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.
Highlights
Entropy theory in dynamics has recently been extended from actions of the integers to actions of sofic groups [1] and arbitrary countable groups [2,3,4]
The claim is that the subset of all transformations T ∈ Aut( X, μ) that have zero entropy contain a dense Gδ subset, so that it is residual in the sense of the Baire category
The first example of this phenomenon is due to Ornstein and Weiss [12]; they showed that the two-shift over the rank two free group factors onto the four-shift
Summary
Entropy theory in dynamics has recently been extended from actions of the integers (and, more generally, amenable groups) to actions of sofic groups [1] and arbitrary countable groups [2,3,4]. The first example of this phenomenon is due to Ornstein and Weiss [12]; they showed that the two-shift over the rank two free group factors onto the four-shift This was generalized in several ways: Ball proved that if Γ is any nonamenable group, there exists some probability space (K, κ ) with |K | < ∞ such that the Bernoulli shift Γy(K, κ )Γ factors onto all Bernoulli shifts over. If Γ is nonamenable and Γy( X, μ) is essentially free, ergodic and e) with zero Rokhlin entropy that probability-measure-preserving, there exists an action Γy( X, μ extends Γy( X, μ). A standard argument shows that, since Γy( X, μ) is a factor of a zero entropy action, it is a limit of zero entropy actions (see Lemma 8), proving that zero entropy actions are dense
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