Abstract
A consequence of Ornstein theory is that the infinite entropy flows associated with Poisson processes and continuous-time irreducible Markov chains on a finite number of states are isomorphic as measure-preserving systems. We give an elementary construction of such an isomorphism that has an additional finitariness property, subject to the additional conditions that the Markov chain has a uniform holding rate and a mixing skeleton.
Similar content being viewed by others
References
K. Ball, Poisson thinning by monotone factors, Electronic Communications in Probability 10 (2005), 60–69.
R. Burton, M. Keane and J. Serafin, Residuality of dynamical morphisms, Colloquium Mathematicum 85 (2000), 307–317.
J. Feldman and M. Smorodinsky, Bernoulli flows with infinite entropy, Annals of Mathematical Statistics 42 (1971), 381–382.
N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Mathematics 5 (1970), 365–394 (1970).
G. Gallavotti and D. S. Ornstein, Billiards and Bernoulli schemes, Communications in Mathematical Physics 38 (1974), 83–101.
A. E. Holroyd, R. Lyons and T. Soo, Poisson splitting by factors, Annals of Probability 39 (2011), 1938–1982.
A. E. Holroyd, R. Pemantle, Y. Peres and O. Schramm, Poisson matching, Annales de l'Institut Henri Poincaré Probabilités et Statistiques 45 (2009), 266–287.
S. Kalikow and B. Weiss, Explicit codes for some infinite entropy Bernoulli shifts, Annals of Probability 20 (1992), 397–402.
A. Katok, Fifty years of entropy in dynamics: 1958–2007, Journal of Modern Dynamics 1 (2007), 545–596.
Y. Katznelson, Ergodic automorphisms of Tn are Bernoulli shifts, Israel Journal of Mathematics 10 (1971), 186–195.
M. Keane and M. Smorodinsky, A class of finitary codes, Israel Journal of Mathematics 26 (1977), 352–371.
M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Annals of Mathematics 109 (1979), 397–406.
M. Keane and M. Smorodinsky, Finitary isomorphisms of irreducible Markov shifts, Israel Journal of Mathematics 34 (1979), 281–286.
A. N. Kolmogorov, Entropy per unit time as a metric invariant of automorphisms, Doklady Akademii Nauk SSSR 124 (1959), 754–755.
D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Mathematics 4 (1970), 337–352.
D. Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Advances in Mathematics 5 (1970), 349–364.
D. S. Ornstein, Imbedding Bernoulli shifts in flows, in Contributions to Ergodic Theory and Probability, Lecture Notes in Mathematics, Vol. 160, Springer, Berlin-Heidelberg, 1970, pp. 178–218.
D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Mathematics 5 (1970), 339–348.
D. S. Ornstein, The isomorphism theorem for Bernoulli flows, Advances in Mathematics 10 (1973), 124–142.
D. S. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, Yale Mathematical Monographs, Vol. 5. Yale University Press, New Haven, CT-London, 1974.
D. Ornstein, Newton's laws and coin tossing, Notices of the American Mathematical Society 60 (2013), 450–459.
D. S. Ornstein and P. C. Shields, Mixing Markov shifts of kernel type are Bernoulli, Advances in Mathematics 10 (1973), 143–146.
D. S. Ornstein and B. Weiss, Statistical properties of chaotic systems, Bulletin of the American Mathematical Society 24 (1991), 11–116.
D. S. Ornstein and B. Weiss, Geodesic flows are Bernoullian, Israel Journal of Mathematics 14 (1973), 184–198.
D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d'Analyse Mathématique 48 (1987), 1–141.
D. J. Rudolph, A characterization of those processes finitarily isomorphic to a Bernoulli shift, in Ergodic Theory and Dynamical Systems, I (College Park, MD, 1979–80), Progress in Mathematics, Vol. 10, Birkhäuser, Boston, MA, 1981, pp. 1–64.
D. J. Rudolph, A mixing Markov chain with exponentially decaying return times is finitarily Bernoulli, Ergodic Theory and Dynamical Systems 2 (1982), 85–97.
Ja. Sinaĭ, On the concept of entropy for a dynamic system, Doklady Akademii Nauk SSSR 124 (1959), 768–771.
T. Soo and A. Wilkens, Finitary isomorphisms of Poisson point processes, Annals of Probability, to appear, https://doi.org/abs/1805.04600.
S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, Vol. 180, Springer-Verlag, New York, 1998.
B. Weiss, The isomorphism problem in ergodic theory, Bulletin of the American Mathematical Society 78 (1972), 668–684.
Acknowledgements
I thank Zemer Kosloff and Amanda Wilkens for their helpful conversations. I also thank the referee for carefully reviewing this article and providing insightful suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Funded in part by a General Research Fund.
Rights and permissions
About this article
Cite this article
Soo, T. Finitary isomorphisms of some infinite entropy Bernoulli flows. Isr. J. Math. 232, 883–897 (2019). https://doi.org/10.1007/s11856-019-1890-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1890-6