## The isomorphism problem in ergodic theory.(English)Zbl 0255.28014

### MSC:

 28D05 Measure-preserving transformations 54H20 Topological dynamics (MSC2010)
Full Text:

### References:

 [1] L. M. Abramov, Metric automorphisms with quasi-discrete spectrum, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 513 – 530 (Russian). [2] R. L. Adler and P. C. Shields, Skew products of Bernoulli shifts with rotations. II, Israel J. Math. 19 (1974), 228 – 236. , https://doi.org/10.1007/BF02757718 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. Roy L. Adler and Benjamin Weiss, Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I., 1970. · Zbl 0307.28014 [3] Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. · Zbl 0184.43301 [4] J. R. Blum and D. L. Hanson, On the isomorphism problem for Bernoulli schemes, Bull. Amer. Math. Soc. 69 (1963), 221 – 223. · Zbl 0121.13601 [5] Rufus Bowen, Markov partitions for Axiom \? diffeomorphisms, Amer. J. Math. 92 (1970), 725 – 747. · Zbl 0208.25901 [6] J. R. Choksi, Non-ergodic transformations with discrete spectrum, Illinois J. Math. 9 (1965), 307 – 320. · Zbl 0151.19101 [7] Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339 – 348 (1970). , https://doi.org/10.1016/0001-8708(70)90008-3 Donald Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math. 5 (1970), 349 – 364 (1970). , https://doi.org/10.1016/0001-8708(70)90009-5 N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1970), 365 – 394 (1970). · Zbl 0203.05801 [8] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1 – 49. · Zbl 0146.28502 [9] Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, no. 3, The Mathematical Society of Japan, 1956. · Zbl 0073.09302 [10] Paul R. Halmos, Recent progress in ergodic theory, Bull. Amer. Math. Soc. 67 (1961), 70 – 80. · Zbl 0161.11401 [11] K. Jacobs, Lecture notes on ergodic theory, 1962/63. Parts I, II, Matematisk Institut, Aarhus Universitet, Aarhus, 1963. · Zbl 0196.31301 [12] S. A. Juzvinskiĭ, Metric properties of the endomorphisms of compact groups, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 1295 – 1328 (Russian). [13] Yitzhak Katznelson, Ergodic automorphisms of \?$$^{n}$$ are Bernoulli shifts, Israel J. Math. 10 (1971), 186 – 195. · Zbl 0219.28014 [14] Wolfgang Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc. 149 (1970), 453 – 464. · Zbl 0204.07904 [15] L. D. Mešalkin, A case of isomorphism of Bernoulli schemes, Dokl. Akad. Nauk SSSR 128 (1959), 41 – 44 (Russian). · Zbl 0099.12301 [16] Donald Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337 – 352. · Zbl 0197.33502 [17] Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339 – 348 (1970). , https://doi.org/10.1016/0001-8708(70)90008-3 Donald Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math. 5 (1970), 349 – 364 (1970). , https://doi.org/10.1016/0001-8708(70)90009-5 N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1970), 365 – 394 (1970). · Zbl 0203.05801 [18] Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339 – 348 (1970). , https://doi.org/10.1016/0001-8708(70)90008-3 Donald Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math. 5 (1970), 349 – 364 (1970). , https://doi.org/10.1016/0001-8708(70)90009-5 N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1970), 365 – 394 (1970). · Zbl 0203.05801 [19] Donald S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Math. 10 (1973), 49 – 62. · Zbl 0245.28011 [20] William Parry, Metric classification of ergodic nilflows and unipotent affines, Amer. J. Math. 93 (1971), 819 – 828. · Zbl 0222.22010 [21] William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0175.34001 [22] William Parry and Peter Walters, Endomorphisms of a Lebesgue space, Bull. Amer. Math. Soc. 78 (1972), 272 – 276. · Zbl 0232.28013 [23] M. Ratner, Markov partitions for Anosov flows on \?-dimensional manifolds, Israel J. Math. 15 (1973), 92 – 114. · Zbl 0269.58010 [24] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477 – 493. · Zbl 0079.08901 [25] V. A. Rohlin, Metric properties of endomorphisms of compact commutative groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 867 – 874 (Russian). [26] V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499 – 530 (Russian). [27] V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 3 – 56 (Russian). [28] V. A. Rohlin and Ja. G. Sinaĭ, The structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR 141 (1961), 1038 – 1041 (Russian). [29] 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. [30] Ja. G. Sinaĭ, A weak isomorphism of transformations with invariant measure, Dokl. Akad. Nauk SSSR 147 (1962), 797 – 800 (Russian). Ja. G. Sinaĭ, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.) 63 (105) (1964), 23 – 42 (Russian). [31] Ja. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 70 – 80 (Loose errata) (Russian). [32] J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. (2) 33 (1932), no. 3, 587 – 642 (German). · Zbl 0005.12203 [33] B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc. 76 (1970), 1266 – 1269. · Zbl 0218.28011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.