On Hausdorff dimension and cusp excursions for Fuchsian groups

Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu.

Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for $k = N-1$, the $k^{th}$ excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space $\mathbb{H}^N$ for any $N \geq 2…

Cusp Excursions on Parameter Spaces

We prove several results for dynamics of $SL(d, \R)$-actions on non-compact parameter spaces by studying associated discrete sets in Euclidean spaces. This allows us to give elementary proofs of logarithm laws for horocycle flows on hyperbolic surfaces and moduli spaces of flat surfaces. We also give applications to quantitative equidistribution and Diophantine approximation.

复连分数、Kleinian理论与尖点远足的极值理论

Complex continued fractions, Kleinian and extremal theory for cusp excursions

本文研究了五个复二次整数欧氏环上的复连分数算法，证明其最大数字服从Fréchet分布，并据此求出相关Bianchi轨形上尖点远足的极限分布。

尖点游荡、流形上的格点与Mizohata-Takeuchi猜想

Cusp Excursions, Lattice Points on Manifolds, and the Mizohata-Takeuchi Conjecture

利用动力系统和数的几何工具，证明了格点空间中尖点游荡的新对数律，并给出了流形附近格点的定量下界，进而为局部Mizohata-Takeuchi猜想提供了带显式误差项的新证明，并表明幂次损失在C^k中是普遍的。

尖点 Hitchin 分量的压力度量

Pressure metrics for cusped Hitchin components

本文研究尖点 Hitchin 分量，针对几何有限 Fuchsian 群到 PSL(d,R) 的表示构造压力度量，并在 d=3 时给出与 Hilbert 长度相关的新压力度量。

A dynamical approach to Maass cusp forms

For nonuniform cofinite Fuchsian groups $Γ$ which satisfy a certain additional geometric condition, we show that the Maass cusp forms for $Γ$ are isomorphic to 1-eigenfunctions of a finite-term transfer operator. The isomorphism is constructive.

除子的体积与双曲流形中测地线的尖点游走

The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds

本文通过将体积函数沿线段的行为与双曲测地线访问尖点相联系，完整描述了Wehler N-流形伪有效锥边界上体积函数的病态振荡行为，回答了Lazarsfeld的问题并反驳了Fujino的猜想。

极环流的极端事件

Extreme events for horocycle flows

本文研究了常负曲率曲面上极环流尖点游历的极端事件，通过研究收缩庞加莱截面的命中时间分布，推广了模曲面的极值定律，并证明极限定律可用 Farey 序列间隙分布的 Hall 公式表达。

Excursions of generic geodesics in right-angled Artin groups and graph products

Motivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group, and study its asymptotic distribution for right-angled Artin groups and graph products. In particular, for any irreducible right-angled Artin group we show that with respect to the counting measure, the maximal excursion of a generic geodesic in any flat tends to $\log n$, where $n$ is the length of the geodesic. In this regard, irreducible RAAGs behave like a free product of groups. In fact, we show that the asymptotic distribution of excurs…

A multifractal analysis for cuspidal windings on hyperbolic surfaces

In this paper we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp winding number. We will completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalise previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph di…

\(R^2\)中加权奇异向量的缓慢发散轨迹

Slowly Divergent Trajectories for Weighted Singular Vectors in R^2

研究\(R^2\)中加权奇异向量，证明满足特定条件的\(w\) - 奇异向量集豪斯多夫维数为\(s_w = 2 - (1 + w_1)^{-1}\)。对不同参数给出加权均匀逼近集维数，用自仿射构造等方法直接证明下包络定理。