Keivan Mallahi-Karai
We will prove an S-arithmetic version of a theorem of Dani-Margulis on the convergence of ergodic averages of a given bounded continuous function, when the initial point is outside certain compact subsets of the singular set associated to the unipotent flow.
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Let $G$ be a Lie group and $Γ$ be a lattice in $G$. We introduce the notion of locally unipotent invariant measures on $G/Γ$. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $G/Γ$ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in \cite{shah1994limit}. The proof relies on Ratner's measure classification theorem, linearization technique for polynomial trajectories near singular sets and a twistin…
We prove analogs of the logarithm laws of Sullivan and Kleinbock-Margulis in the context of unipotent flows. In particular, we prove results for horospherical actions on homogeneous spaces $G/Γ$. We describe some relations with multi-dimensional diophantine approximation.
We consider a family of smooth perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$ which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of…
We prove an avoidance principle for expanding translates of unipotent orbits for some semisimple homogeneous spaces. In addition, we prove a quantitative isolation result of closed orbits and give an upper bound on the number of closed orbits of bounded volume. The proofs of our results rely on the construction of a Margulis function and the theory of finite dimensional representations of semisimple Lie groups.
We study shrinking targets problems for discrete time flows on a homogenous space $Γ\backslash G$ with $G$ a semisimple group and $Γ$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.
In honor of S.G.Dani's 65th birthday, we describe some of his many contributions to the theory and applications of dynamical systems on homogeneous spaces, with emphasis on unipotent flows.
Directional measure rigidity for time-changed first row unipotent actions
本文研究高秩阿贝尔幂零作用时间改变的定向流，在有限体积齐性流形上证明Ratner型刚性：定量余循环条件迫使流为代数且测度齐性，并给出紧致情形下光滑共轭的充要条件及精确渐近。
Quantitative behavior of unipotent flows and an effective avoidance principle
本文基于Dani-Margulis线性化方法，给出幂单轨道在算术商中接近齐次子簇时间的有效界，并证明长时间适度接近必导致非常接近另一子簇，为后续定量稠密性研究奠定基础。
Effective equidistribution of unipotent orbits in homogeneous spaces of $\SL(2,\R)\ltimes(\R^2)^{k}$
研究了$\SL(2,\R)\ltimes(\R^2)^{k}$齐性空间中单侧轨道的等分布性质，利用圆方法的delta符号版本证明了多项式有效渐近等分布结果。
Let G=SL(n,R) with n>5. We construct examples of lattices Gamma of G, subgroup A of the diagonal group and points x in G/Gamma such that the closure of the orbit Ax is not homogeneous but does not factors through the action of a one-parameter non-unipotent group. This contradicts a conjecture of Margulis.
We prove Sarnak's Möbius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $Γ\subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\inΓ\backslash G$ and every continuous, bounded function $f$ on $Γ\backslash G$, the sequence $f(xu^{n})$ cannot correlate with the Möbius function on average.
Effective equidistribution for some one parameter unipotent flows
本文针对 SL₂(ℂ) 及 SL₂(ℝ)×SL₂(ℝ) 的算术商空间，利用 Margulis 函数、关联几何工具和谱间隙，证明了 SL₂(ℝ) 幂幺子群轨道的有效均匀分布定理，并给出多项式误差率。
Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors
本文提出了一种多项式速率的有效等分布定理，针对单变量上三角子群在SL₂因子积中的等分布问题，通过引入中间群和障碍物分析来克服等分布的挑战。
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.
Let $G$ be a real algebraic group defined over $\mathbb{Q}$, $Γ$ be an arithmetic subgroup of $G$, and $T$ be a maximal $\mathbb{R}$-split torus. A trajectory in $G/Γ$ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which account for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in $T$, we study the existence of non-obvious divergent trajectories under its action in $G/Γ$. We get a sufficient condition for the existence of a non-obvious divergence trajectory i…
Polynomially effective equidistribution for certain unipotent subgroups in quotients of perfect Lie groups
本文证明了完美李群商中某些单位元子群轨道的有效等分布定理，并应用该结果得到不定二次型的有效Oppenheim猜想版本。
We classify measures on a homogeneous space which are invariant under a certain solvable subgroup and ergodic under its unipotent radical. Our treatment is independent of characteristic. As a result we get the first measure classification for the action of semisimple subgroups without any characteristic restriction.
We provide a new proof that the horocycle flow preserving the Margulis measure on a variable negative curvature surface is standard. This was first proved by Ratner. The main purpose of this note is to provide a simplified case of the arguments for Kakutani equivalence of unipotent flows on homogeneous spaces, which have similar but more complicated structures, as well as illustrate the versatility of the method by applying it to a non-homogeneous flow.
Effective multi-equidistribution for translates of unipotent flows and Central limit theorems in inhomogeneous Diophantine approximation
本文在非利ouville位移条件下，证明了非均匀迪奥菲安逼近的中心极限定理，通过将问题转化为同质空间上的流问题，结合Kim和Shi的结果，利用Björklund和Gorodnik的方法得出结论。
Consider an homogeneous space under a locally compact group G and a lattice in G. Then the lattice naturally acts on the homogeneous space. Looking at a dense orbit, one may wonder how to describe its repartition. One then adopt a dynamical point of view and compare the asymptotic distribution of points in the orbits with the natural measure on the space. In the setting of Lie groups and their homogeneous spaces, several results showed an equidistribution of points in the orbits using Ratner's rigidity of polynomial dynamics in homogeneous spaces. We address here this problem in the setting of…
We define globally hypoelliptic smooth $\mathbb R^k$ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $k=1$, strong global rigidity is conjectured for such actions by Greenfield-Wallach and Katok: every such action is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces \cite{FFRH}. In this paper we conjecture that among homogeneous $\mathbb R^k$ actions ($k\ge 2$) on homogeneous spaces globally hypoelliptic actions exist only on nilman…
This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, techniques from partially hyperbolic dynamics, and the geometry and pseudo-Riemannian structure of the homogeneous space.
Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
研究了在算术同构空间中格点的大翻译的有效等分布，应用于3×3整矩阵的渐近计数公式，改进了Eskin-Mozes-Shah的经典定理。
The purpose of this article is to describe and characterize the limit distributions of translates of a bounded open "piece of orbit" of a reductive subgroup on a space of S-arithmetic lattices. This is accomplished under a mild assumption of "analytic stability" on the sequence of translates. It is important to note however that it is not necessary to assume that the reductive subgroup or its centralizer are algebraic. Moreover, it is also not necessary to assume that the initial orbit is of finite measure or even closed. This article thus provides important generalizations on previously known…
We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.
We study invariant probability measures on the homogeneous space $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ for the action of subgroups of $\mathrm{SL}_n(\mathbb R)$ of the form $SF$ where $F$ is generated by one parameter unipotent groups and $S$ is a one parameter $\mathbb R$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains an element with only one eigenvalue less than one (counted with multiplicity) and others bigger than one we prove that all the $SF$ invariant and ergodic probability measures on $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ are h…
Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms
本文在绝对秩2的拟分裂几乎单线性代数群商空间中建立了带多项式误差率的幂幺轨道有效等分布定理，并据此给出不定三元二次型整点取值分布的定量结果，从而有效且定量地证明了Oppenheim猜想。
We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb R)\times\operatorname{SL}_2(\mathbb R)$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.
Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics
本文研究约化李群在有限维向量空间上的线性作用，以自然的群论或几何条件解决由数论齐次动力学问题转化而来的线性动力学问题。
Let G be a semisimple linear algebraic group defined over rational numbers, K be a maximal compact subgroup of its real points and Γ be an arithmetic lattice. One can associate a probability measure μ(H) on Γ\G for each subgroup H of G defined over Q with no non-trivial rational characters. As G acts on Γ\G from the right, we can push-forward this measure by elements from G. By pushing down these measures to Γ\G/K, we call them homogeneous. It is a natural question to ask what are the possible weak-* limits of homogeneous measures. In the non-divergent case this has been answered by Eskin--Moz…
Let $G/Γ$ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups $H$ of $G$ that is large enough, the orbits of $H$ on $G/Γ$ intersect nontrivially with a fixed compact set. As a consequence, we deduce finiteness result for totally geodesic submanifolds of arithmetic quotients of symmetric spaces that do not admit nontrivial deformation and with bounded volume. Our work generalizes previous work of Tomanov--Weiss and Oh on this topic.
We show that joinings of higher rank torus actions on S-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.
We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribut…
Nondivergence of Reductive group action on Homogeneous Spaces
研究了约当群在齐性空间上的非分歧性，给出了存在固定紧集与每个轨道相交的代数条件，并在实秩为一时补充了轨道返回单个紧集的条件。
In the present article we study the following problem. Let G be a linear algebraic group over Q, $Γ$ be an arithmetic lattice and H be an observable Q-subgroup. There is a H-invariant measure $μ_H$ supported on the closed submanifold $HΓ/Γ$. Given a sequence $g_n$ in G we study the limiting behavior of $(g_n)_*μ_H$. In the non-divergent case we give a rather complete classification. We further supplement this by giving criterion of non-divergence and prove non-divergence for arbitrary sequence $g_n$ for certain H. This work can be viewed as a natural extension of the work of Eskin--Mozes--Shah…
We prove measure rigidity for the action of (maximal) horospherical subgroups on homogeneous spaces obtained by quotient by a uniform (nonuniform) arithmetic lattices over a field of positive characteristic.
Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)
本文用任意特征下的代数方法证明阿贝尔簇拟幂等自同构多项式体积增长公式，并得到 Tate 空间作用、Néron–Severi 空间拟幂等性及 Jordan 块尺寸的新结果。
We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner's seminal work.
We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie and later reappeared as an illustrative special case of Ratner's work on the equidistribution of unipotent flows in homogeneous spaces. We also prove an interesting probabilistic result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, …
In this paper we prove that the space of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space is compact. As an application, we explain some consequences for the distribution of weakly special subvarieties of Shimura varieties.
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 $Γ\backslash G$.
Asymptotics of integral points, equivariant compactifications and equidistributions for homogeneous spaces
针对线性代数群在Q上的齐次簇整点渐近计数问题，借助幂零流测度刚性将其约化为等变双有理几何问题，证明了特定条件下齐次空间的强Hardy-Littlewood性质及整点与体积渐近性的关系，并给出具体例子说明方法。
Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces
研究多项式有界o-最小曲线在齐性空间中的等分布性质，证明非收缩曲线轨迹的等分布定理及相关测度的不变性。
We prove a quantitative equidistribution statement for certain adelic homogeneous subsets in positive characteristic. As an application, we describe a proof of property tau for arithmetic groups in this context.
Ergodic decompositions of geometric measures on Anosov homogeneous spaces
本文研究Anosov齐性空间上几何测度的遍历分解，给出Burger-Roblin测度的$N$-分解与Bowen-Margulis-Sullivan测度的$A$-分解，并强化了非平凡不变遍历Radon测度空间的分类结果。
Let $Γ$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $Γ$ is arithmetic. This answers a question of Reid for hyperbolic $n$-manifolds and, independently, McMullen for hyperbolic $3$-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics and our main result also admits a formulation in that language.
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral …
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