We show that, whenever is a countable abelian group and is a finitely-generated subgroup of , a generic measure-preserving action of on a standard atomless probability space extends to a free measure-preserving action of on . This extends a result of Ageev, corresponding to the case when is infinite cyclic.
Nous établissons que, pour tout groupe dénombrable abélien et tout sous-groupe finiment engendré de , l’ensemble des actions de sur un espace de probabilités standard qui peuvent être étendues en une action libre de sur est générique (au sens de Baire). Ce résultat étend un théorème d’Ageev, qui correspond au cas où est un groupe cyclique infini.
Keywords: Measure-preserving action, Baire category, Polish group
Mot clés : action préservant une mesure de probabilité, méthodes de Baire, groupe polonais
@article{AIF_2014__64_2_607_0, author = {Melleray, Julien}, title = {Extensions of generic measure-preserving actions}, journal = {Annales de l'Institut Fourier}, pages = {607--623}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {2}, year = {2014}, doi = {10.5802/aif.2859}, zbl = {06387286}, mrnumber = {3330916}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2859/} }
TY - JOUR AU - Melleray, Julien TI - Extensions of generic measure-preserving actions JO - Annales de l'Institut Fourier PY - 2014 SP - 607 EP - 623 VL - 64 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2859/ DO - 10.5802/aif.2859 LA - en ID - AIF_2014__64_2_607_0 ER -
%0 Journal Article %A Melleray, Julien %T Extensions of generic measure-preserving actions %J Annales de l'Institut Fourier %D 2014 %P 607-623 %V 64 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2859/ %R 10.5802/aif.2859 %G en %F AIF_2014__64_2_607_0
Melleray, Julien. Extensions of generic measure-preserving actions. Annales de l'Institut Fourier, Volume 64 (2014) no. 2, pp. 607-623. doi : 10.5802/aif.2859. http://www.numdam.org/articles/10.5802/aif.2859/
[1] Conjugacy of a group action to its inverse, Mat. Zametki, Volume 45 (1989) no. 3, p. 3-11, 127 | MR | Zbl
[2] The generic automorphism of a Lebesgue space conjugate to a -extension for any finite abelian group , Dokl. Akad. Nauk, Volume 374 (2000) no. 4, pp. 439-442 | MR | Zbl
[3] On the genericity of some nonasymptotic dynamic properties, Uspekhi Mat. Nauk, Volume 58 (2003) no. 1(349), pp. 177-178 | MR | Zbl
[4] Commuting transformations and mixing, Proc. Amer. Math. Soc., Volume 24 (1970), pp. 637-642 | DOI | MR | Zbl
[5] Commuting point transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 11 (1969), pp. 277-287 | DOI | MR | Zbl
[6] An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc. (JEMS), Volume 6 (2004) no. 3, pp. 277-292 | DOI | MR | Zbl
[7] Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), 293, CRC Press, Boca Raton, FL, 2009, pp. xiv+383 | MR | Zbl
[8] Every countable group has the weak Rohlin property, Bull. London Math. Soc., Volume 38 (2006) no. 6, pp. 932-936 | DOI | MR | Zbl
[9] A zero-one law for dynamical properties, Topological dynamics and applications (Minneapolis, MN, 1995) (Contemp. Math.), Volume 215, Amer. Math. Soc., Providence, RI, 1998, pp. 231-242 | MR | Zbl
[10] On infinite soluble groups. IV., J. Lond. Math. Soc., Volume 27 (1952), pp. 81-85 | DOI | MR | Zbl
[11] Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995, pp. xviii+402 | MR | Zbl
[12] Global aspects of ergodic group actions, Mathematical Surveys and Monographs, 160, American Mathematical Society, Providence, RI, 2010, pp. xii+237 | MR | Zbl
[13] The commutant is the weak closure of the powers, for rank- transformations, Ergodic Theory Dynam. Systems, Volume 6 (1986) no. 3, pp. 363-384 | DOI | MR | Zbl
[14] The generic transformation has roots of all orders, Colloq. Math., Volume 84/85 (2000) no. part 2, pp. 521-547 (Dedicated to the memory of Anzelm Iwanik) | MR | Zbl
[15] Generic representations of abelian groups and extreme amenability (2011) (To appear in Isr. J. Math, preprint available at http://front.math.ucdavis.edu/1107.1698 , Disponible “Online first”, DOI 10.1007/s11856-013-0036-5) | MR | Zbl
[16] A course in the theory of groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1996, pp. xviii+499 | MR | Zbl
[17] Une transformation générique peut être insérée dans un flot, Ann. Inst. H. Poincaré Probab. Statist., Volume 39 (2003) no. 1, pp. 121-134 | Numdam | MR | Zbl
[18] Closed subgroups generated by generic measure automorphisms (2012) (to appear in Ergodic Theory and Dynamical Systems, preprint available at http://www.math.uiuc.edu/~ssolecki/papers/gen_meas_aut_fin2.pdf) | MR
[19] Nonuniqueness of an inclusion in a flow and the vastness of a centralizer for a generic measure-preserving transformation, Mat. Sb., Volume 195 (2004) no. 12, pp. 95-108 | MR | Zbl
[20] Embeddings of lattice actions in flows with multidimensional time, Mat. Sb., Volume 197 (2006) no. 1, pp. 97-132 | MR | Zbl
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