A new dynamical proof of the Shmerkin–Wu theorem

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ a &lt; b $\end{document}</tex-math></inline-formula> be multiplicatively independent integers, both at least <inline-formula><tex-math id="M2">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M3">\begin{document}$ A,B $\end{document}</tex-math></inline-formula> be closed subsets of <inline-formula><tex-math id="M4">\begin{document}$ [0,1] $\end{document}</tex-math></inline-formula> that are forward invariant under multiplication by <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> respectively, and let <inline-formula><tex-math id="M7">\begin{document}$ C : = A\times B $\end{document}</tex-math></inline-formula>. An old conjecture of Furstenberg asserted that any planar line <inline-formula><tex-math id="M8">\begin{document}$ L $\end{document}</tex-math></inline-formula> not parallel to either axis must intersect <inline-formula><tex-math id="M9">\begin{document}$ C $\end{document}</tex-math></inline-formula> in Hausdorff dimension at most <inline-formula><tex-math id="M10">\begin{document}$ \max\{\dim C,1\} - 1 $\end{document}</tex-math></inline-formula>. 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.</p>

A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system

<p style='text-indent:20px;'>We show the existence, over an arbitrary infinite ergodic <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-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.</p>

On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao

<p style='text-indent:20px;'>It is shown that in a class of counterexamples to Elliott's conjecture by Matomäki, Radziwiłł, and Tao [<xref ref-type="bibr" rid="b23">23</xref>] the Chowla conjecture holds along a subsequence.</p>

Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

<p style='text-indent:20px;'>We prove that for any partially hyperbolic diffeomorphism having neutral center behavior on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Möbius band or a plane.</p><p style='text-indent:20px;'>Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-map (for <inline-formula><tex-math id="M2">\begin{document}$ m&gt;0 $\end{document}</tex-math></inline-formula> large) of a non-transitive Anosov flow <inline-formula><tex-math id="M3">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula> on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation is given by a topological Anosov flow which is topologically equivalent to <inline-formula><tex-math id="M4">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula>. We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.</p>

Computing the Rabinowitz Floer homology of tentacular hyperboloids

<p style='text-indent:20px;'>We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids <inline-formula><tex-math id="M1">\begin{document}$ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $\end{document}</tex-math></inline-formula>. Using an embedding of a compact sphere <inline-formula><tex-math id="M2">\begin{document}$ \Sigma_0\simeq S^{2k-1} $\end{document}</tex-math></inline-formula> into the hypersurface <inline-formula><tex-math id="M3">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>, we construct a chain map from the Floer complex of <inline-formula><tex-math id="M4">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> to the Floer complex of <inline-formula><tex-math id="M5">\begin{document}$ \Sigma_0 $\end{document}</tex-math></inline-formula>. In contrast to the compact case, the Rabinowitz Floer homology groups of <inline-formula><tex-math id="M6">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> 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.</p>

Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{M} $\end{document}</tex-math></inline-formula> be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.</p>

Dynamics of transcendental Hénon maps III: Infinite entropy

<p style='text-indent:20px;'>Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.</p>

Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations

<p style='text-indent:20px;'>We extend the notion of Rauzy induction of interval exchange transformations to the case of toral <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation, i.e., <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action defined by rotations on a 2-torus. If <inline-formula><tex-math id="M3">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> denotes the symbolic dynamical system corresponding to a partition <inline-formula><tex-math id="M4">\begin{document}$ \mathscr{P} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula> is Cartesian on a sub-domain <inline-formula><tex-math id="M8">\begin{document}$ W $\end{document}</tex-math></inline-formula>, we express the 2-dimensional configurations in <inline-formula><tex-math id="M9">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> as the image under a <inline-formula><tex-math id="M10">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-dimensional morphism (up to a shift) of a configuration in <inline-formula><tex-math id="M11">\begin{document}$ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M12">\begin{document}$ \widehat{\mathscr{P}}|_W $\end{document}</tex-math></inline-formula> is the induced partition and <inline-formula><tex-math id="M13">\begin{document}$ \widehat{R}|_W $\end{document}</tex-math></inline-formula> is the induced <inline-formula><tex-math id="M14">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action on <inline-formula><tex-math id="M15">\begin{document}$ W $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We focus on one example, <inline-formula><tex-math id="M16">\begin{document}$ \mathscr{X}_{\mathscr{P}_0, R_0} $\end{document}</tex-math></inline-formula>, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift <inline-formula><tex-math id="M17">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, <inline-formula><tex-math id="M18">\begin{document}$ {\mathscr{P}}_0 $\end{document}</tex-math></inline-formula> is a Markov partition for the associated toral <inline-formula><tex-math id="M19">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M20">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>. It also implies that the subshift <inline-formula><tex-math id="M21">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> is uniquely ergodic and is isomorphic to the toral <inline-formula><tex-math id="M22">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M23">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.</p>

Horospherically invariant measures and finitely generated Kleinian groups

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Gamma &lt; {\rm{PSL}}_2( \mathbb{C}) $\end{document}</tex-math></inline-formula> be a Zariski dense finitely generated Kleinian group. We show all Radon measures on <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PSL}}_2( \mathbb{C}) / \Gamma $\end{document}</tex-math></inline-formula> 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 [<xref ref-type="bibr" rid="b18">18</xref>] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [<xref ref-type="bibr" rid="b2">2</xref>] and Calegari-Gabai [<xref ref-type="bibr" rid="b10">10</xref>].</p>