Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Joinings in Ergodic Theory

  • Thierry de la Rue
Reference work entry

Article Outline


Definition of the Subject


Joinings of Two or More Dynamical Systems


Some Applications and Future Directions



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  1. 1.
    Ahn Y-H, Lemańczyk M (2003) An algebraic property of joinings. Proc Amer Math Soc 131(6):1711–1716 (electronic)Google Scholar
  2. 2.
    Bułatek W, Lemańczyk M, Lesigne E (2005) On the filtering problem for stationary random \({\mathbb{Z}\sp 2}\)-fields. IEEE Trans Inform Theory 51(10):3586–3593Google Scholar
  3. 3.
    Burton R, Rothstein A (1977) Isomorphism theorems in ergodic theory. Technical report, Oregon State UniversityGoogle Scholar
  4. 4.
    Chacon RV (1969) Weakly mixing transformations which are not strongly mixing. Proc Amer Math Soc 22:559–562MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    de la Rue T (2006) 2‑fold and 3‑fold mixing: why 3-dot-type counterexamples are impossible in one dimension. Bull Braz Math Soc (NS) 37(4):503–521MATHCrossRefGoogle Scholar
  6. 6.
    de la Rue T (2006) An introduction to joinings in ergodic theory. Discret Contin Dyn Syst 15(1):121–142MATHCrossRefGoogle Scholar
  7. 7.
    del Junco A (1983) A family of counterexamples in ergodic theory. Isr J Math 44(2):160–188MATHCrossRefGoogle Scholar
  8. 8.
    del Junco A, Lemańczyk M, Mentzen MK (1995) Semisimplicity, joinings and group extensions. Studia Math 112(2):141–164Google Scholar
  9. 9.
    del Junco A, Rahe M, Swanson L (1980) Chacon's automorphism has minimal self‐joinings. J Analyse Math 37:276–284MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    del Junco A, Rudolph DJ (1987) On ergodic actions whose self‐joinings are graphs. Ergod Theory Dynam Syst 7(4):531–557MATHGoogle Scholar
  11. 11.
    Derriennic Y, Frączek K, Lemańczyk M, Parreau F (2008) Ergodic automorphisms whose weak closure of off‐diagonal measures consists of ergodic self‐joinings. Colloq Math 110:81–115Google Scholar
  12. 12.
    Ferenczi S (1997) Systems of finite rank. Colloq Math 73(1):35–65MathSciNetMATHGoogle Scholar
  13. 13.
    Frączek K, Lemańczyk M (2004) A class of special flows over irrational rotations which is disjoint from mixing flows. Ergod Theory Dynam Syst 24(4):1083–1095Google Scholar
  14. 14.
    Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Syst Theory 1:1–49MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Furstenberg H (1981) Recurrence in ergodic theory and combinatorial number theory. In: M.B. Porter Lectures. Princeton University Press, PrincetonGoogle Scholar
  16. 16.
    Furstenberg H, Peres Y, Weiss B (1995) Perfect filtering and double disjointness. Ann Inst H Poincaré Probab Stat 31(3):453–465MathSciNetMATHGoogle Scholar
  17. 17.
    Garsia AM (1970) Topics in almost everywhere convergence. In: Lectures in Advanced Mathematics, vol 4. Markham Publishing Co, Chicago, ILGoogle Scholar
  18. 18.
    Glasner E (2003) Ergodic theory via joinings. In: Mathematical Surveys and Monographs, vol 101. American Mathematical Society, ProvidenceGoogle Scholar
  19. 19.
    Glasner E, Host B, Rudolph DJ (1992) Simple systems and their higher order self‐joinings. Isr J Math 78(1):131–142MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Glasner E, Thouvenot J-P, Weiss B (2000) Entropy theory without a past. Ergod Theory Dynam Syst 20(5):1355–1370MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Glasner S, Weiss B (1983) Minimal transformations with no common factor need not be disjoint. Isr J Math 45(1):1–8MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Goodson GR (2000) Joining properties of ergodic dynamical systems having simple spectrum. Sankhyā Ser A 62(3):307–317, Ergodic theory and harmonic analysis (Mumbai, 1999)MathSciNetMATHGoogle Scholar
  23. 23.
    Host B (1991) Mixing of all orders and pairwise independent joinings of systems with singular spectrum. Isr J Math 76(3):289–298MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Janvresse É, de la Rue T (2007) On a class of pairwise‐independent joinings. Ergod Theory Dynam Syst (to appear)Google Scholar
  25. 25.
    Kalikow SA (1984) Twofold mixing implies threefold mixing for rank one transformations. Ergod Theory Dynam Syst 4(2):237–259MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    King J (1986) The commutant is the weak closure of the powers, for rank-1 transformations. Ergod Theory Dynam Syst 6(3):363–384MATHCrossRefGoogle Scholar
  27. 27.
    King J (1988) Joining‐rank and the structure of finite rank mixing transformations. J Anal Math 51:182–227MATHCrossRefGoogle Scholar
  28. 28.
    Krieger W (1970) On entropy and generators of measure‐preserving transformations. Trans Amer Math Soc 149:453–464MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Lemańczyk M, Parreau F, Thouvenot J-P (2000) Gaussian automorphisms whose ergodic self‐joinings are Gaussian. Fund Math 164(3):253–293Google Scholar
  30. 30.
    Lemańczyk M, Thouvenot J-P, Weiss B (2002) Relative discrete spectrum and joinings. Monatsh Math 137(1):57–75Google Scholar
  31. 31.
    Lesigne E, Rittaud B, and de la Rue T (2003) Weak disjointness of measure‐preserving dynamical systems. Ergod Theory Dynam Syst 23(4):1173–1198MATHCrossRefGoogle Scholar
  32. 32.
    Nadkarni MG (1998) Basic ergodic theory. In: Birkhäuser Advanced Texts: Basler Lehrbücher, 2nd edn. Birkhäuser, BaselGoogle Scholar
  33. 33.
    Nadkarni MG (1998) Spectral theory of dynamical systems. In: Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, BaselGoogle Scholar
  34. 34.
    Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4:337–352MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Ornstein DS (1972) On the root problem in ergodic theory. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, 1970/1971, vol II: Probability theory. Univ California Press, Berkeley, pp 347–356Google Scholar
  36. 36.
    Ornstein DS (1974) Ergodic theory, randomness, and dynamical systems. In: James K Whittemore Lectures in Mathematics given at Yale University, Yale Mathematical Monographs, vol 5. Yale University Press, New HavenGoogle Scholar
  37. 37.
    Parreau F, Roy E (2007) Poisson joinings of Poisson suspension. PreprintGoogle Scholar
  38. 38.
    Ratner M (1983) Horocycle flows, joinings and rigidity of products. Ann of Math 118(2):277–313MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Rohlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340MathSciNetGoogle Scholar
  40. 40.
    Roy E (2007) Poisson suspensions and infinite ergodic theory. Ergod Theory Dynam Syst(to appear)Google Scholar
  41. 41.
    Rudolph DJ (1979) An example of a measure preserving map with minimal self‐joinings, and applications. J Anal Math 35:97–122MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Rudolph DJ (1990) The title is Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces. Oxford University Press, New YorkGoogle Scholar
  43. 43.
    Rudolph DJ (1994) A joinings proof of Bourgain's return time theorem. Ergod Theory Dynam Syst 14(1):197–203MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Ryzhikov VV (1992) Stochastic wreath products and joinings of dynamical systems. Mat Zametki 52(3):130–140, 160MathSciNetGoogle Scholar
  45. 45.
    Ryzhikov VV (1992) Mixing, rank and minimal self‐joining of actions with invariant measure. Mat Sb 183(3):133–160MathSciNetMATHGoogle Scholar
  46. 46.
    Ryzhikov VV (1993) Joinings and multiple mixing of the actions of finite rank. Funktsional Anal Prilozhen 27(2):63–78, 96MathSciNetGoogle Scholar
  47. 47.
    Ryzhikov VV (1993) Joinings, wreath products, factors and mixing properties of dynamical systems. Izv Ross Akad Nauk Ser Mat 57(1):102–128Google Scholar
  48. 48.
    Thorisson H (2000) Coupling, stationarity, and regeneration. In: Probability and its Applications (New York). Springer, New YorkGoogle Scholar
  49. 49.
    Thouvenot J-P (1995) Some properties and applications of joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis, Alexandria, 1993. London Math Soc Lecture Note Ser, vol 205. Cambridge Univ Press, Cambridge, pp 207–235Google Scholar
  50. 50.
    Thouvenot J-P (1987) The metrical structure of some Gaussian processes. In: Proceedings of the conference on ergodic theory and related topics, II, Georgenthal, 1986. Teubner-Texte Math, vol 94. Teubner, Leipzig, pp 195–198Google Scholar
  51. 51.
    Veech WA (1982) A criterion for a process to be prime. Monatsh Math 94(4):335–341MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Thierry de la Rue
    • 1
  1. 1.Laboratoire de Mathématiques Raphaël SalemCNRS – Université de RouenSaint Étienne du RouvrayFrance