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 [
Fix an integral Soddy sphere packing
We give effective estimates for the number of saddle connections on a translation surface that have length
We present a new proof of the following theorem of Benoist-Quint: Let
An integer is called almost-prime if it has fewer than a fixed number of prime factors. In this paper, we study the asymptotic distribution of almost-prime entries in horospherical flows on
The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings that preserve underlying timing factors. The classes are canonically isomorphic by a continuous map that takes synchronous and anti-synchronous factor maps to synchronous and anti-synchronous factor maps, synchronous and anti-synchronous measure-isomorphisms to synchronous and anti-synchronous measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. The first class includes all finite entropy ergodic transformations that have an odometer factor. By results in [
We review the impact of Mike Hochman's work on mutlidimensional symbolic dynamics and Borel dynamics.
M. Hochman's work on the dimension of self-similar sets has given impetus to resolving other questions regarding fractal dimension. We describe Hochman's approach and its influence on the subsequent resolution by P. Shmerkin of the conjecture on the dimension of the intersection of
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