# Bernoulli automorphism

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 .

How to Cite This Entry:
Bernoulli automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_automorphism&oldid=17876
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article