Séminaire de Géométrie, Groupes et Dynamique

Brandon Seward (Courant Institute) : Bernoulli shifts with bases of equal entropy are isomorphic

Europe/Paris
Description
We prove that if G is a countably infinite group and (L, \lambda) and (K, \kappa) are probability spaces having equal Shannon entropy, then the Bernoulli shifts (L^G, \lambda^G) and (K^G, \kappa^G) are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both \lambda and \kappa have at least 3 points in their support. We furthermore produce finitary isomorphisms in the case where both L and K are finite.