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Probability Theory and Related Fields

, Volume 167, Issue 3–4, pp 1117–1136 | Cite as

A monotone isomorphism theorem

  • Terry SooEmail author
Article

Abstract

In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary sequence to another that is coordinatewise smaller than or equal to the original sequence.

Keywords

Sinai factor theorem Ornstein theorem Stochastic domination Monotone coupling Burton–Rothstein 

Mathematics Subject Classification

37A35 60G10 60E15 

Notes

Acknowledgments

I thank Zemer Kosloff for his help with Example 1. I also thank the referee for the careful reading of this paper and useful suggestions.

References

  1. 1.
    Ball, K.: Monotone factors of i.i.d. processes. Isr. J. Math. 150, 205–227 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blum, J.R., Hanson, D.L.: On the isomorphism problem for Bernoulli schemes. Bull. Am. Math. Soc. 69, 221–223 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burton, R., Keane, M., Serafin, J.: Residuality of dynamical morphisms. Colloq. Math. 85, 307–317 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Burton, R., Rothstein, A.: Isomorphism theorems in ergodic theory. Technical report, Oregon State University (1977)Google Scholar
  5. 5.
    Downarowicz, T.: Entropy in Dynamical Systems. New Mathematical Monographs, vol. 18. Cambridge University Press, Cambridge (2011)Google Scholar
  6. 6.
    Gurel-Gurevich, O., Peled, R.: Poisson thickening. Isr. J. Math. 196(1), 215–234 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    del Junco, A.: Finitary codes between one-sided Bernoulli shifts. Ergod. Theory Dyn. Syst. 1(3), 285–301 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    del Junco, A.: Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic. Ergod. Theory Dyn. Syst. 10(4), 687–715 (1990)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Katok, A.: Fifty years of entropy in dynamics: 1958–2007. J. Mod. Dyn. 1(4), 545–596 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Keane, M., Smorodinsky, M.: A class of finitary codes. Isr. J. Math. 26, 352–371 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Keane, M., Smorodinsky, M.: Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. Math. 2(109), 397–406 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    King, J.L.F.: Entropy in ergodic theory. In: Mathematics of Complexity and Dynamical Systems, vols. 1–3, pp. 205–224. Springer, New York (2012)Google Scholar
  13. 13.
    Lyons, R.: Factors of iid on trees. Combin. Probab. Comput. (to appear). arXiv:1401.4197
  14. 14.
    Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, Cambridge (2014) (in preparation). Current version available at http://mypage.iu.edu/ rdlyons/
  15. 15.
    Mešalkin, L.D.: A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk SSSR 128, 41–44 (1959)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ornstein, D.: Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4, 337–352 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ornstein, D.: Newton’s laws and coin tossing. Not. Am. Math. Soc. 60(4), 450–459 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Quas, A., Soo, T.: A monotone Sinai theorem. Ann. Probab. 44(1), 107–130 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    de la Rue, T.: An introduction to joinings in ergodic theory. Discrete Contin. Dyn. Syst. 15, 121–142 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Serafin, J.: Finitary codes, a short survey. Dynamics & stochastics. IMS Lecture Notes in Monograph Series, vol. 48, pp. 262–273. Institute of Mathematical Statistics, Beachwood (2006)Google Scholar
  21. 21.
    Sinaĭ, J.G.: On a weak isomorphism of transformations with invariant measure. Mat. Sb. (N.S.) 63(105), 23–42 (1964)Google Scholar
  22. 22.
    Sinai, Y.G.: Selecta. Volume I. Ergodic Theory and Dynamical Systems. Springer, New York (2010)Google Scholar
  23. 23.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Weiss, B.: The isomorphism problem in ergodic theory. Bull. Am. Math. Soc. 78, 668–684 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA

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