Krieger’s Finite Generator Theorem for Actions of Countable Groups Ii
نویسنده
چکیده
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 Abért–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.
منابع مشابه
Krieger’s Finite Generator Theorem for Actions of Countable Groups I
For an ergodic p.m.p. action G y (X,μ) of a countable group G, we define the Rokhlin entropy hRok G (X,μ) to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov– Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Un...
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