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Transactions of the American Mathematical Society

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A $K$ counterexample machine

Author: Christopher Hoffman
Journal: Trans. Amer. Math. Soc. 351 (1999), 4263-4280
MSC (1991): Primary 28D05; Secondary 28D20
Published electronically: July 1, 1999
MathSciNet review: 1650089
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Abstract | References | Similar Articles | Additional Information

Abstract: We present a general method for constructing families of measure preserving transformations which are $K$ and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two $K$ transformations which are weakly isomorphic but not isomorphic, and a $K$ transformation with no roots. Ornstein's isomorphism theorem says families of Bernoulli shifts cannot have these properties. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations.

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  • 1. Clark, Jack, A Kolmolgorov Shift with no Roots. Ph.D. Dissertation, Stanford University (1972).
  • 2. J. Feldman, New 𝐾-automorphisms and a problem of Kakutani, Israel J. Math. 24 (1976), no. 1, 16–38. MR 0409763
  • 3. Marlies Gerber, A zero-entropy mixing transformation whose product with itself is loosely Bernoulli, Israel J. Math. 38 (1981), no. 1-2, 1–22. MR 599470, 10.1007/BF02761843
  • 4. Hoffman, Christopher, A loosely Bernoulli counterexample machine. to appear in Israel J. Math. (1997).
  • 5. Hoffman, Christopher, The behavior of Bernoulli shifts relative to their factors. to appear in ETDS (1997).
  • 6. Donald Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337–352. MR 0257322
  • 7. Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339–348 (1970). MR 0274716
  • 8. Donald S. Ornstein, On the root problem in ergodic theory, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 347–356. MR 0399415
  • 9. Donald S. Ornstein, Ergodic theory, randomness, and dynamical systems, Yale University Press, New Haven, Conn.-London, 1974. James K. Whittemore Lectures in Mathematics given at Yale University; Yale Mathematical Monographs, No. 5. MR 0447525
  • 10. Donald S. Ornstein and Paul C. Shields, An uncountable family of 𝐾-automorphisms, Advances in Math. 10 (1973), 63–88. MR 0382598
  • 11. D. Rudolph, Two nonisomorphic 𝐾-automorphisms all of whose powers beyond one are isomorphic, Israel J. Math. 27 (1977), no. 3–4, 277–298. MR 0444907
  • 12. Daniel J. Rudolph, Two nonisomorphic 𝐾-automorphisms with isomorphic squares, Israel J. Math. 23 (1976), no. 3-4, 274–287. MR 0414826
  • 13. Daniel J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97–122. MR 555301, 10.1007/BF02791063
  • 14. Jean-Paul Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli, Israel J. Math. 21 (1975), no. 2-3, 177–207 (French, with English summary). Conference on Ergodic Theory and Topological Dynamics (Kibbutz, Lavi, 1974). MR 0399419

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Additional Information

Christopher Hoffman
Affiliation: The Hebrew University, Institute of Mathematics, Jerusalem, Israel
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Received by editor(s): March 31, 1997
Published electronically: July 1, 1999
Article copyright: © Copyright 1999 American Mathematical Society