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The conjugacy problem in ergodic theory. (English) Zbl 1243.37006

Classifying mathematical objects is a fundamental endeavor, and in ergodic theory, this classification distinguishes measurable dynamical systems up to isomorphism. There are many papers that establish isomorphism invariants within a certain class of systems: one famous example is D. Ornstein’s result [Adv. Math. 5 (1970), 339–348 (1971; Zbl 0227.28014)] that two Bernoulli systems are isomorphic if and only if they are of the same entropy.
In this paper, the authors show that in the entire collection of measure-preserving transformations, no such classification is possible. They show that, setting \(E\) equal to the collection of ergodic transformations, the set of isomorphic pairs \((S,T)\subset E\times E\) is not Borel, i.e., there is no way to reliably distinguish between non-isomorphic measure-preserving transformations using a countable number of steps. In fact, they show that this set has maximal complexity by proving that even the collection of transformations isomorphic to their inverse is maximally complicated.
The main idea of the proof involves constructing a continuous, one-to-one map between trees (certain subsets of the finite sequences of elements from a countable set) and ergodic measures in such a way that the collection of trees which have an infinite branch maps to those measures which are isomorphic to their inverses. It is known [A. S. Kechris, Classical descriptive set theory. Graduate Texts in Mathematics. 156. Berlin: Springer (1995; Zbl 0819.04002)] that such a collection of trees is not Borel, and thus the image is also not Borel.
In the final section of the paper, the authors show that when restricted to the rank-one transformations, the isomorphism relation is Borel.

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
28D05 Measure-preserving transformations
28D20 Entropy and other invariants
37A05 Dynamical aspects of measure-preserving transformations
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