The conjugacy problem in ergodic theory


All common probability preserving transformations can be represented as elements of MPT, the group of measure preserving transformations of the unit interval with Lebesgue measure. This group has a natural Polish topology and the induced topology on the set of ergodic transformations is also Polish. Our main result is that the set of ergodic elements $T$ in MPT that are isomorphic to their inverse is a complete analytic set. This has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not Borel. This is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is Borel.
This result explains, perhaps, why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable. The construction that we use is general enough to show that the set of ergodic $T$’s with nontrivial centralizer is also complete analytic.
On the positive side we show that the isomorphism relation is Borel when restricted to the rank one transformations, which form a generic subset of MPT. It remains an open problem to find a good explicit method of checking when two rank one transformations are isomorphic.


Matthew Foreman

Department of Mathematics
University of California, Irvine
Irvine, CA 92697

Daniel J. Rudolph

Department of Mathematics
Colorado State University
Fort Collins, CO 80523

Benjamin Weiss

Einstein Institute of Mathematics
The Hebrew University of Jerusalem
Jerusalem 91904