Minju Lee

Abstract Let $\Gamma $ 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 $\Gamma \backslash G$, 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 that normalizes $N$. One of the main ingredients is to

Let G be a connected semisimple real algebraic group, and Γ<G a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients ⟨(exptv). f1,f2⟩ in L2(Γ∖G) as t→∞ for any f1,f2Cc(Γ∖G) and any vector v in them interior of the limit cone of Γ. These asymptotics involve higher-rank analogues of Burger–Roblin measures, which are introduced in this paper. As an application, for any affine symmetric subgroup Hof G, we obtain a b

We establish an extension of the Hopf–Tsuji–Sullivan dichotomy to any Zariski dense discrete subgroup of a semisimple real algebraic group $ G $. We then apply this dichotomy to Anosov subgroups of $ G $, which surprisingly presents a different phenomenon depending on the rank of the ambient group $ G $.

Let $G$ be a connected semisimple real algebraic group, and $Γ

In the late seventies, Sullivan showed that, for a convex cocompact subgroup Γ of SO∘(n,1) with critical exponent δ>0, any Γ-conformal measure on ∂Hn 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 Γ 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∖G including Haar

We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\operatorname{SO}(d,1)$ acting on the space $\Gamma\backslash \operatorname{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash \mathbb H^d$ is a convex cocompact manifold with Fuchsian ends. For $d=3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of int

Let $G$ be a connected semisimple real algebraic group. We establish an analogue of the Hopf-Tsuji-Sullivan dichotomy for any regular Zariski dense discrete subgroup of $G$. We deduce the following surprising consequence for $\Gamma$ Anosov: the localized Poincare series diverges for each interior direction $v$ of the limit cone of $\Gamma$ if and only if $\operatorname{rank} G\le 3$. This result implies that the Burger-Roblin measure $m^{\operatorname{BR}}_{v}$ on $\Gamma\backslash G$ is suppor

Let $ G = \prod_{i = 1}^{\mathsf r} G_i $ be a product of simple real algebraic groups of rank one and $ \Gamma $ an Anosov subgroup of $ G $ with respect to a minimal parabolic subgroup. For each $ \mathsf v $ in the interior of a positive Weyl chamber, let $ \mathscr R_ \mathsf v\subset \Gamma\backslash G $ denote the Borel subset of all points with recurrent $ \exp (\mathbb R_+ \mathsf v) $-orbits. For a maximal horospherical subgroup $ N $ of $ G $, we show that the $ N $-action on $ {\maths

Let $G$ be a connected semisimple real algebraic group and $Γ$ a Zariski dense Anosov subgroup of $G$ with respect to a minimal parabolic subgroup $P$. Let $N$ be the maximal horospherical subgroup of $G$ given by the unipotent radical of $P$. We describe the $N$-ergodic decompositions of all Burger-Roblin measures as well as the $A$-ergodic decompositions of all Bowen-Margulis-Sullivan measures on $Γ\backslash G$. As a consequence, we obtain the following refinement of the main result of [LO]:

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 $NM$-erg

For any $d\geq 1$, we obtain counting and equidistribution results for tori with small volume for a class of $d$-dimensional torus packings, invariant under a self-joining $Γ_ρ<\prod_{i=1}^d\mathrm{PSL}_2(\mathbb{C})$ of a Kleinian group $Γ$ formed by a $d$-tuple of convex cocompact representations $ρ=(ρ_1, \cdots, ρ_d)$. More precisely, if $\mathcal P$ is a $Γ_ρ$-admissible $d$-dimensional torus packing, then for any bounded subset $E\subset \mathbb{C}^d$ with $\partial E$ contained in a pro

We establish an extension of the Hopf-Tsuji-Sullivan dichotomy to any Zariski dense discrete subgroup of a semisimple real algebraic group $G$. We then apply this dichotomy to Anosov subgroups of $G$, which surprisingly presents a different phenomenon depending on the rank of the ambient group $G$.

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> be a higher rank simple real algebraic group, or more generally, any semisimple real algebraic group with no rank one factors and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> the associated Riemannian symmetric space. For any Zariski dense discrete subgroup <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>&lt;</mml

We present a new proof of Benoist-Quint's finite or dense dichotomy for orbits of Zariski dense subgroups acting on the quotient space of SO(d,1) by a cocompact lattice. Our proof is topological. We use ideas from the study of dynamics of unipotent flows on homogeneous spaces of infinite volume. We also use Ratner's orbit closure theorem for a one-parameter unipotent semigroup action on homogeneous spaces of finite volume.