Journal of Modern Dynamics最新文献

Let begin{document}$ a < b $end{document} be multiplicatively independent integers, both at least begin{document}$ 2 $end{document}. Let begin{document}$ A,B $end{document} be closed subsets of begin{document}$ [0,1] $end{document} that are forward invariant under multiplication by begin{document}$ a $end{document}, begin{document}$ b $end{document} respectively, and let begin{document}$ C : = Atimes B $end{document}. An old conjecture of Furstenberg asserted that any planar line begin{document}$ L $end{document} not parallel to either axis must intersect begin{document}$ C $end{document} in Hausdorff dimension at most begin{document}$ max{dim C,1} - 1 $end{document}. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.

Let begin{document}$ Gamma < {rm{PSL}}_2( mathbb{C}) $end{document} be a Zariski dense finitely generated Kleinian group. We show all Radon measures on begin{document}$ {rm{PSL}}_2( mathbb{C}) / Gamma $end{document} which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [18] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [2] and Calegari-Gabai [10].

We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids begin{document}$ Sigmasimeq S^{n+k-1}timesmathbb{R}^{n-k} $end{document}. Using an embedding of a compact sphere begin{document}$ Sigma_0simeq S^{2k-1} $end{document} into the hypersurface begin{document}$ Sigma $end{document}, we construct a chain map from the Floer complex of begin{document}$ Sigma $end{document} to the Floer complex of begin{document}$ Sigma_0 $end{document}. In contrast to the compact case, the Rabinowitz Floer homology groups of begin{document}$ Sigma $end{document} are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where begin{document}$ k $end{document} critical points bifurcate independently, with begin{document}$ k $end{document} up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior. As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.

It is shown that in a class of counterexamples to Elliott's conjecture by Matomäki, Radziwiłł, and Tao [23] the Chowla conjecture holds along a subsequence.

We prove a rigidity result for cocycles from higher rank lattices to begin{document}$ mathrm{Out}(F_N) $end{document} and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let begin{document}$ G $end{document} be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let begin{document}$ G curvearrowright X $end{document} be an ergodic measure-preserving action on a standard probability space, and let begin{document}$ H $end{document} be a torsion-free hyperbolic group. We prove that every Borel cocycle begin{document}$ Gtimes Xto mathrm{Out}(H) $end{document} is cohomologous to a cocycle with values in a finite subgroup of begin{document}$ mathrm{Out}(H) $end{document}. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every homomorphism from begin{document}$ G $end{document} to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image.

The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.

We show the existence, over an arbitrary infinite ergodic begin{document}$ mathbb{Z} $end{document}-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.