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Entropy 2016, 18(6), 220; doi:10.3390/e18060220

Zero Entropy Is Generic

Department of Mathematics, College of Natural Sciences, University of Texas at Austin, Austin, TX 78712, USA
Academic Editor: Tomasz Downarowicz
Received: 16 April 2016 / Revised: 19 May 2016 / Accepted: 31 May 2016 / Published: 4 June 2016
(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)
View Full-Text   |   Download PDF [319 KB, uploaded 4 June 2016]

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. View Full-Text
Keywords: entropy; measure-preserving actions entropy; measure-preserving actions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Bowen, L. Zero Entropy Is Generic. Entropy 2016, 18, 220.

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