# 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 $A$ 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)