Ergodicity and type of nonsingular Bernoulli actions

Abstract

We determine the Krieger type of nonsingular Bernoulli actions G g∈G ({0, 1}, µ g ). When G is abelian, we do this for arbitrary marginal measures µ g . We prove in particular that the action is never of type II ∞ if G is abelian and not locally finite, answering Krengel's question for G = Z. When G is locally finite, we prove that type II ∞ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II 1 or III 1 . When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group G admits a Bernoulli action of type III 1 if and only if G has nontrivial first L 2 -cohomology.

The orbital equivalence of Bernoulli actions and their Sinai factors

“…Let g ∈ G = g k k∈N and n := n(g ) ∈ N such that g = g n . We will show that the action is nonsingular by applying Kakutani's theorem on the equivalence of product measures; furthermore we will obtain that there exists K (λ) ≥ 6 λ such that…”

“…The question arises whether Theorem 8 holds when the product measure does not satisfy the Doeblin condition, as in the countable case. When A = {0, 1} the following question arising from [6] is still open. 2.2.1.…”

“…We give a positive (partial) answer to this question in Theorem 3 by producing type-III λ Bernoulli action of G for λ ∈ (0, 1). The case where λ = 1 is known from [6,57].…”

“…In a recent article of Björklund, Kosloff, and Vaes [6], they proved that in the very special case when G is a locally finite group, then for every λ ∈ (0, 1) there is a type-III λ Bernoulli action on {0, 1} G ; their construction makes use of the locally finite assumption and when applied to Bernoulli shifts of Z, and most of the other amenable groups, the resulting Bernoulli action is dissipative, hence not ergodic.…”

Some factors of nonsingular Bernoulli shifts

“…Let λ ∈ (0, 1). Let ρ and ν correspond to the type-III λ Bernoulli shifts as given in (5), where the marginals are defined on the disjoint sets [−1, 0) and [0, 1]. Then the measure…”