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

Présenté par : Gilles Pisier

Foreman, Matthew D.1 ; Rudolph, Daniel J.2 ; Weiss, Benjamin3
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, Tome 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
Foreman, Matthew D. 1 ; Rudolph, Daniel J. 2 ; Weiss, Benjamin 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 
@article{CRMATH_2006__343_10_653_0,
     author = {Foreman, Matthew D. and Rudolph, Daniel J. and Weiss, Benjamin},
     title = {On the conjugacy relation in ergodic theory},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {653--656},
     publisher = {Elsevier},
     volume = {343},
     number = {10},
     year = {2006},
     doi = {10.1016/j.crma.2006.09.011},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.09.011/}
}
TY  - JOUR
AU  - Foreman, Matthew D.
AU  - Rudolph, Daniel J.
AU  - Weiss, Benjamin
TI  - On the conjugacy relation in ergodic theory
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 653
EP  - 656
VL  - 343
IS  - 10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.09.011/
DO  - 10.1016/j.crma.2006.09.011
LA  - en
ID  - CRMATH_2006__343_10_653_0
ER  - 
%0 Journal Article
%A Foreman, Matthew D.
%A Rudolph, Daniel J.
%A Weiss, Benjamin
%T On the conjugacy relation in ergodic theory
%J Comptes Rendus. Mathématique
%D 2006
%P 653-656
%V 343
%N 10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.09.011/
%R 10.1016/j.crma.2006.09.011
%G en
%F CRMATH_2006__343_10_653_0
Foreman, Matthew D.; Rudolph, Daniel J.; Weiss, Benjamin. On the conjugacy relation in ergodic theory. Comptes Rendus. Mathématique, Tome 343 (2006) no. 10, pp. 653-656. doi : 10.1016/j.crma.2006.09.011. http://www.numdam.org/articles/10.1016/j.crma.2006.09.011/

[1] Feldman, J. Borel structures and invariants for measurable transformations, Proc. Amer. Math. Soc., Volume 46 (1974), pp. 383-394

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

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

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

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

Cité par Sources :