Journal of Modern Dynamics
2019 , Volume 15
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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.
We prove a global topological rigidity theorem for locally
We study the phenomenon of robust bifurcations in the space of holomorphic maps of
We present the achievements of Lewis Bowen, or, more precisely, his breakthrough works after which a theory started to develop. The focus will therefore be made here on the isomorphism problem for Bernoulli actions of countable non-amenable groups which he solved brilliantly in two remarkable papers. Here two invariants were introduced, which led to many developments.
We prove that there are infinitely many linearly independent homogeneous quasimorphisms on
We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on
Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow
Using ideas due to Entov-Polterovich [
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