View source for Bernoulli automorphism

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155801.png" /> be the collection of all possible outcomes of a trial, and let the probability of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155802.png" /> be given by the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155803.png" />; for a countable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155804.png" />, denote its elements by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155805.png" /> and their probabilities by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155806.png" />. The phase space of a Bernoulli automorphism is the direct product of a countable number of copies of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155807.png" />, i.e. the points of the phase space are infinite sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155808.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b0155809.png" /> runs through the set of integers and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558010.png" />. The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558011.png" /> consists in shifting all members of each sequence one place to the left: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558012.png" />. The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558013.png" /> is defined as the direct product of a countable number of measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558014.png" />; thus if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558015.png" /> is countable, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558016.png" /></td> </tr></table>

In this case, the [[Entropy|entropy]] of the Bernoulli automorphism is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558017.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558018.png" />-automorphism but there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558019.png" />-automorphisms which are metrically non-isomorphic to a Bernoulli automorphism, even though many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015580/b01558020.png" />-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 [[#References|[2]]].

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



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Keane,   M. Smorodinsky,   "Bernoulli schemes with the same entropy are finitarily isomorphic"  ''Ann. of Math.'' , '''109'''  (1979)  pp. 397–406</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Smorodinsky,   "Ergodic theory, entropy" , Springer  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Ornstein,   "Ergodic theory, randomness, and dynamical systems" , Yale Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I.P. [I.P. Kornfel'd] Cornfel'd,   S.V. Fomin,   Ya.G. Sinai,   "Ergodic theory" , Springer  (1982)  (Translated from Russian)</TD></TR></table>