Some Open Problems in Ergodic Theory

Donald S. Ornstein

Publications mathématiques et informatique de Rennes (1975)

  • Issue: S4, page 1-28

How to cite


Ornstein, Donald S.. "Some Open Problems in Ergodic Theory." Publications mathématiques et informatique de Rennes (1975): 1-28. <>.

author = {Ornstein, Donald S.},
journal = {Publications mathématiques et informatique de Rennes},
language = {eng},
number = {S4},
pages = {1-28},
publisher = {Département de Mathématiques et Informatique, Université de Rennes},
title = {Some Open Problems in Ergodic Theory},
url = {},
year = {1975},

AU - Ornstein, Donald S.
TI - Some Open Problems in Ergodic Theory
JO - Publications mathématiques et informatique de Rennes
PY - 1975
PB - Département de Mathématiques et Informatique, Université de Rennes
IS - S4
SP - 1
EP - 28
LA - eng
UR -
ER -


  1. [1] M. Smorodinsky, Ergodic Theory, Entropy. SpringerLecture Notes214 (1970) Zbl0213.07502MR422582
  2. [2] P. Shields, The Theory of Bernoulli Shifts. University of Chicago Press, Chicago and London, 1973 Zbl0308.28011MR442198
  3. [3] D. S. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, (James K. Whittemore Lectures in Mathematics given at Yale University) New Haven and London, Yale University Press, 1974 Zbl0296.28016MR447525
  4. [4] D. S. Ornstein, "Some new results in the Kolmogorov-Sinal theory of entropy and ergodic theory", Bull. Amer. Math. Soc.77, No. 6 . November, 1971, 878-890 Zbl0269.60032MR288233
  5. [5] D. S. Ornstein, What does it mean for mechanical systems to be Bernoulli? To appear Battelle Raconties, 1974 
  6. [6] D. S. Ornstein, "A mixing transformation for which Pinsker's conjecture fails", Advances in Math.10 (1973), 103-123 Zbl0248.28011MR399416
  7. [7] D. S. Ornstein, "A K-automorphism with no square root and Pinsker's conjecture", Advances in Math.10 (1973), 89-102 Zbl0248.28010MR330416
  8. [8] P. Shields and D. S. Ornstein, "An uncountable family of K-automorphism", Advances in Math.10 (1973), 63-88 Zbl0251.28004MR382598
  9. [9] J. Clark, "A K-automorphism with no roots" (to appear in AMS Transactions) 
  10. [10] M. Smorodinsky, "Construction of K-flows" (to appear) Zbl0293.28014MR355008
  11. [11] J.-P. Thouvenot, "Quelques Propriétés des Systèmes Dynamiques Qui se Décomposent en un Produit de Leux Systèmes dont L'un est Schéma de Bernoulle", (to appear) Zbl0329.28008
  12. [12] J.-P. Thouvenot, "Une Classe de Systemes Pour Lesquels la Conjecture de Pinsker est Vraie (to appear) Zbl0329.28009MR382602
  13. [13] J.-P. Thouvenot and P. Shields, "Entropy Zero x Bernoulli Processes are Closed in the d ¯ -metric" (to appear) Zbl0333.28007MR385072
  14. [14] J.-P. Thouvenot, "Remarques sur les systemes dynamiques donnes avec plusieurs facteurs", October, 1974 Zbl0331.28012MR399420
  15. [15] D.S. Ornstein, "Factors of Bernoulli Shifts (to appear in Israel J. Math.) Zbl0326.28029MR382599
  16. [16] R. Bowen, "Smooth Partitions of Anosov Diffeomorphisms are Weak Bernoulli", (to appear) Zbl0315.58020MR385927
  17. [17] K. Berg, On the Conjugacy problem for K-systems. Ph.D. thesis, University of Minnesota, 1967 MR2616688
  18. [18] R. L. Adler and B. Weiss, "Similarity of Automorphisms of the torus", IBM Research, August 5, 1969, 1-50 Zbl0195.06104MR257315
  19. [19] Y. Katznelson and B. Weiss, "Commuting Measure Preserving Transformations", Israel J. Math.12 (1972), 161-173 Zbl0239.28014MR316680
  20. [20] D. S. Ornstein and B. Weiss, " Z n Bernoulli actions", (to appear in Israel J. of Math.) Zbl0915.58076
  21. [21] G. Gallavotti, "Ising Model and Bernoulli Schemes in one Dimension", Commun. Math. Physics, 23, &#35;2, 183-190, 1973 Zbl0262.60061MR356801
  22. [22] G. Gallavotti, F. de Liberto, and L. Russo, "Markov processes, Bernoulli schemes and Ising model", Communications Math. Physics Zbl0333.60099
  23. [23] D. Lind, Locally Compact Measure Preserving Flows. Ph.D. thesis, Stanford University, 1973 Zbl0293.28012MR2623727
  24. [24] D. Lind, "Isomorphism theorem for R n (to appear) 
  25. [24a] J.-P. Thouvenot, "Convergence en moyenne de l’information pour l’action de Z 2 . Z . Wahrscheinlichkeitstheorie verw. Gebiete24, 2, 135-137 (1972) Zbl0266.60037MR321612
  26. [24b] R. Esposito and G. Gallavotti, "Approximate Symmetries and their Spontaneous Breakdown" (preprint) MR389108
  27. [24c] W. Krieger, "On the entropy of groups of measure-preserving transformations",(to appear) Zbl0239.28013
  28. [24d] J. C. Kieffer, "The Isomorphism Theorem for Generalized Bernoulli Schenes", submitted to Adv. in Math; Zbl0443.28012
  29. J. C. Kieffer, "A Generalized Shannon-McMillan Theorem for the Action and Amenable group on a Probability Space", to appear in Ann. of Math. Zbl0322.60032MR393422
  30. [25] V. A. Rokhlin, "Metric properties of endomorphisms of compact commutative groups", Izv. Akad. Nauk SSSR Ser. Mat.28 (1964), 867-874 Zbl0126.06101MR168697
  31. [26] S. A. Juzvinskii, "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 &#35;2798 Zbl0206.03602MR194588
  32. [27] Y. Katznelson, "Ergodic automorphism of T n are Bernoulli shifts", Israel J. Math.10 (1971), 186-195 Zbl0219.28014MR294602
  33. [28] D. Lind, "Ergodic automorphisms of the infinite torus are Bernoulli" Zbl0284.28007
  34. [29] N. Aoki and H. Totoki, "Ergodic automorphisms of T are Bernoulli transformations", Zbl0319.22007
  35. [30] Ya. G. Sinai, "Geodesic flows on compact surfaces of negative curvature", Dokl. Akad. Nauk SSR, 136 (3):549-552 (1961) Zbl0133.11002MR123678
  36. [31] D. V. Anosov, "Geodesic flows on closed Riemannian manifolds with negative curvature", Proc. of Steklov Inst.90 (1967). Zbl0176.19101MR224110
  37. [32] L. A. Bunimovich, Imbedding of Bernoulli shifts in certain special flows (in Russian). Uspehi mat. Nauk28 n° 3 (1973), 171-172 Zbl0285.28016MR460591
  38. [33] M. Ratner, "Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math. (to appear) Zbl0304.28011MR374387
  39. [34] R. Bowen and D. Ruelle, "The ergodic theory of axiom A flows" (to appear) Zbl0311.58010MR380889
  40. [35] R. Bowen, Bernoulli equilibrium states for a xiom A diffeomorphisms, Math. Systems Theory (to appear) Zbl0304.28012
  41. [36] D. Ruelle, "A measure associated with Axiom A attractors Zbl0355.58010
  42. [37] Ja. G. Sinai, "On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics", Dokl. Akad. Nauk SSSR153 (1963), 1261-1264 = Soviet Math. Dokl.4 (1963), 1818-1822.MR 35 &#35;5576 MR214727
  43. [38] Ja. G. Sinai, "Dynamical systems with elastic reflections", Uspekhi Mat. Nauk27 (1962), 137f. Zbl0252.58005
  44. [39] G. Gallavotti and D. S. Ornstein, "Billiards and Bernoulli schemes", Commun. Math. Physics38, (1974) 83-101 Zbl0313.58017MR355003
  45. [39a] I. Kubo, "Perturbed billiard systems I . The ergodicity of the motion of a particle in a compound central field" (preprint) Zbl0348.58008MR433510
  46. [39b] I. Kubo and H. Murata, "Perturbed billiard systems II. The Bernoulli property" Zbl0458.58014
  47. [40] J. Lebowitz, S. Goldstein, and M. Aizenman, "Ergodic peoperties of infinite systems", (to appear Battelle Recontres, summer 1974) Zbl0316.28008MR376039
  48. [41] S. Goldstein, "Space-time ergodic properties of systems of infinitely many independent particles", Battelle Recontres,summer 1974 Zbl0361.60088MR376040
  49. [42] O. Lanford III and J. Lebowitz, "Time evolution and ergodic properties of harmonic systems", Battelle Rencontres, summer 1974 Zbl0338.28011MR459441
  50. [42a] M. Aizenman, S. Goldsteir, and J. Lebowitz, "Ergodic properties of an infinite one dimensional hard rod system" Zbl0352.60073
  51. [42b] S. Goldstein and J. Lebowitz, "Ergodic properties of an infinite system of particles moving independently in a periodic field", Commun. math. Phys.371-18 (1974) Zbl0316.28008MR356802
  52. [42c] G. Caldiera and E. Presutti, 'Gibbs Processes and Generalized Bernoulli Flows for Hard-core One-dimensional Systems", Commun. Math. Phys.35, 279-286 (1974) © by Springer-Verlag1974 MR340542
  53. [43] M. Smorodinsky, "β-Automorphisms are Bernoulli shifts", Acta MathematicaAcademiae Scientiarum Hungaricae Tomus 24 (3-4), (1973) 273-278 Zbl0268.28007MR346133
  54. [44] K. M. Wilkinson, "Ergodic properties of certain linear mod one transformations", Advances in Math.141974,64-72 Zbl0286.28009MR344421
  55. [45] S. M. Rudolfer and K. M. Wilkinson, "A number-theoretic class of weak Bernoulli transformations", Math. Systems Theory Vol. 7, &#35;1, © 1973 by Springer-VerlagNew York Inc. Zbl0258.10031MR323744
  56. [46] S. Ito, H. Murata, and H. Totoki, "A remark on the isomorphism theorem", "On the isomorphism theorem for weak Bernoulli transformations in general case" (to appear) Zbl0246.28012
  57. [46a] Wilkinson, K. M., "Ergodic properties of a class of piecewise lineartrans. Zeitschrift fur Wahrscheinlichkeitstheorie, 303-328, 31-4, 1975 Zbl0299.28015MR374390
  58. [46b] R. Adler and B. Weiss, "Notes on β transformations, to appear 
  59. [46c] R. Adler, Paper given at the Conference on Recent Advances in Topological Dynamics. Publ. by Springer, 318 
  60. [47] V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Benjamin, New York, 1968. MR 38 &#35;1233 Zbl0167.22901MR232910
  61. [48] D. Ornstein and B. Weiss, "Every Transformation is Bilaterally Deterministic" (to appear in Israel J. Math.) Zbl0325.28016MR382600
  62. [49] Y. Katznelson and B. Weiss, "Ergodic automorphisms of the Solenoid are Bernoulli" 
  63. [50] R. Gray (1975), "Sliding-Block Source Coding" (to appear in IEEE Trans. on Info. Theory) Zbl0308.94015MR376230
  64. [51] R. Gray, D. Neuhoff and D. Ornstein, (1975) "Nonblock Source Coding with a Fidelity Criterion" (to appear in Annals of Prob.) Zbl0313.94006MR376239
  65. [52] R. Gray, D. Neuhoff and P. Shields (1975) "A Generalization of Ornstein's d-Distance with Applications to Information Theory", (&gt;to appear in Annals of Prob.) Zbl0304.94025MR368127
  66. [53] R. Gray and D. Ornstein, "Sliding-block joint source/noisy-channel coding theorems" (to appear) Zbl0348.94019MR530088
  67. [54] Ja. G. Sinai, "A weak isomorphism of transformations with an invariant measure", Dokl. Akad- Nauk SSSR147 (1962), 797-800 = Soviet Math. Dokl.3 (1962), 1725-1729. MR 28 &#35;5164a: 28 &#35;1247 Zbl0205.13501MR161960
  68. [55] D. Ornstein and B. Weiss, "Unilateral codings of Bernoulli systems" (to appear in Israel J. Math.) Zbl0323.28008MR412386
  69. [56] W. Parry and P. Walters, "Endomorphisms of a Lebesgue Space" (to appear) Zbl0232.28013MR294604
  70. [57] R. Fellgett and W. Parry, "Endomorphisms of a Lebesgue space II" (to appear) Zbl0305.28009MR382590
  71. [58] W. Parry, "Endomorphisms of a Lebesgue space III" (to appear) Zbl0312.28018MR382591
  72. [59] I. Kubo, H. Murata and H. Totoki, "On the Isomorphism Problem for Endomorphisms of Lebesgue Spaces, I", Publ. RIMS, Kyoto Univ.9, &#35;2 (1974), 285-296 Zbl0277.28005MR340551
  73. [60] I. Kubo, H. Murata and H. Totoki, "On the Isomorphism Problem for Endomorphisms of Lebesgue Spaces, II", Publ.RIMS, Kyoto Univ.9, &#35;2 (1974), 297-303 Zbl0277.28005
  74. [61] I. Kubo, H. Murata and H. Totoki, "On the Isomorphism Problem for Endomorphisms of Lebesgue Spaces, III", Publ. RIMS, Kyoto Univ.9, &#35;2 (1974), 305-317 Zbl0277.28005
  75. [62] L. D. Mesalkin, "A case of isomorphism of Bernoulli schemes", Dokl. Akad. Nauk SSSR128 (1959), 41-44. (Russian) MR 22 &#35;1650 Zbl0099.12301MR110782
  76. [63] M. Rosenblatt, "Stationary Processes as Shifts of Functions of Independent Random Variables", J. Math. and Mech, 8, No. 5., Sept. 1959, 665-682 Zbl0092.33601MR114249
  77. [64] M. Rosenblatt, "Stationary Markov Chains and Independent Random Variables", J. Math. and Mech., 9, No. 6, Nov. 1960, 945-950 Zbl0096.34004MR166839
  78. [65] M. Rosenblatt, "Chapter VI (called Nonlinear Representations in Terms of Independent Random Variables)" Markov Processes: Structure and Asymptotic Behavior, Springer1971. 
  79. [66] R. Adler and P. Shields, (A) "Skew products of Bernoulli shifts with rotations", Israel J. Math.12 (1972), 215-222; Zbl0246.28009MR315090
  80. R. Adler and P. Shields, (B) "Skew products of Bernoulli shifts with rotations, II", Israel J. Math. (to appear) Zbl0246.28009MR377013

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.