Comptes Rendus
no. 10
Dynamical Systems
On the conjugacy relation in ergodic theory
[Sur la relation de conjugation dans la théorie ergodique]

Présenté par : Gilles Pisier

Matthew D. Foreman1 ; Daniel J. Rudolph2 ; Benjamin Weiss3
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
Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 653-656.

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.

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.

Reçu le :
Accepté le :
Publié le :
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 
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/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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