American Institute of Mathematical Sciences

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 \begin{document}$ C^2 $\end{document}-non-discrete subgroups of \begin{document}$ {\rm {Diff}}^{\omega} (S^1) $\end{document}.

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.

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.

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.

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.

Fix an integral Soddy sphere packing \begin{document}$ \mathscr{P} $\end{document}. Let \begin{document}$ \mathscr{B} $\end{document} be the set of all bends in \begin{document}$ \mathscr{P} $\end{document}. A number \begin{document}$ n $\end{document} is called represented if \begin{document}$ n\in \mathscr{B} $\end{document}, that is, if there is a sphere in \begin{document}$ \mathscr{P} $\end{document} with bend equal to \begin{document}$ n $\end{document}. A number \begin{document}$ n $\end{document} is called admissible if it is everywhere locally represented, meaning that \begin{document}$ n\in \mathscr{B}( \operatorname{mod} q) $\end{document} for all \begin{document}$ q $\end{document}. It is shown that every sufficiently large admissible number is represented.

We give effective estimates for the number of saddle connections on a translation surface that have length \begin{document}$ \leq L $\end{document} and are in a prescribed homology class modulo \begin{document}$ q $\end{document}. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur–Veech measure on the stratum.

We present a new proof of the following theorem of Benoist-Quint: Let \begin{document}$ G: = \operatorname{SO}^\circ(d, 1) $\end{document}, \begin{document}$ d\ge 2 $\end{document} and \begin{document}$ \Delta<G $\end{document} a cocompact lattice. Any orbit of a Zariski dense subgroup \begin{document}$ \Gamma $\end{document} of \begin{document}$ G $\end{document} is either finite or dense in \begin{document}$ \Delta \backslash G $\end{document}. While Benoist and Quint's proof is based on the classification of stationary measures, our proof is topological, using ideas from the study of dynamics of unipotent flows on the infinite volume homogeneous space \begin{document}$ \Gamma \backslash G $\end{document}.

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 \begin{document}$ \Gamma\backslash {{\rm{SL}}}_n(\mathbb{R}) $\end{document}, where \begin{document}$ \Gamma $\end{document} is either \begin{document}$ {{\rm{SL}}}_n(\mathbb{Z}) $\end{document} or a cocompact lattice. In the cocompact case, we obtain a result that implies density for almost-primes in horospherical flows where the number of prime factors is independent of basepoint, and in the space of lattices we show the density of almost-primes in abelian horospherical orbits of points satisfying a certain Diophantine condition. Along the way we give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, as well as an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows.

For any \begin{document}$ \mathrm{SL}(2, \mathbb{R}) $\end{document} invariant and ergodic probability measure on any stratum of flat surfaces, almost every flat surface has the property that its non-decreasing sequence of saddle connection lengths is uniformly distributed mod one.

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 [6], the second class contains all transformations realizable as diffeomorphisms using the untwisted Anosov–Katok method. An application of the main result will appear in a forthcoming paper [7] that shows that the diffeomorphisms of the torus are inherently unclassifiable up to measure-isomorphism. Other consequences include the existence of measure distal diffeomorphisms of arbitrary countable distal height.

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 \begin{document}$ \times p $\end{document}- and \begin{document}$ \times q $\end{document}-Cantor sets.