Bernoulli automorphism

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An automorphism of a measure space, which describes Bernoulli trials and their generalization — a sequence of independent trials with the same result and with the same probability distribution.

Let be the collection of all possible outcomes of a trial, and let the probability of the event be given by the measure ; for a countable set , denote its elements by and their probabilities by . The phase space of a Bernoulli automorphism is the direct product of a countable number of copies of the set , i.e. the points of the phase space are infinite sequences , where runs through the set of integers and each . The transformation consists in shifting all members of each sequence one place to the left: . The measure is defined as the direct product of a countable number of measures ; thus if is countable, then

In this case, the entropy of the Bernoulli automorphism is .

In ergodic theory, Bernoulli automorphisms (or, more exactly, the cascades generated by iteration of them) play the role of a standard example of a dynamical system, the behaviour of which displays statistical features. A Bernoulli automorphism is a -automorphism but there exist -automorphisms which are metrically non-isomorphic to a Bernoulli automorphism, even though many -automorphisms are metrically isomorphic to a Bernoulli automorphism. Two Bernoulli automorphisms are metrically isomorphic if and only if they have the same entropy . A Bernoulli automorphism is a quotient automorphism of any ergodic automorphism of a Lebesgue space with a larger entropy [2].


[1a] D. Ornstein, "Bernoulli shifts with the same entropy are isomorphic" Adv. Math. , 4 (1970) pp. 337–352
[1b] D.Ornstein, "A Kolmogorov automorphism that is not a Bernoulli shift" Matematika , 15 : 1 (1971) pp. 131–150 (In Russian)
[2] Ya.G. Sinai, "On weak isomorphism of transformations with invariant measure" Mat. Sb. , 63 (105) : 1 (1964) pp. 23–42 (In Russian)


A metric isomorphism between two Bernoulli automorphisms can be given by means of a finitary code ([a1], see also [a4]).


[a1] M. Keane, M. Smorodinsky, "Bernoulli schemes with the same entropy are finitarily isomorphic" Ann. of Math. , 109 (1979) pp. 397–406
[a2] M. Smorodinsky, "Ergodic theory, entropy" , Springer (1971)
[a3] D. Ornstein, "Ergodic theory, randomness, and dynamical systems" , Yale Univ. Press (1974)
[a4] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)
How to Cite This Entry:
Bernoulli automorphism. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article