Some Open Problems in Ergodic Theory

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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 $\overline{d}$-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, #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 #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 $T^{\infty }$ 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 #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, #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 #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", (>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 #5164a: 28 #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, #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, #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, #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 #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