Bernoulli scheme

<p>
In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Bernoulli scheme or Bernoulli shift is a generalization of the <a href="/facts/Bernoulli_process/AF9E3jpu">Bernoulli process</a> to more than two possible outcomes. Bernoulli schemes appear naturally in <a href="/facts/Symbolic_dynamics/kApuvAj8">symbolic dynamics</a>, and are thus important in the study of <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a>. Many important dynamical systems (such as <a href="/facts/Axiom_A_system/IHHhw0DP">Axiom A systems</a>) exhibit a <a href="/facts/Repellor/FAx5Db4w">repellor</a> that is the product of the <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a> and a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a>, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the <a href="/facts/Markov_partition/VfqVylWR">Markov partition</a>. The term <i>shift</i> is in reference to the <a href="/facts/Shift_operator/6c6lc0ks">shift operator</a>, which may be used to study Bernoulli schemes.  The <a href="/facts/Ornstein_isomorphism_theorem/fGlAFh0G">Ornstein isomorphism theorem</a> shows that Bernoulli shifts are isomorphic when their <a href="/facts/Kolmogorov_entropy/acIdlpqt">entropy</a> is equal.
</p>

Bernoulli scheme open-in-new

Bernoulli scheme