Weyl chamber flow on irreducible quotients of products of PSL(2,R)

Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of $\text{PSL}(2,\mathbb R)$, with appendices

In this note, we consider the orbits $\{pu(n^{1+γ})|n\in\mathbb N\}$ in $Γ\backslash\text{PSL}(2,\mathbb R)$, where $Γ$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $u(t)$ is the standard unipotent group in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on the intial point $p$, we can prove that $\{pu(n^{1+γ})|n\in\mathbb N\}$ is equidistributed in $Γ\backslash\text{PSL}(2,\mathbb R)$ for small $γ>0$, which generalizes a result of Venkatesh (Ann.of Math. 2010). We will compute Hausdorff dimensions of subsets of non-Diophantine points in Appendix A, using results of latti…

关于$PSL_2(\mathbb{R})$的商空间中多项式序列的等分布

On equidistribution of polynomial sequences in quotients of $PSL_2(\mathbb{R})$

该研究证明$PSL_2(\mathbb{R})$中任意格对应的商空间上，带指数$1+\gamma$（$\gamma$小于某正常数）的horocycle流多项式序列等分布，推广了余紧格情形的已有结果，推进了Shah猜想的研究。

Weyl 腔空间中周期环面的均匀分布与计数

Equidistribution and counting of periodic tori in the space of Weyl chambers

本文研究半单李群中正则 Weyl 腔流的周期环面，证明了其以指数速度均匀分布于 Haar 测度商空间，并由此推导出高秩素数测地线定理。

Weyl腔流元素附近的中心化子刚性

Local Centralizer Rigidity near Elements of the Weyl Chamber Flow

本文证明了半单李群上Weyl腔流元素附近的中心化子刚性，即体积保持扰动要么中心化子维数为0或1，要么光滑共轭于Weyl腔流元素，并给出了部分双曲微分同胚中心化子为李群的一般条件。

Equidistribution and counting of periodic flat tori

Let $G$ be a semisimple Lie group without compact factor and $Γ< G$ 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 Weyl chambers. We prove that these flat periodic tori equidistribute exponentially fast towards the quotient of the Haar measure. From the equidistribution formula, we deduce a higher rank prime geodesic theorem. These counting and equidistribution results also hold in the non cocompact, finite covolume case for $G=\mathrm{SL}(d,\mathbb{R})$ and $Γ<\mathrm{SL}(d,…

On Discrete Subgroups of automorphism of $P^2_C$

We study the geometry and dynamics of discrete subgroups $Γ$ of $\PSL(3,\mathbb{C})$ with an open invariant set $Ω\subset \PC^2$ where the action is properly discontinuous and the quotient $Ω/Γ$ contains a connected component whicis compact. We call such groups {\it quasi-cocompact}. In this case $Ω/Γ$ is a compact complex projective orbifold and $Ω$ is a {\it divisible set}. Our first theorem refines classical work by Kobayashi-Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify…