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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: downar@pwr.wroc.pl, serafin@pwr.wroc.pl)
J. SERAFIN
Affiliation:
Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland (email: downar@pwr.wroc.pl, serafin@pwr.wroc.pl)
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Abstract

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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.

Information

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

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 (1977), unpublished manuscript.Google Scholar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