Some Open Problems in Ergodic Theory

Ornstein, Donald S.

Ornstein, Donald S.

Publications des séminaires de mathématiques et informatique de Rennes, International Conference on Dynamical Systems in Mathematical Physics, no. S4 (1975), article no. 19, 28 p.

Export
Comment citer

@article{PSMIR_1975___S4_A19_0,
     author = {Ornstein, Donald S.},
     title = {Some {Open} {Problems} in {Ergodic} {Theory}},
     journal = {Publications des s\'eminaires de math\'ematiques et informatique de Rennes},
     eid = {19},
     pages = {1--28},
     publisher = {D\'epartement de Math\'ematiques et Informatique, Universit\'e de Rennes},
     number = {S4},
     year = {1975},
     language = {en},
     url = {http://www.numdam.org/item/PSMIR_1975___S4_A19_0/}
}

TY  - JOUR
AU  - Ornstein, Donald S.
TI  - Some Open Problems in Ergodic Theory
JO  - Publications des séminaires de mathématiques et informatique de Rennes
PY  - 1975
SP  - 1
EP  - 28
IS  - S4
PB  - Département de Mathématiques et Informatique, Université de Rennes
UR  - http://www.numdam.org/item/PSMIR_1975___S4_A19_0/
LA  - en
ID  - PSMIR_1975___S4_A19_0
ER  -

%0 Journal Article
%A Ornstein, Donald S.
%T Some Open Problems in Ergodic Theory
%J Publications des séminaires de mathématiques et informatique de Rennes
%D 1975
%P 1-28
%N S4
%I Département de Mathématiques et Informatique, Université de Rennes
%U http://www.numdam.org/item/PSMIR_1975___S4_A19_0/
%G en
%F PSMIR_1975___S4_A19_0

Ornstein, Donald S. Some Open Problems in Ergodic Theory. Publications des séminaires de mathématiques et informatique de Rennes, International Conference on Dynamical Systems in Mathematical Physics, no. S4 (1975), article  no. 19, 28 p. http://www.numdam.org/item/PSMIR_1975___S4_A19_0/

Bibliographie
Cité par

[1] M. Smorodinsky, Ergodic Theory, Entropy. Springer Lecture Notes 214 (1970) | MR | Zbl

[2] P. Shields, The Theory of Bernoulli Shifts. University of Chicago Press, Chicago and London, 1973 | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[5] D. S. Ornstein, What does it mean for mechanical systems to be Bernoulli? To appear Battelle Raconties, 1974

[6] D. S. Ornstein, "A mixing transformation for which Pinsker's conjecture fails", Advances in Math. 10 (1973), 103-123 | MR | Zbl

[7] D. S. Ornstein, "A K-automorphism with no square root and Pinsker's conjecture", Advances in Math. 10 (1973), 89-102 | MR | Zbl

[8] P. Shields and D. S. Ornstein, "An uncountable family of K-automorphism", Advances in Math. 10 (1973), 63-88 | MR | Zbl

[9] J. Clark, "A K-automorphism with no roots" (to appear in AMS Transactions)

[10] M. Smorodinsky, "Construction of K-flows" (to appear) | MR | Zbl

[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) | Zbl

[12] J.-P. Thouvenot, "Une Classe de Systemes Pour Lesquels la Conjecture de Pinsker est Vraie (to appear) | MR | Zbl

[13] J.-P. Thouvenot and P. Shields, "Entropy Zero x Bernoulli Processes are Closed in the $\bar{d}$ -metric" (to appear) | MR | Zbl

[14] J.-P. Thouvenot, "Remarques sur les systemes dynamiques donnes avec plusieurs facteurs", October, 1974 | MR | Zbl

[15] D.S. Ornstein, "Factors of Bernoulli Shifts (to appear in Israel J. Math.) | MR | Zbl

[16] R. Bowen, "Smooth Partitions of Anosov Diffeomorphisms are Weak Bernoulli", (to appear) | MR | Zbl

[17] K. Berg, On the Conjugacy problem for K-systems. Ph.D. thesis, University of Minnesota, 1967 | MR

[18] R. L. Adler and B. Weiss, "Similarity of Automorphisms of the torus", IBM Research, August 5, 1969, 1-50 | MR | Zbl

[19] Y. Katznelson and B. Weiss, "Commuting Measure Preserving Transformations", Israel J. Math. 12 (1972), 161-173 | MR | Zbl

[20] D. S. Ornstein and B. Weiss, " $Z^{n}$ Bernoulli actions", (to appear in Israel J. of Math.)

[21] G. Gallavotti, "Ising Model and Bernoulli Schemes in one Dimension", Commun. Math. Physics, 23, #2, 183-190, 1973 | MR | Zbl

[22] G. Gallavotti, F. De Liberto, and L. Russo, "Markov processes, Bernoulli schemes and Ising model", Communications Math. Physics | Zbl

[23] D. Lind, Locally Compact Measure Preserving Flows. Ph.D. thesis, Stanford University, 1973 | MR | Zbl

[24] D. Lind, "Isomorphism theorem for $R^{n}$ (to appear)

[24a] J.-P. Thouvenot, "Convergence en moyenne de l’information pour l’action de $Z^{2}$ . Z . Wahrscheinlichkeitstheorie verw. Gebiete 24, 2, 135-137 (1972) | MR | Zbl

[24b] R. Esposito and G. Gallavotti, "Approximate Symmetries and their Spontaneous Breakdown" (preprint) | Numdam | MR

[24c] W. Krieger, "On the entropy of groups of measure-preserving transformations",(to appear)

[24d] J. C. Kieffer, "The Isomorphism Theorem for Generalized Bernoulli Schenes", submitted to Adv. in Math; | Zbl

J. C. Kieffer, "A Generalized Shannon-McMillan Theorem for the Action and Amenable group on a Probability Space", to appear in Ann. of Math. | MR | Zbl

[25] V. A. Rokhlin, "Metric properties of endomorphisms of compact commutative groups", Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 867-874 | MR | Zbl

[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 | MR | Zbl

[27] Y. Katznelson, "Ergodic automorphism of $T^{n}$ are Bernoulli shifts", Israel J. Math. 10 (1971), 186-195 | MR | Zbl

[28] D. Lind, "Ergodic automorphisms of the infinite torus are Bernoulli" | Zbl

[29] N. Aoki and H. Totoki, "Ergodic automorphisms of $T^{\infty}$ are Bernoulli transformations", | Zbl

[30] Ya. G. Sinai, "Geodesic flows on compact surfaces of negative curvature", Dokl. Akad. Nauk SSR, 136 (3):549-552 (1961) | MR | Zbl

[31] D. V. Anosov, "Geodesic flows on closed Riemannian manifolds with negative curvature", Proc. of Steklov Inst. 90 (1967). | MR | Zbl

[32] L. A. Bunimovich, Imbedding of Bernoulli shifts in certain special flows (in Russian). Uspehi mat. Nauk 28 n° 3 (1973), 171-172 | MR | Zbl

[33] M. Ratner, "Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math. (to appear) | MR | Zbl

[34] R. Bowen and D. Ruelle, "The ergodic theory of axiom A flows" (to appear) | MR | Zbl

[35] R. Bowen, Bernoulli equilibrium states for a xiom A diffeomorphisms, Math. Systems Theory (to appear) | Zbl

[36] D. Ruelle, "A measure associated with Axiom A attractors | Zbl

[37] Ja. G. Sinai, "On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics", Dokl. Akad. Nauk SSSR 153 (1963), 1261-1264 = Soviet Math. Dokl. 4 (1963), 1818-1822.MR 35 #5576 | MR

[38] Ja. G. Sinai, "Dynamical systems with elastic reflections", Uspekhi Mat. Nauk 27 (1962), 137f. | Zbl

[39] G. Gallavotti and D. S. Ornstein, "Billiards and Bernoulli schemes", Commun. Math. Physics 38, (1974) 83-101 | MR | Zbl

[39a] I. Kubo, "Perturbed billiard systems I . The ergodicity of the motion of a particle in a compound central field" (preprint) | MR | Zbl

[39b] I. Kubo and H. Murata, "Perturbed billiard systems II. The Bernoulli property" | Zbl

[40] J. Lebowitz, S. Goldstein, and M. Aizenman, "Ergodic peoperties of infinite systems", (to appear Battelle Recontres, summer 1974) | MR | Zbl

[41] S. Goldstein, "Space-time ergodic properties of systems of infinitely many independent particles", Battelle Recontres,summer 1974 | MR | Zbl

[42] O. Lanford Iii and J. Lebowitz, "Time evolution and ergodic properties of harmonic systems", Battelle Rencontres, summer 1974 | MR | Zbl

[42a] M. Aizenman, S. Goldsteir, and J. Lebowitz, "Ergodic properties of an infinite one dimensional hard rod system" | Zbl

[42b] S. Goldstein and J. Lebowitz, "Ergodic properties of an infinite system of particles moving independently in a periodic field", Commun. math. Phys. 37 1-18 (1974) | MR | Zbl

[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-Verlag 1974 | MR

[43] M. Smorodinsky, "β-Automorphisms are Bernoulli shifts", Acta Mathematica Academiae Scientiarum Hungaricae Tomus 24 (3-4), (1973) 273-278 | MR | Zbl

[44] K. M. Wilkinson, "Ergodic properties of certain linear mod one transformations", Advances in Math. 14 1974,64-72 | MR | Zbl

[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) | Zbl

[46a] Wilkinson, K. M., "Ergodic properties of a class of piecewise linear trans. Zeitschrift fur Wahrscheinlichkeitstheorie, 303-328, 31-4, 1975 | MR | Zbl

[46b] R. Adler and B. Weiss, "Notes on β transformations, to appear

[46c] R. Adler, Paper given at the Conference on Recent Advances in Topological Dynamics. Publ. by Springer, 318

[47] V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Benjamin, New York, 1968. MR 38 #1233 | MR | Zbl

[48] D. Ornstein and B. Weiss, "Every Transformation is Bilaterally Deterministic" (to appear in Israel J. Math.) | MR | Zbl

[49] Y. Katznelson and B. Weiss, "Ergodic automorphisms of the Solenoid are Bernoulli"

[50] R. Gray (1975), "Sliding-Block Source Coding" (to appear in IEEE Trans. on Info. Theory) | MR | Zbl

[51] R. Gray, D. Neuhoff and D. Ornstein, (1975) "Nonblock Source Coding with a Fidelity Criterion" (to appear in Annals of Prob.) | MR | Zbl

[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.) | MR | Zbl

[53] R. Gray and D. Ornstein, "Sliding-block joint source/noisy-channel coding theorems" (to appear) | MR | Zbl

[54] Ja. G. Sinai, "A weak isomorphism of transformations with an invariant measure", Dokl. Akad- Nauk SSSR 147 (1962), 797-800 = Soviet Math. Dokl. 3 (1962), 1725-1729. MR 28 #5164a: 28 #1247 | MR | Zbl

[55] D. Ornstein and B. Weiss, "Unilateral codings of Bernoulli systems" (to appear in Israel J. Math.) | MR | Zbl

[56] W. Parry and P. Walters, "Endomorphisms of a Lebesgue Space" (to appear) | MR | Zbl

[57] R. Fellgett and W. Parry, "Endomorphisms of a Lebesgue space II" (to appear) | MR | Zbl

[58] W. Parry, "Endomorphisms of a Lebesgue space III" (to appear) | MR | Zbl

[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 | MR | Zbl

[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 | Zbl

[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 | Zbl

[62] L. D. Mesalkin, "A case of isomorphism of Bernoulli schemes", Dokl. Akad. Nauk SSSR 128 (1959), 41-44. (Russian) MR 22 #1650 | MR | Zbl

[63] M. Rosenblatt, "Stationary Processes as Shifts of Functions of Independent Random Variables", J. Math. and Mech, 8, No. 5., Sept. 1959, 665-682 | MR | Zbl

[64] M. Rosenblatt, "Stationary Markov Chains and Independent Random Variables", J. Math. and Mech., 9, No. 6, Nov. 1960, 945-950 | MR | Zbl

[65] M. Rosenblatt, "Chapter VI (called Nonlinear Representations in Terms of Independent Random Variables)" Markov Processes: Structure and Asymptotic Behavior, Springer 1971.

[66] R. Adler and P. Shields, (A) "Skew products of Bernoulli shifts with rotations", Israel J. Math. 12 (1972), 215-222; | MR | Zbl

R. Adler and P. Shields, (B) "Skew products of Bernoulli shifts with rotations, II", Israel J. Math. (to appear) | MR | Zbl