Zero Entropy Is Generic

Dan Rudolph showed that for an amenable group, Γ, the generic measure-preserving action of Γ on a Lebesgue space has zero entropy. Here, this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in th...

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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Switzerland), 2016-06, Vol.18 (6), p.220-220
Main Author: Bowen, Lewis
Format: Article
Language:English
Subjects:
Entropy
measure-preserving actions
Theorem proving
Theorems
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Summary:Dan Rudolph showed that for an amenable group, Γ, the generic measure-preserving action of Γ on a Lebesgue space has zero entropy. Here, this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward's recent generalization of Sinai's Factor Theorem, the Gaboriau-Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other.
ISSN:1099-4300
1099-4300
DOI:10.3390/e18060220