Abstract
We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abert–Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (ⅰ) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ⅱ) Bernoulli shifts have completely positive Rokhlin entropy; and (ⅲ) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.
Highlights
Let (X, μ) be a standard probability space, meaning X is a standard Borel space and μ is a Borel probability measure
Both statements are known to be true when G is a countably infinite sofic group since sofic entropy is a lower bound to Rokhlin entropy
In the case H(L, λ) = ∞ we obtain a result stronger than Theorem 1.10. This is surprising from a historical perspective, since when Kolmogorov defined entropy in 1958 he could only handle Bernoulli shifts with a finite Shannon entropy base [24, 25]
Summary
Let (X, μ) be a standard probability space, meaning X is a standard Borel space and μ is a Borel probability measure. Kerr that Bernoulli shifts over sofic groups have completely positive sofic entropy [20] Along these lines, we obtain the following corollary of Theorem 1.3. Both statements are known to be true when G is a countably infinite sofic group since sofic entropy is a lower bound to Rokhlin entropy. In the case H(L, λ) = ∞ we obtain a result stronger than Theorem 1.10 This is surprising from a historical perspective, since when Kolmogorov defined entropy in 1958 he could only handle Bernoulli shifts with a finite Shannon entropy base [24, 25]. (ii) Every Bernoulli shift over any countably infinite group has completely positive Rokhlin entropy;. For convenience to the reader we summarize the implications we uncovered in the two lines below: INF ⇒ RBS ⇒ INV + CPE + GOT + KAP (∀G POS) ⇒ (∀G INF)
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