Bernoulli scheme

In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.^[1][2] Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift.^[3] This is essentially the Markov partition. The term shift is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem^[4][5] shows that Bernoulli shifts are isomorphic when their entropy is equal.