Comptes Rendus
Dynamical Systems
On the conjugacy relation in ergodic theory
Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 653-656.

The set of pairs of transformations on the interval [0,1] can be equipped with a standard Borel structure. We prove that the relation of conjugacy is not a Borel subset of this space, in fact it is complete analytic. Moreover, our construction proves that the two sets, {T:Tis conjugate ofT−1}, and {T:the centralizer ofTis non-trivial} are complete analytic sets.

L'ensemble des paires de transformations ergodiques de l'intervalle [0,1] peut être muni d'une structure borélienne standard. Nous montrons que la relation de conjugaison n'est pas borélienne dans cet espace, en fait est analytique complète. Notre construction montre aussi que les ensembles {T:Test conjugué deT−1} et {T:la centralisateur deTest non-trivial} sont des analytiques complets.

Published online:
DOI: 10.1016/j.crma.2006.09.011

Matthew D. Foreman 1; Daniel J. Rudolph 2; Benjamin Weiss 3

1 Mathematics Department, UC Irvine, Irvine, CA 92697, USA
2 Mathematics Department, Colorado State University, Fort Collins, CO 80523, USA
3 Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
     author = {Matthew D. Foreman and Daniel J. Rudolph and Benjamin Weiss},
     title = {On the conjugacy relation in ergodic theory},
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Matthew D. Foreman; Daniel J. Rudolph; Benjamin Weiss. On the conjugacy relation in ergodic theory. Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 653-656. doi : 10.1016/j.crma.2006.09.011.

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[2] M. Foreman; B. Weiss An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc., Volume 6 (2004), pp. 277-292

[3] P.R. Halmos Ergodic Theory, Chelsea Publishing Co, New York, NY, 1956

[4] G. Hjorth On invariants for measure preserving transformations, Fund. Math., Volume 169 (2001), pp. 51-84

[5] D. Ornstein Ergodic Theory, Randomness, and Dynamical Systems, Yale Mathematical Monographs, vol. 5, Yale University Press, New Haven, CT, London, 1974

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