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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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##### References:
 [1] H. Anzai, ”On an example of a measure preserving transformation which is not conjugate to its inverse,” Proc. Japan Acad., vol. 27, pp. 517-522, 1951. · Zbl 0044.12502 · doi:10.3792/pja/1195571227 · projecteuclid.org [2] H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Group Actions, Cambridge: Cambridge Univ. Press, 1996, vol. 232. · Zbl 0949.54052 · doi:10.1017/CBO9780511735264 [3] M. Foreman, Examples of ergodic transformations with complicated centralizers. · Zbl 0962.03043 [4] M. Foreman, ”A descriptive view of ergodic theory,” in Descriptive Set Theory and Dynamical Systems, Cambridge, 2000, pp. 87-171. · Zbl 0962.03043 [5] M. Foreman, D. J. Rudolph, and B. Weiss, Models for measure preserving transformations. · Zbl 1063.37004 [6] M. Foreman and B. Weiss, ”An anti-classification theorem for ergodic measure preserving transformations,” J. Eur. Math. Soc. $$($$JEMS$$)$$, vol. 6, iss. 3, pp. 277-292, 2004. · Zbl 1063.37004 · doi:10.4171/JEMS/10 · link.springer.de [7] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton, N.J.: Princeton Univ. Press, 1981. · Zbl 0459.28023 [8] E. Glasner, Ergodic Theory via Joinings, Providence, RI: Amer. Math. Soc., 2003, vol. 101. · Zbl 1038.37002 [9] P. R. Halmos and J. von Neumann, ”Operator methods in classical mechanics. II,” Ann. of Math., vol. 43, pp. 332-350, 1942. · Zbl 0063.01888 · doi:10.2307/1968872 [10] G. Hjorth, ”On invariants for measure preserving transformations,” Fund. Math., vol. 169, iss. 1, pp. 51-84, 2001. · Zbl 0990.03036 · doi:10.4064/fm169-1-2 [11] G. Hjorth, Classification and Orbit Equivalence Relations, Providence, RI: Amer. Math. Soc., 2000, vol. 75. · Zbl 0942.03056 [12] A. S. Kechris, Classical Descriptive Set Theory, New York: Springer-Verlag, 1995, vol. 156. · Zbl 0819.04002 [13] A. S. Kechris, ”On the concept of $${\Pi}^1_1$$-completeness,” Proc. Amer. Math. Soc., vol. 125, iss. 6, pp. 1811-1814, 1997. · Zbl 0864.03034 · doi:10.1090/S0002-9939-97-03770-2 [14] J. King, ”The commutant is the weak closure of the powers, for rank-$$1$$ transformations,” Ergodic Theory Dynam. Systems, vol. 6, iss. 3, pp. 363-384, 1986. · Zbl 0595.47005 · doi:10.1017/S0143385700003552 [15] A. Louveau and J. Saint-Raymond, ”Borel classes and closed games: Wadge-type and Hurewicz-type results,” Trans. Amer. Math. Soc., vol. 304, iss. 2, pp. 431-467, 1987. · Zbl 0655.04001 · doi:10.2307/2000725 [16] D. S. Ornstein, ”On the root problem in ergodic theory,” in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory, Berkeley, CA, 1972, pp. 347-356. · Zbl 0262.28009 [17] D. S. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, New Haven, Conn.: Yale Univ. Press, 1974, vol. 5. · Zbl 0296.28016 [18] D. J. Rudolph, Fundamentals of Measurable Dynamics, New York: The Clarendon Press Oxford University Press, 1990. · Zbl 0718.28008 [19] G. Sarton, A History of Science. Hellenistic Science and Culture in the Last Three Centuries B.C., Cambridge, MA: Harvard Univ. Press, 1959. · Zbl 0092.24201 [20] M. Suslin, ”Sur une définition des ensembles B sans nombres transfinis,” Compte Rendu Acad Science, vol. 164, pp. 88-91, 1917. · JFM 46.0296.01 [21] J. von Neumann, ”Zur Operatorenmethode in der klassischen Mechanik,” Ann. of Math., vol. 33, iss. 3, pp. 587-642, 1932. · Zbl 0005.12203 · doi:10.2307/1968537 [22] S. Wagon, The Banach-Tarski Paradox, Cambridge: Cambridge Univ. Press, 1985, vol. 24. · Zbl 0569.43001 [23] P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982, vol. 79. · Zbl 0475.28009
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