For a fixed irrational α > 0 we say that probability measure-preserving transformations S and T are α-equivalent if they can be realized as cross-sections in a common flow such that the return time functions on the cross-sections both take values in {1, 1 +α} and have equal integrals. Similarly we call two flows F and G α-equivalent if F has a cross-section S and G has a cross-section T isomorphic to S and again both the return time functions take values in {1, 1 + α} and have equal integrals. The integer kα(S), equal to the least positive such such that exp2πikα-1 belongs to the point spectrum of S, is an invariant of α-equivalence.

We obtain a characterization of a-equivalence as a particular type of restricted orbit equivalence and use this to prove that within the class of loosely Bernoulli maps ka(S) together with the entropy h(S) are complete invariants of α-equivalence. There is a corresponding a-equivalence theorem for flows which has as a consequence, for example, that up to an obvious entropy restriction, any weakly mixing cross-section of a loosely Bernoulli flow can also be realized as a cross-section with a {1,1 + α}-valued return time function.

For the proof of the α-equivalence theorem we develop a relative Kakutani equivalence theorem for compact group extensions which is of interest in its own right. Finally, an example of Fieldsteel and Rudolph is used to show that in general kα(S) is not a complete invariant of α-equivalence within a given even Kakutani equivalence class.