The conjugacy problem in ergodic theory

Abstract

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.In Memoriam: Prior to the final proofs of our paper our dear friend, Dan Rudolph, died just short of his 60th birthday, after a long and valiant struggle with ALS. He took an active part in this work to the very end despite his illness. This is not the place to describe in detail his lasting contributions to the modern theory of measurable dynamics; suffice it to say that his loss will be deeply felt not only by his close collaborators but by all who have an interest in this field. We want to dedicate our part of this paper to his memory.

The conjugacy relation on unitary representations

“…The purpose of this paper is to prove that this is indeed the case: This also answers a question raised by Kechris in [9, p. 123, (IIIb) and (IVb)], who asked if unitary equivalence of probability measure preserving actions of Γ is Borel. We note that Theorem 1.1 stands in contrast to the recent result of Foreman et al [4], who have shown that the conjugacy relation on ergodic transformations in the group of measure preserving transformations Aut(X, μ) is complete analytic, and so cannot be Borel, when (X, μ) is a standard non-atomic probability space. The paper is organized as follows: First, in Section 2, we will recall various basic notions from the theory of von Neumann algebras and the direct integral theory of unitary representations, and establish a simple lemma that will facilitate hierarchy complexity calculations later.…”

From odometers to circular systems: A global structure theorem

“…This question was answered by Foreman, Rudolph and Weiss in [6], where they gave a negative answer. This answer can be interpreted as saying that determining isomorphism between ergodic transformations is inaccessible to countable methods that use countable amounts of information.…”

“…However we were faced at first with the following difficulty. The transformations built in [6] were based on odometers (in the sense that the Kronecker factor was an odometer). It is a well known open problem whether it is possible to have any smooth transformation on a compact manifold that has a non-trivial odometer factor.…”

Book Review: Entropy in dynamical systems

“…In other words, a Bernoulli shift can be effectively recoded into an arbitrary second Bernoulli shift of the same entropy. Ornstein's ideas also played an important role in the construction of counterexamples, such as non-Bernoulli K-automorphisms [Or73] and the recent anti-classification theorems [FW04,FRW11] that study the complexity of the classification problem for measurepreserving transformations up to measure-conjugacy. For example, [FW04] proves it is impossible to classify all measure-preserving transformations up to measureconjugacy by countable structures.…”