Proximality and equidistribution on the Furstenberg boundary

Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure

Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also establish a Ledrappier-Young type formula for its dimension. The general strategy of the proof is based on the argument given by Feng for the exact dimensionality of self-affine measures.

Equidistribution and measure rigidity under $\times p,\times q$

We show that equidistribution of irrational orbits on the unit circle implies Furstenberg's conjecture.

Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

Let X be a symmetric space of noncompact type and Γa lattice in the isometry group of X. We study the distribution of orbits of Γacting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in X and b in X(\infty), we investigate the distribution of the set {(yγ,bγ^{-1}):γ\inΓ} in X\times X(\infty). It is proved, in particular, that the orbits of Γin the Furstenberg boundary are equidistributed, and that the orbits of Γin X are equidistributed in ``sectors'' defined with respect to a Cartan decomposition. We also discuss an application to the Patterson-Sulliv…

Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups

Let $G$ be a connected nilpotent Lie group. Given probability-preserving $G$-actions $(X_i,Σ_i,μ_i,u_i)$, $i=0,1,...,k$, and also polynomial maps $ϕ_i:\mathbb{R}\to G$, $i=1,...,k$, we consider the trajectory of a joining $λ$ of the systems $(X_i,Σ_i,μ_i,u_i)$ under the `off-diagonal' flow \[(t,(x_0,x_1,x_2,...,x_k))\mapsto (x_0,u_1^{ϕ_1(t)}x_1,u_2^{ϕ_2(t)}x_2,...,u_k^{ϕ_k(t)}x_k).\] It is proved that any joining $λ$ is equidistributed under this flow with respect to some limit joining $λ'$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, r…

复分析与算子代数中Furstenberg的$\times 2 \times 3$猜想的表述形式

Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras

该研究针对遍历论核心开放问题Furstenberg的$\times 2 \times 3$猜想及两个相关命题，分别在复分析和算子代数框架下给出等价表述，形成九组三联猜想，明确了猜想在不同数学领域的对应形式。

有限字段上乘法群的动力学与Følner-Kloosterman求和

Dynamics of multiplicative groups over fields and Folner-Kloosterman sums

研究无限域上乘法群的动力学及Følner-Kloosterman求和的等分布性，揭示其乘法不对称性及组合结论。