Consequences of Arnoux, Ornstein, Weiss Theorem.

Question

The theorem states that any invertible, aperiodic, measure-preserving system on a Borel probability space is isomorphic to a cutting and stacking transformation. My question is, why is this useful? Are there any interesting consequences for example?

Answer 1

Well, it is not that it is "useful" in itself, but one of the main problems of ergodic theory is whether transformations with specific properties (say ergodicity, to give a simple example) can be realized in compact smooth manifolds, using preferably $C^\infty$ maps or flows. In this respect the theorem is of no help, since the "applications" never consider compact manifolds. A more down to earth expert opinion on the paper would be that it is a step towards this program, other than the quite nice properties that it implies (although apparently all of interest only from the internal point of view of theory).

Answer 2

I don't know about this theorem, but here is a general comment:

The important consequence of a theorem of this form is that if you want to study the whole family of invertible aperiodic transformations on Borel spaces, you can focus on their concrete models as cutting and stacking transformations. For example, in order to prove that every invertible aperiodic transformation on a Borel space satisfies a property $Q$, you only need to prove every (invertible and aperiodic) cutting and stacking transformation satisfies property $Q$, or if you are looking for an example of an invertible aperiodic transformation on a Borel space having property $Q$, you only need to look for it among cutting and stacking transformations. A theorem like this is a great gift for experts in cutting and stacking transformations.