Searcharxiv⌕ Search

arXiv subjects

Or Landesberg

Publications and source records attributed to Or Landesberg.

8 recordsLinked to original sources
arXiv2025

Weaving Geodesics and New Phenomena in Horocyclic Dynamics

We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite conservative horocyclic invariant measures which are singular with respect to the geodesic flow. Other examples include surfaces supporting horocyclic orbit closures of arbitrary Hausdorff dimension in $(1,2)$.

math.DS↗
arXiv2024

Classification of horocycle orbit closures in $ \mathbb{Z} $-covers

We fully describe all horocycle orbit closures in $ \mathbb{Z} $-covers of compact hyperbolic surfaces. Our results rely on a careful analysis of the efficiency of all distance minimizing geodesic rays in the cover. As a corollary we obtain in this setting that all non-maximal horocycle orbit closures, while fractal, have integer Hausdorff dimension.

math.DS↗
arXiv2024

On denseness of horospheres in higher rank homogeneous spaces

Let $ G $ be a connected, semisimple real algebraic group and $Γ< G$ be a Zariski dense discrete subgroup. Let $N$ denote a maximal horospherical subgroup of $G$, and $P=MAN$ the minimal parabolic subgroup which is the normalizer of $N$. Let $\mathcal{E}$ denote the unique $P$-minimal subset of $Γ\backslash G$ and let $\mathcal{E}_0$ be a $P^\circ$-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary $ G/P $ and show that the following are equivalent for any $[g]\in \mathcal{E}_0$: (1) $gP\in G/P$ is a horospherical limit point; (2) $[g]NM$ is dense in $\mathcal{E}$; (3) $[g]N$ is dense in $\mathcal{E}_0$. The equivalence of (1) and (2) is due to Dal'bo in the rank one case. We also observe that unlike convex cocompact groups of rank one Lie groups, the $NM$-minimality of $\mathcal{E}$ does not hold in a general Anosov homogeneous space.

math.DS↗
arXiv2023

Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps

We expose a connection between distance minimizing laminations and horospherical orbit closures in $\mathbb{Z}$-covers of compact hyperbolic manifolds. For surfaces, we provide novel constructions of $\mathbb{Z}$-covers with prescribed geometric and dynamical properties, in which an explicit description of all horocycle orbit closures is given. We further show that even the slightest of perturbations to the hyperbolic metric on a $\mathbb{Z}$-cover can lead to drastic topological changes to horocycle orbit closures.

math.DS↗
arXiv2022

Horospherical invariant measures and a rank dichotomy for Anosov groups

Let $G=\prod_{i=1}^{r} G_i$ be a product of simple real algebraic groups of rank one and $Γ$ an Anosov subgroup of $G$ with respect to a minimal parabolic subgroup. For each $v$ in the interior of a positive Weyl chamber, let $\mathcal R_v\subsetΓ\backslash G$ denote the Borel subset of all points with recurrent $\exp (\mathbb R_+ v)$-orbits. For a maximal horospherical subgroup $N$ of $G$, we show that the $N$-action on ${\mathcal R}_v$ is uniquely ergodic if $r={rank}(G)\le 3$ and $v$ belongs to the interior of the limit cone of $Γ$, and that there exists no $N$-invariant {Radon} measure on $\mathcal R_v$ otherwise.

math.DS↗
arXiv2021

Horospherically invariant measures and finitely generated Kleinian groups

Let $ Γ< PSL_2(\mathbb{C}) $ be a Zariski dense finitely generated Kleinian group. We show all Radon measures on $ PSL_2(\mathbb{C}) / Γ$ 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 together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol and Calegari-Gabai.

math.DS↗
arXiv2020

On Radon measures invariant under horospherical flows on geometrically infinite quotients

We consider a locally finite (Radon) measure on $ SO^+(d,1)/ Γ$ invariant under a horospherical subgroup of $ SO^+(d,1) $ where $ Γ$ is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not observe any additional invariance properties then it must be supported on a set of points with geometrically degenerate trajectories under the corresponding contracting $ 1 $-parameter diagonalizable flow (geodesic flow).

math.DS↗