This note presents a proof that shift invariant measures with entropy ≥ c generate a Bernoulli shift, indicating foundational insights into measure theory.
Twenty years ago Dan Rudolph gave a seminar talk at the University of Maryland in which he discussed his discovery that if one restricts the shift invariant measures on a finite alphabet shift space to have entropy greater than or equal to c > 0, then the generic measure there defines a process that has entropy c and is isomorphic to a Bernoulli shift. He never published a proof of this theorem. In the following note we will present a proof of this result.
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Thouvenot et al. (2025) studied this question.
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