Zero entropy is generic

Lewis Bowen

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arXiv · 1603.02621

Zero entropy is generic

Lewis Bowen

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Abstract

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.

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Lewis Bowen. 2016-05-19. Zero entropy is generic. https://arxiv.org/abs/1603.02621

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