arXiv subjects

Hee Oh

Publications and source records attributed to Hee Oh.

At least 55 records · Page 3Linked to original sources

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↗

arXiv2022

Isolations of geodesic planes in the frame bundle of a hyperbolic $3$-manifold

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical exponents.

math.DS↗

arXiv2022

Closed geodesics and holonomies for Kleinian manifolds

For a rank one Lie group G and a Zariski dense and geometrically finite subgroup $Γ$ of G, we establish equidistribution of holonomy classes about closed geodesics for the associated locally symmetric space. Our result is given in a quantitative form for real hyperbolic geometrically finite manifolds whose critical exponents are big enough. In the case when G=PSL(2, C), our results can be interpreted as the equidistribution of eigenvalues of $Γ$ in the complex plane. When $Γ$ is a lattice, this result was proved by Sarnak and Wakayama in 1999.

math.DS↗

arXiv2022

Invariant measures for horospherical actions and Anosov groups

Let $Γ$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $Γ\backslash G$, up to proportionality, is homeomorphic to ${\mathbb R}^{\text{rank}\,G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup which normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger-Roblin measures.

math.DS↗

arXiv2022

Euclidean traveller in hyperbolic worlds

We will discuss all possible closures of a Euclidean line in various geometric spaces. Imagine the Euclidean traveller, who travels only along a Euclidean line. She will be travelling to many different geometric worlds, and our question will be "what places does she get to see in each world?". Here is the itinerary of our Euclidean traveller: In 1884, she travels to the torus of any dimension, guided by Kronecker. In 1936, she travels to the world, called a closed hyperbolic surface, guided by Hedlund. In 1991, she then travels to a closed hyperbolic manifold of higher dimension $n\ge 3$ guided by Ratner. Finally, she adventures into hyperbolic manifolds of infinite volume guided by Dal'bo in dimension $2$ in 2000, by McMullen-Mohammadi-Oh in dimension $3$ in 2016 and by Lee-Oh in all higher dimensions in 2019.

math.GT↗

arXiv2022

Uniqueness of conformal measures and local mixing for Anosov groups

In the late seventies, Sullivan showed that for a convex cocompact subgroup $Γ$ of $\operatorname{SO}^\circ(n,1)$ with critical exponent $δ>0$, any $Γ$-conformal measure on $\partial \mathbb{H}^n$ of dimension $δ$ is necessarily supported on the limit set $Λ$ and that the conformal measure of dimension $δ$ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$ of rank at most $3$. We also obtain the local mixing for generalized BMS measures on $Γ\backslash G$ including Haar measures.

math.DS↗

arXiv2021

Local mixing and invariant measures for horospherical subgroups on abelian covers

Abelian covers of hyperbolic $3$-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic $3$-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as well as to other counting and equidistribution problems. Our results are proved for any abelian cover of a homogeneous space $Γ_0\backslash G$ where $G$ is a rank one simple Lie group and $Γ_0<G$ is a convex cocompact Zariski dense subgroup.

math.DS↗

arXiv2021

Geodesic planes in the convex core of an acylindrical 3-manifold

Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$. In this paper we show that any geodesic plane in $M^*$ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theorems for planes in convex cocompact 3-manifolds of infinite volume that depend only on the topology of M.

math.DS↗

arXiv2021

Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

Let $\mathcal{M}$ 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.

math.DS↗

arXiv2021

Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds

Let $\mathcal{M}=Γ\backslash\mathbb{H}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2(\mathrm{T}^1(\mathcal{M}))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(\mathcal{M})$ due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varju on expanders, this implies uniform exponential mixing for congruence covers of $\mathcal{M}$ when $Γ$ is a Zariski dense subgroup contained in an arithmetic subgroup of $\mathrm{SO}^{\circ}(d+1,1)$.

math.DS↗

arXiv2019

Topological proof of Benoist-Quint's orbit closure theorem for SO(d,1)

We present a new proof of the following theorem of Benoist-Quint: Let $G:=SO^\circ(d,1)$, $d\ge 2$ and $Δ<G$ a cocompact lattice. Any orbit of a Zariski dense subgroup $Γ$ of $G$ is either finite or dense in $Δ\backslash G$. 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 $Γ\backslash G$.

math.DS↗

arXiv2019

Ergodicity of unipotent flows and Kleinian groups

Let M be a non-elementary convex cocompact hyperbolic 3 manifold and delta the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure provided that delta>1.

math.DS↗

arXiv2018

Geodesic planes in geometrically finite acylindrical 3-manifolds

Let $M$ be a geometrically finite acylindrical hyperbolic 3-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained earlier by McMullen, Mohammadi, and the second named author when M is convex cocompact. As a corollary we obtain that when $M$ covers an arithmetic hyperbolic 3-manifold $M_0$, the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$. We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.

math.DS↗

arXiv2017

Uniform congruence counting for Schottky semigroups in $\mathrm{SL}_2(\mathbf{Z})$

Let $Γ$ be a Schottky semigroup in $\mathrm{SL}_2(\mathbf{Z})$, and for $q\in \mathbf N$, let $Γ(q):=\{γ\in Γ: γ= e \text{ (mod $q$)}\}$ be its congruence subsemigroup of level $q$. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls $B_R$ in $M_2(\mathbf{R})$ of radius $R$: for all $q$ with no small prime factors, $ (Γ(q) \cap B_R )= c_Γ\frac{R^{2δ}}{ (\mathrm{SL}_2(\mathbf{Z}/q\mathbf{Z}))} +O(q^C R^{2δ-ε})$ as $R\to \infty$ for some $c_Γ>0, C>0, ε>0$ which are independent of $q$. Our technique also applies to give a similar counting result for the continued fractions semigroup of $\mathrm{SL}_2(\mathbf{Z})$, which arises in the study of Zaremba's conjecture on continued fractions.

math.NT↗

arXiv2016

Prime number theorems and holonomies for hyperbolic rational maps

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of $S^1$. We will show that these zeta functions are non-vanishing on a half plane $\Re(s) > δ- ε$, where $δ$ is the Hausdorff dimension of the Julia set of f.

math.DS↗

arXiv2015

Invariant Radon measures for Unipotent flows and products of Kleinian groups

Let G=PSL(2, F) where F= R or C, and consider the space Z=(Γ_1 x Γ_2)\ (G x G) where Γ_1<G is a co-compact lattice and Γ_2<G is a finitely generated discrete Zariski dense subgroup. The work of Benoist-Quint gives a classification of all ergodic invariant Radon measures on Z for the diagonal G-action. In this paper, for a horospherical subgroup N of G, we classify all ergodic, conservative, invariant Radon measures on Z for the diagonal N-action, under the additional assumption that Γ_2 is geometrically finite.

math.DS↗

arXiv2015

Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$

Let $Γ<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $Γ(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(Γ(q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term for all square-free $q$ with no small prime divisors. As an application we establish a uniform resonance-free half plane for the resolvent of the Laplacian on $Γ(q) \backslash \mathbb{H}^2$ over $q$ as above. Our approach is to extend Dolgopyat's dynamical proof of exponential mixing of the geodesic flow uniformly over congruence covers, by establishing uniform spectral bounds for congruence transfer operators associated to the geodesic flow. One of the key ingredients is the expander theory due to Bourgain-Gamburd-Sarnak.

math.DS↗

arXiv2015

Classification of joinings for Kleinian groups

We classify all locally finite joinings of a horospherical subgroup action on Γ\ G when Γis a Zariski dense geometrically finite subgroup of G=PSL_2(R) or PSL_2(C). This generalizes Ratner's 1983 joining theorem for the case when Γis a lattice in G. One of the main ingredients is equidistribution of non-closed horospherical orbits with respect to the Burger-Roblin measure which we prove in a greater generality where G is the connected component of the identity in SO(n,1) for n at least 2 and Γis any Zariski dense geometrically finite subgroup of G.

math.DS↗