Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • 1. Clark, Jack, A Kolmolgorov Shift with no Roots. Ph.D. Dissertation, Stanford University (1972).
  • 2. Feldman, Jacob, New $K$-automorphisms and a problem of Kakutani. Israel J. Math. 24 (1976), no. 1, 16-38. MR 53:13515
  • 3. Gerber, Marlies, A zero-entropy mixing transformation whose product with itself is loosely Bernoulli. Israel J. Math. 38 (1981), 1-22. MR 82d:28013
  • 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. Ornstein, Donald, Bernoulli shifts with the same entropy are isomorphic. Advances in Math. 4 (1970), 337-352. MR 41:1973
  • 7. Ornstein, Donald, Two Bernoulli shifts with infinite entropy are isomorphic. Advances in Math. 5 (1970), 339-348. MR 43:478a
  • 8. Ornstein, Donald, On the Root Problem in Ergodic Theory. Proc. Sixth Berkeley Symp. Math. Stat. Prob., Vol. II, University of California Press, 1967, pp. 347-356. MR 53:3259
  • 9. Ornstein, Donald, Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, New Haven, (1974). MR 56:5836
  • 10. Ornstein, Donald; Shields, Paul, An uncountable family of $K$-automorphisms. Advances in Math. 10, (1973), 63-88. MR 52:3480
  • 11. Rudolph, Daniel, Two nonisomorphic $K$-automorphisms all of whose powers beyond one are isomorphic. Israel J. Math. 27 (1977), no. 3-4, 277-298. MR 56:3253
  • 12. Rudolph, Daniel, Two nonisomorphic $K$-automorphisms with isomorphic squares. Israel J. Math. 23 (1976), no. 3-4, 274-287. MR 54:2918
  • 13. Rudolph, Daniel, An example of a measure preserving map with minimal self-joinings, and applications. J. Analyse Math. 35 (1979), 97-122. MR 81e:28011
  • 14. Thouvenot, Jean-Paul, 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. MR 53:3263

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 28D05, 28D20

Retrieve articles in all journals with MSC (1991): 28D05, 28D20

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

American Mathematical Society