Zero Entropy Is Generic

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.

Examples in the entropy theory of countable group actions

“…The next lemma is the key step. The proof given here is simpler than the one in [Bow16] (which was written before Theorem 5.2 was known).…”

Harmonic models and Bernoullicity

“…It is also used in Bowen's generalization of the Gaboriau–Lyons theorem to all Bernoulli shifts (see [Bow19, § 9]). It has applications to percolation theory, particularly in the study of random induced subgraphs of the Cayley graph (see, for example, [CI10, § 4], as well as [Hou12, Mar15]), and to the entropy theory of probability measure-preserving actions of general groups [Bow16].…”

“…If the acting group is assumed sofic, we can also say that any action with zero sofic entropy with respect to some sofic approximation is not a factor of a Bernoulli shift. By a recent result of Bowen [Bow16], this implies that a generic action of a sofic group is not a factor of a Bernoulli shift. Special to the sofic case, we can also exhibit actions that are inverse limits of Bernoulli shifts but have zero entropy and are thus not factors of a Bernoulli shift (see [Bow16, Corollary 4.4]).…”

“…By a recent result of Bowen [Bow16], this implies that a generic action of a sofic group is not a factor of a Bernoulli shift. Special to the sofic case, we can also exhibit actions that are inverse limits of Bernoulli shifts but have zero entropy and are thus not factors of a Bernoulli shift (see [Bow16, Corollary 4.4]). Moreover, if and if is a probability measure-preserving action which is not a factor of a Bernoulli shift over then the coinduction of to gives an action which is not a factor of a Bernoulli shift.…”

Krieger’s finite generator theorem for actions of countable groups III

“…Specifically, any free ergodic action of positive Rokhlin entropy factors onto all Bernoulli shifts of lesser or equal entropy. Rokhlin entropy has also been studied in [1,3,7,25].…”