Science China Mathematics

, Volume 62, Issue 1, pp 69–72 | Cite as

Relative weak mixing is generic

  • Eli GlasnerEmail author
  • Benjamin Weiss


A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos’ result to the collection of ergodic extensions of a fixed, but arbitrary, aperiodic transformation T0. We then use a result of Ornstein and Weiss to extend this relative theorem to the general (countable) amenable group.


relative weak mixing Rokhlin’s lemma amenable groups 


37A25 37A05 37A15 37A20 


  1. 1.
    Connes A, Feldman J, Weiss B. An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynam Systems, 1981, 1: 431–450MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Furstenberg H. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Anal Math, 1977, 31: 204–256CrossRefzbMATHGoogle Scholar
  3. 3.
    Glasner E. Ergodic Theory via Joinings. Mathematical Surveys and Monographs. Providence: Amer Math Soc, 2003CrossRefzbMATHGoogle Scholar
  4. 4.
    Glasner E, Weiss B. Relative weak mixing is generic. ArXiv:1707.06425, 2017Google Scholar
  5. 5.
    Halmos P R. In general a measure preserving transformation is mixing. Ann of Math (2), 1944, 45: 786–792MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Halmos P R. Lectures on Ergodic Theory. Publications of the Mathematical Society of Japan, No. 3. Tokyo: Math Soc Japan, 1956Google Scholar
  7. 7.
    Ornstein D S, Weiss B. Ergodic theory of amenable group actions, I: The Rohlin lemma. Bull Amer Math Soc (NS), 1980, 2: 161–164MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Oxtoby J C. Measure and Category, 2nd ed. A Survey of the Analogies Between Topological and Measure Spaces. Graduate Texts in Mathematics, vol. 2. New York-Berlin: Springer-Verlag, 1980Google Scholar
  9. 9.
    Rokhlin V. A “general” measure-preserving transformation is not mixing (in Russian). Dokl Akad Nauk SSSR (NS), 1948, 60: 349–351Google Scholar
  10. 10.
    Rudolph D J. Classifying the isometric extensions of a Bernoulli shift. J Anal Math, 1978, 34: 36–60MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rudolph D J, Weiss B. Entropy and mixing for amenable group actions. Ann of Math (2), 2000, 151: 1119–1150MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schnurr M. Generic properties of extensions. Ergodic Theory Dynam Systems, 2018, in pressGoogle Scholar
  13. 13.
    Thouvenot J-P. Quelques propriétés des systemes dynamiques qui se décomposent en un produit de deux systemes dont l’un est un schéma de Bernoulli. Israel J Math, 1975, 21: 177–207MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zimmer R J. Extensions of ergodic group actions. Illinois J Math, 1976, 20: 373–409MathSciNetzbMATHGoogle Scholar
  15. 15.
    Zimmer R J. Ergodic actions with generalized discrete spectrum. Illinois J Math, 1976, 20: 555–588MathSciNetzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTel Aviv UniversityTel AvivIsrael
  2. 2.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations