Bulletin of the American Mathematical Society

The isomorphism problem in ergodic theory

Benjamin Weiss

Full-text: Open access

Article information

Source
Bull. Amer. Math. Soc., Volume 78, Number 5 (1972), 668-684.

Dates
First available in Project Euclid: 4 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.bams/1183533966

Mathematical Reviews number (MathSciNet)
MR0304616

Zentralblatt MATH identifier
0255.28014

Subjects
Primary: 28A65
Secondary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

Citation

Weiss, Benjamin. The isomorphism problem in ergodic theory. Bull. Amer. Math. Soc. 78 (1972), no. 5, 668--684. https://projecteuclid.org/euclid.bams/1183533966


Export citation

References

  • 1. L. M. Abramov, Metric automorphisms with a quasi-discrete spectrum, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 513-530; English transl., Amer. Math. Soc. Transl. (2) 39 (1964), 37-56. MR 26 #606.
  • 2. R. L. Adler and P. Shields, Skew products of Bernoulli shifts with rotations, Israel J. Math, (to appear).
  • 3a. R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 1573-1576. MR 35 #3031.
  • 3b. R. L. Adler and B. Weiss, Similarity of automorphisms of the torus, Mem. Amer. Math. Soc. No. 98 (1970). MR 41 # 1966.
  • 4. P. Billingsley, Ergodic theory and information, Wiley, New York, 1965. MR 33 # 254.
  • 5. J. R. Blum and D. L. Hanson, On the isomorphism problem for Bernoulli schemes, Bull. Amer. Math. Soc. 69 (1963), 221-223. MR 26 #1412.
  • 6. R. Bowen, Markov partitions for axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747. MR 43 #2740.
  • 7. J. R. Choksi, Non-ergodic transformations with discrete spectrum, Illinois J. Math. 9 (1965), 307-320. MR 30 # 4903.
  • 8. N. Friedman and D. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1970), 365-394. MR 43 #478c.
  • 9. H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1-49. MR 35 #4369.
  • 10. P. R. Halmos, Lectures on ergodic theory, Publ. Math. Soc. Japan, no. 3, Math. Soc. Japan, Tokyo, 1956. MR 20 #3958.
  • 11. P. R. Halmos, Recent progress in ergodic theory, Bull. Amer. Math. Soc. 67 (1961), 70-80. MR 23#A290.
  • 12. K. Jacobs, Lecture notes on ergodic theory. Parts I, II, Matematisk Institut, Aarhus Universitet, Aarhus, 1963. MR 28 #3238; errata, p. 1247.
  • 13. S. A. Juzvinskiĭ, Metric properties of endomorphisms of compact groups, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 1295-1328; English transl., Amer. Math. Soc. Transl. (2) 66 (1968), 63-98. MR 33 #2798.
  • 14. Y. Katznelson, Ergodic automorphisms of Tn are Bernoulli shifts, Israel J. Math. 10 (1971), 186-195.
  • 15. W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc. 149 (1970), 453-464. MR 41 # 3710.
  • 16. L. D. Mešalkin, A case of isomorphism of Bernouilli schemes, Dokl. Akad. Nauk SSSR 128 (1959), 41-44. (Russian) MR 22 # 1650.
  • 17. D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337-352. MR 41 #1973.
  • 18. D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339-348. MR 42 # 478a.
  • 19. D. Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math. 5 (1970), 349-364. MR 42 # 478b.
  • 20. D. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift (to appear).
  • 21. W. Parry, Metric classification of ergodic nilflows and unipotent affines, Amer. J. Math. 93 (1971), 819-828.
  • 22. W. Parry, Entropy and generators in ergodic theory, Benjamin, New York, 1969. MR 41 #7071.
  • 23. W. Parry and P. Walters, Endomorphisms of a Lebesgue space, Bull. Amer. Math. Soc. 78 (1972), 272-276.
  • 24. M. Ratner, Markov partitions for Anosov flows on n-dimensional manifolds, Math. Systems (to appear).
  • 25. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. MR 20 #3843.
  • 26. V. A. Rohlin, Metric properties of endomorphisms of compact commutative groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 867-874; English transl., Amer. Math. Soc. Transl. (2) 64 (1962), 244-252. MR 29 #5955.
  • 27. V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499-530; English transl., Amer. Math. Soc. Transl. (2) 39 (1964), 1-36. MR 26 # 1423.
  • 28. V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 3-56; Russian Math. Surveys 22 (1967), no. 5, 1-52. MR 36 #349.
  • 29. V. A. Rohlin and Ja. G. Sinaĭ, Construction and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR 141 (1961), 1038-1041=Soviet Math. Dokl. 2 (1961), 1611-1614. MR 27#2604.
  • 30. S. Rudolfer, The independence properties of certain number-theoretic endomorphisms, Proc. Sympos. on Topology, Dynamics and Ergodic Theory, University of Kentucky, Lexington, Ky., 1971, 68-69.
  • 31. Ja. G. Sinaĭ, Weak isomorphism of transformations with invariant measure, Mat. Sb. 63 (105) (1964), 23-42; English transl., Amer. Math. Soc. Transl. (2) 57 (1966), 123-143. MR 28 # 5164b.
  • 32. Ja. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 70-80. (Russian) MR 40 #3591.
  • 33. J. von Neumann, Zur operatoren Methoden in der klassischen Mechanik, Ann. of Math. 33 (1932), 587-642.
  • 34. B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc. 76 (1970), 1266-1269. MR 42 # 1978.

See also

  • Errata: Corrigendum, Volume 78. Bull. Amer. Math. Soc., Volume 79, Number 3 (1973), 614--614.