Related papers: Expanding curves in $\mathrm{T}^1(\mathbb{H}^n)$ u…

Given a homogeneous space $X = G/\Gamma$ with $G$ containing the group $H = (\mathrm{SO}(n,1))^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an analytic curve $\phi: I=[a,b] \rightarrow H$, we will show that if $\phi$ satisfies…

In this article, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(n\times n, \mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric conditions, then for almost every…

In this paper, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric condition, then for almost every…

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is…

Given a lattice \Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of \Gamma on the homogeneous space V=H\G. For an increasing family of finite subsets {\Gamma_T: T>0}, a dense orbit v\Gamma, v\in V,…

Let $H = \mathrm{SO}(n,1)$ and $A = \{a(t): t \in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $\Gamma \subset H \times H$ be a nonuniform lattice in $H \times H$ and $X_{\Gamma} : = H \times H/ \Gamma$. Let…

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic…

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of $G=\mathrm{SO}(n_1,1)\times\cdots\times\mathrm{SO}(n_k,1)$, where $G$ acts on a finite volume homogeneous space…

We extend Ratner's theorem on equidistribution of individual orbits of unipotent flows on finite volume homogeneous spaces of Lie groups to trajectories of non-contracting curves definable in polynomially bounded o-minimal structures. To be…

We study geodesics of the form $\gamma(t)=\pi(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\pi:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous…

Let $H < G$ both be noncompact connected semisimple real algebraic groups where the former is maximal proper and $\Gamma < G$ be a lattice. Building on the work of Gorodnik-Weiss, we refine their techniques and obtain effective results.…

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact…

Let $G=SL_2(\mathbb R)^d$ and $\Gamma=\Gamma_0^d$ with $\Gamma_0$ a lattice in $SL_2(\mathbb R)$. Let $S$ be any "curved" submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an $\mathbb R$-diagonalizable…

Let $\Gamma$ be a lattice of a semisimple Lie group $L$. Suppose that one parameter Ad-diagonalizable subgroup $\{g_t\}$ of $L$ acts ergodically on $L/\Gamma$ with respect to the probability Haar measure $\mu$. For certain proper subgroup…

Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the quotient map. For a semi-algebraic…

Let $G$ be a semisimple Lie group and $\Gamma$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $\Gamma\backslash G$.

Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a Lie group $G$ and $\Gamma < G$ is a lattice. We study the dimension of bounded and divergent orbits of $g_t$ emanating from a class of curves lying on leaves of the…

Let $G$ be a semisimple Lie group without compact factor and $\Gamma < G$ a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on maximal flat periodic tori of the space of…

We prove effective equidistribution of horospherical flows in $\operatorname{SO}(n,1)^\circ / \Gamma$ when $\Gamma$ is geometrically finite and the frame flow is exponentially mixing for the Bowen-Margulis-Sullivan measure. We also discuss…

Let X be a symmetric space of noncompact type and \Gamma a lattice in the isometry group of X. We study the distribution of orbits of \Gamma acting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in…