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Bernoulli shifts with bases of equal entropy are isomorphic

The author was partially supported by ERC grant 306494 and Simons Foundation grant 328027 (P.I. Tim Austin)

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  • 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 $ G \curvearrowright (L^G, \lambda^G) $ and $ G \curvearrowright (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.

    Mathematics Subject Classification: Primary: 37A35; Secondary: 28D20.


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