Bernoulli automorphism

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 $ A $ be the collection of all possible outcomes of a trial, and let the probability of the event $ B \subset A $ be given by the measure $ \nu $; for a countable set $ A $, denote its elements by $ a _ {i} $ and their probabilities by $ p _ {i} = \nu (a _ {i} ) $. The phase space of a Bernoulli automorphism is the direct product of a countable number of copies of the set $ A $, i.e. the points of the phase space are infinite sequences $ b = \{ b _ {k} \} $, where $ k $ runs through the set of integers and each $ b _ {k} \in A $. The transformation $ T $ consists in shifting all members of each sequence one place to the left: $ T \{ b _ {k} \} = \{ b _ {k+1 } \} $. The measure $ \mu $ is defined as the direct product of a countable number of measures $ \nu $; thus if $ A $ is countable, then

$$ \mu \{ {b } : {b _ {i _ {1} } = a _ {j _ {1} } \dots b _ {i _ {k} } = a _ {j _ {k} } } \} = \ p _ {j _ {1} } \dots p _ {j _ {k} } . $$

In this case, the entropy of the Bernoulli automorphism is $ - \sum p _ {i} \mathop{\rm log} p _ {i} $.

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 $ K $- automorphism but there exist $ K $- automorphisms which are metrically non-isomorphic to a Bernoulli automorphism, even though many $ K $- 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].

References

[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)

Comments

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

References

[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)