某些齐性空间上随机游走的多切片与有效均匀分布

Markov Random Walks on Homogeneous Spaces and Diophantine Approximation on Fractals

In a first part, using the recent measure classification results of Eskin--Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space $G/Γ$. Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of Simmons--Weiss, we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of gene…

Linear random walks on the torus

We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption.

随机的birational映射乘积：曲线预像的等分布

Random products of birational maps: Equidistribution of preimages of curves

本文研究了随机birational变换乘积对平面中曲线预像的等分布性质，基于Diller和Roeder最近的工作，探讨了Cremona群在所有平面模型的当前空间逆极限上的作用，并证明了随机有限支持行走的等分布性。

Equidistribution of Dense Subgroups on Nilpotent Lie Groups

Let $Γ$ be a dense subgroup of a simply connected nilpotent Lie group $G$ generated by a finite symmetric set $S$. We consider the $n$-ball $S_n$ for the word metric induced by $S$ on $Γ$. We show that $S_n$ (with uniform measure) becomes equidistributed on $G$ with respect to the Haar measure as n tends to infinity. We give rates and also prove the analogous result for random walk averages (i.e. the local limit theorem).

环面上的半单随机游走

Semisimple random walks on the torus

本文研究环面上的线性随机游走，在作用群Zariski闭包半单的假设下，证明了定量的等分布结论。

Equidistribution of Horocyclic Flows on Complete Hyperbolic Surfaces of Finite Area

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, …

Random walks in negative curvature and boundary representations

In this paper we establish a version of the Margulis Roblin equidistribution theorem's for harmonic measures. As a consequence a von Neumann type theorem is obtained for boundary actions and the irreducibility of the associated quasi-regular representations is deduced.

齐性空间上非退化随机游走的有效密度

Effective density of non-degenerate random walks on homogeneous spaces

本文在矩阵具有代数元素且生成稠密子群的条件下，证明齐性空间上随机游走的有效密度，核心是利用卷积算子的谱间隙实现从高维到有效均匀分布的过渡。

自相似测度像的Khintchine二分法与几乎随机游走

Khintchine dichotomy for images of self-similar measures and almost random walks

该研究在Bénard-He-Zhang成果基础上，证明自相似测度经一致C¹⁺ᵅ映射推前像的Khintchine二分法，还证明SL(d+1,ℝ)/SL(d+1,ℤ)上几乎随机游走的有效均匀分布。

关于模空间上随机游走的收敛性

On Convergence of Random Walks on Moduli Space

本文证明了阿贝尔微分模空间上随机游走在两种模式下的收敛性，包括n步分布收敛和几乎必然路径等分布，推广了齐次空间上的已有结果。

分形地毯的测度刚性与等分布

Measure rigidity and equidistribution for fractal carpets

研究由有限个收缩有理仿射膨胀生成的随机游走的平稳伯努利测度，证明在无质量逃逸条件下，沿发散对角矩阵序列的推进测度等分布，并推广到加权坏逼近向量和Dirichlet可改进向量的零测性质。

Random walks with bounded first moment on finite-volume spaces

Let $G$ be a real Lie group, $Λ\leq G$ a lattice, and $Ω=G/Λ$. We study the equidistribution properties of the left random walk on $Ω$ induced by a probability measure $μ$ on $G$. It is assumed that $μ$ has a finite first moment, and that the Zariski closure of the group generated by the support of $μ$ in the adjoint representation is semisimple without compact factors. We show that for every starting point $x\in Ω$, the $μ$-walk with origin $x$ has no escape of mass, and equidistributes in Cesàro averages toward some homogeneous measure. This extends several fundamental results due to Benoist…

环面上的仿射随机游走

Affine random walks on the torus

该数学研究针对环面上的仿射随机游走，在线性部分生成群的Zariski闭包满足强不可约等条件下，给出等分布速率定量估计，证明其依律等分布于Haar测度当且仅当不陷入有限轨道。

幂零李群上随机游走的局部极限定理

Local limit theorems for random walks on nilpotent Lie groups

该研究建立了任意单连通幂零李群上非中心化独立同分布随机变量乘积的非格局部极限定理，改进了中心化情形的相关结果，并获得了等分布定理与Choquet-Deny性质新证明等应用。

Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…

扩张测度：齐性空间上的随机游走与刚性

Expanding measures: Random walks and rigidity on homogeneous spaces

本文研究齐性空间上的扩张随机游走，通过构造 Lyapunov/Margulis 函数证明平稳测度齐性、轨迹等分布及轨道闭包描述，并将 Birkhoff 通用性推广到非共形加权分形上的丢番图逼近。

亏格2模空间上极限圆轨道的分离

Separation of horocycle orbits on moduli space in genus 2

本文通过构造 Margulis 函数度量极限圆轨道横截分离的分形维数，证明了亏格2阿贝尔微分模空间上极限圆流的定量闭合引理，并由此推出小横截分离蕴含附近存在 Veech 群含伪阿诺索夫元素的点。