# Importance of Ornstein's isomorphism theorem

"Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B_{ct}$ , or $B_{t}^{\infty}$) because it allows us to prove that certain concrete systems are Bernoulli."[Quote from scholarpedia][1] [1]: http://www.scholarpedia.org/article/Ornstein_theory

Firstly, can you mention some profound applications and a bit of the intuition of the power of this theorem?

So say I have a dynamical system with action T isomorphic (by $\phi$) to a Bernouli system with action S, then $S(x)=\phi(T)(x)$. Any interesting examples with explicit $\phi$?

Can you mention some interesting applications of this theorem especially in the field of Probability intersecting with Statistical mechanics models like SLE and Spin glasses?

Thanks

• You should probably read some of the earlier papers (Adler-Weiss,Ornstein-Weiss), where they've constructed the Markov partitions for Toral automorphisms and geodesic flows respectively. – Asaf Sep 19 '15 at 17:19