equivalent in the sense of Kakutani [5]. Below we show, as a consequence of Rudolph's work, that in this case (T, f) is isomorphic to (T, g) where g has finite range. In other words without altering the base transformation it is possible to make the function quite simple. Iff is unbounded then g can be chosen with countable range. I am grateful to William Parry for discussions about this subject. Received by the editors June 20, 1977. AMS (MOS) subject classifications (1970). Primary 28A65.

Representing ergodic flows as flows built under functions with finite range