# American Institute of Mathematical Sciences

ISSN:
1930-5311

eISSN:
1930-532X

All Issues

## Journal of Modern Dynamics

2019 , Volume 15

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2019, 15: 1-39 doi: 10.3934/jmd.2019012 +[Abstract](444) +[HTML](270) +[PDF](347.81KB)
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 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.

2019, 15: 41-93 doi: 10.3934/jmd.2019013 +[Abstract](350) +[HTML](220) +[PDF](432.62KB)
Abstract:

We prove a global topological rigidity theorem for locally \begin{document}$C^2$\end{document}-non-discrete subgroups of \begin{document}${\rm {Diff}}^{\omega} (S^1)$\end{document}.

2019, 15: 95-130 doi: 10.3934/jmd.2019014 +[Abstract](162) +[HTML](53) +[PDF](462.94KB)
Abstract:

We study the phenomenon of robust bifurcations in the space of holomorphic maps of \begin{document}$\mathbb{P}^2(\mathbb{C})$\end{document}. We prove that any Lattès example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattès map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in \begin{document}$\mathbb{C}^2$\end{document} with a well-oriented complex curve. Then we show that any Lattès map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry.

2019, 15: 131-132 doi: 10.3934/jmd.2019015 +[Abstract](28) +[HTML](9) +[PDF](1902.58KB)
Abstract:
2019, 15: 133-141 doi: 10.3934/jmd.2019016 +[Abstract](15) +[HTML](6) +[PDF](151.84KB)
Abstract:

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.

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