Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:30:14.516Z Has data issue: false hasContentIssue false
  • English
  • Français

A short proof of the Ornstein theorem

Published online by Cambridge University Press:  14 June 2011

T. DOWNAROWICZ  and
J. SERAFIN
T. DOWNAROWICZ
Affiliation:
Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland (email: [email protected], [email protected])
J. SERAFIN
Affiliation:
Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a short (and fairly elementary) proof of the ‘residual Sinai theorem’ of which the Ornstein theorem is an immediate consequence. In principle, we follow the lines of [BKS] but our proof avoids any substantial quotations, in particular, invoking any characterizations of systems isomorphic to Bernoulli shifts. The core of the proof is five pages long and it relies only on standard facts in ergodic theory.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 587 - 597
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[BKS]Burton, R. M., Keane, M. S. and Serafin, J.. Residuality of dynamical morphisms. Colloq. Math. 84/85 (2000), 307–317.Google Scholar
[BR]Burton, R. and Rothstein, A.. Isomorphism theorems in ergodic theory. (1977), unpublished manuscript.Google Scholar
[D]Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18). Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
[KS]Keane, M. and Smorodinsky, M.. Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109 (1979), 397–406.CrossRefGoogle Scholar
[O1]Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337–352.CrossRefGoogle Scholar
[O2]Ornstein, D.. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven, CT, 1974.Google Scholar
[P]Petersen, K.. Ergodic Theory. Cambridge University Press, Cambridge, 1983.Google Scholar
[R]Rudolph, D.. Fundamentals of Measurable Dynamics. Oxford University Press, Oxford, 1990.Google Scholar
[Sh]Shields, P.. The Theory of Bernoulli Shifts. University of Chicago Press, Chicago, IL, 1973.Google Scholar
[S]Sinai, Ya.. A weak isomorphism of transformations with invariant measure. Dokl. Akad. Nauk SSSR 147 (1962), 797–800.Google Scholar