Criteria for 𝑑̄-continuity

Coelho, Zaqueu; Quas, Anthony

doi:10.1090/S0002-9947-98-01923-0

Criteria for $\bar {d}$-continuity
HTML articles powered by AMS MathViewer

by Zaqueu Coelho and Anthony N. Quas PDF

Trans. Amer. Math. Soc. 350 (1998), 3257-3268 Request permission

Abstract:

Bernoullicity is the strongest mixing property that a measure-theoretic dynamical system can have. This is known to be intimately connected to the so-called $\bar d$ metric on processes, introduced by Ornstein. In this paper, we consider families of measures arising in a number of contexts and give conditions under which the measures depend $\bar d$-continuously on the parameters. At points where there is $\bar d$-continuity, it is often straightforward to establish that the measures have the Bernoulli property.

References

Henry Berbee, Chains with infinite connections: uniqueness and Markov representation, Probab. Theory Related Fields 76 (1987), no. 2, 243–253. MR 906777, DOI 10.1007/BF00319986
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
Maury Bramson and Steven Kalikow, Nonuniqueness in $g$-functions, Israel J. Math. 84 (1993), no. 1-2, 153–160. MR 1244665, DOI 10.1007/BF02761697
R. L. Dobrušin, The problem of uniqueness of a Gibbsian random field and the problem of phase transitions, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 44–57 (Russian). MR 0250631
G. Gallavotti, Ising model and Bernoulli schemes in one dimension, Comm. Math. Phys. 32 (1973), 183–190. MR 356801
Hans-Otto Georgii, Gibbs measures and phase transitions, De Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. MR 956646, DOI 10.1515/9783110850147
G. R. Grimmett and D. R. Stirzaker, Probability and random processes, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1992. MR 1199812
Michael Keane, Strongly mixing $g$-measures, Invent. Math. 16 (1972), 309–324. MR 310193, DOI 10.1007/BF01425715
F. Ledrappier, Mesures d’équilibre sur un reseau, Comm. Math. Phys. 33 (1973), 119–128 (French, with English summary). MR 348077
R. A. Minlos and G. M. Natapov, Uniqueness of the limit Gibbs distribution in one-dimensional classical systems, Teoret. Mat. Fiz. 24 (1975), no. 1, 100–108 (Russian, with English summary). MR 489567
Donald S. Ornstein, Ergodic theory, randomness, and dynamical systems, Yale Mathematical Monographs, No. 5, Yale University Press, New Haven, Conn.-London, 1974. James K. Whittemore Lectures in Mathematics given at Yale University. MR 0447525
William Parry and Peter Walters, Endomorphisms of a Lebesgue space, Bull. Amer. Math. Soc. 78 (1972), 272–276. MR 294604, DOI 10.1090/S0002-9904-1972-12954-9
Anthony N. Quas, Non-ergodicity for $C^1$ expanding maps and $g$-measures, Ergodic Theory Dynam. Systems 16 (1996), no. 3, 531–543. MR 1395051, DOI 10.1017/s0143385700008956
Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631
David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
Paul Shields, The theory of Bernoulli shifts, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1973. MR 0442198
Peter Walters, Ruelle’s operator theorem and $g$-measures, Trans. Amer. Math. Soc. 214 (1975), 375–387. MR 412389, DOI 10.1090/S0002-9947-1975-0412389-8
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108