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Journal of Modern Dynamics

 2019 , Volume 15

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Krieger's finite generator theorem for actions of countable groups Ⅱ
Brandon Seward
2019, 15: 1-39 doi: 10.3934/jmd.2019012 +[Abstract](935) +[HTML](385) +[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.

Global rigidity of conjugations for locally non-discrete subgroups of $ {\rm {Diff}}^{\omega} (S^1) $
Anas Eskif and Julio C. Rebelo
2019, 15: 41-93 doi: 10.3934/jmd.2019013 +[Abstract](826) +[HTML](332) +[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}.

Lattès maps and the interior of the bifurcation locus
Sébastien Biebler
2019, 15: 95-130 doi: 10.3934/jmd.2019014 +[Abstract](619) +[HTML](175) +[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.

The 2017 Michael Brin Prize in Dynamical Systems
The Editors
2019, 15: 131-132 doi: 10.3934/jmd.2019015 +[Abstract](306) +[HTML](141) +[PDF](1902.58KB)
Abstract:
The work of Lewis Bowen on the entropy theory of non-amenable group actions
Jean-Paul Thouvenot
2019, 15: 133-141 doi: 10.3934/jmd.2019016 +[Abstract](258) +[HTML](137) +[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.

Entropy and quasimorphisms
Michael Brandenbursky and Michał Marcinkowski
2019, 15: 143-163 doi: 10.3934/jmd.2019017 +[Abstract](179) +[HTML](92) +[PDF](233.91KB)
Abstract:

Let \begin{document}$ S $\end{document} be a compact oriented surface. We construct homogeneous quasimorphisms on \begin{document}$ {\rm Diff}(S, \operatorname{area}) $\end{document}, on \begin{document}$ {\rm Diff}_0(S, \operatorname{area}) $\end{document}, and on \begin{document}$ {\rm Ham}(S) $\end{document}, generalizing the constructions of Gambaudo-Ghys and Polterovich.

We prove that there are infinitely many linearly independent homogeneous quasimorphisms on \begin{document}$ {\rm Diff}(S, \operatorname{area}) $\end{document}, on \begin{document}$ {\rm Diff}_0(S, \operatorname{area}) $\end{document}, and on \begin{document}$ {\rm Ham}(S) $\end{document} whose absolute values bound from below the topological entropy. In cases when \begin{document}$ S $\end{document} has a positive genus, the quasimorphisms we construct on \begin{document}$ {\rm Ham}(S) $\end{document} are \begin{document}$ C^0 $\end{document}-continuous.

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 \begin{document}$ {\rm Ham}(S) $\end{document} is unbounded.

Mather theory and symplectic rigidity
Mads R. Bisgaard
2019, 15: 165-207 doi: 10.3934/jmd.2019018 +[Abstract](80) +[HTML](32) +[PDF](395.93KB)
Abstract:

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow \begin{document}$ \phi_H $\end{document} of a Hamiltonian \begin{document}$ H\in C^{\infty}(M) $\end{document} on a symplectic manifold \begin{document}$ (M, \omega) $\end{document}. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures, and we construct an example showing that their support can be extremely unstable when \begin{document}$ H $\end{document} fails to be convex, even for nearly integrable \begin{document}$ H $\end{document}. Parts of these results extend work by Viterbo [54] and Vichery [52].

Using ideas due to Entov-Polterovich [22,40], we also detect interesting invariant measures for \begin{document}$ \phi_H $\end{document} by studying a generalization of the symplectic shape of sublevel sets of \begin{document}$ H $\end{document}. This approach differs from the first one in that it works also for \begin{document}$ (M, \omega) $\end{document} in which every compact subset can be displaced. We present applications to Hamiltonian systems on \begin{document}$ \mathbb R^{2n} $\end{document} and twisted cotangent bundles.

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