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.

  1. ^ P. Shields, The theory of Bernoulli shifts, Univ. Chicago Press (1973)
  2. ^ Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). ISBN 0-19-853390-X
  3. ^ Pierre Gaspard, Chaos, scattering and statistical mechanics (1998), Cambridge University press
  4. ^ Ornstein, Donald (1970). "Bernoulli shifts with the same entropy are isomorphic". Advances in Mathematics. 4: 337–352. doi:10.1016/0001-8708(70)90029-0.
  5. ^ D.S. Ornstein (2001) [1994], "Ornstein isomorphism theorem", Encyclopedia of Mathematics, EMS Press