测地线到hyperbolic Riemann surface子集的偏移时间和返回时间

Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials

For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. In the Teichmuller setting, we consider invariant measures for the SL(2,R) action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani, these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teic…

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…

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。

测地流在θ群商上的极值定理

Extreme value theorem for geodesic flow on the quotient of the theta group

本文研究了θ群商上测地流的极值定理，通过广义连续分数算法和谱性质推导出极值定律，描述测地线的最大咽喉游离。

Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

Let $\mathcal{M}$ be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.

Weil--Petersson geodesics on the modular surface

We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow …

闭hyperbolic Riemann surfaces上的闭磁测地线

Closed magnetic geodesics on closed hyperbolic Riemann surfaces

证明了在闭双曲曲面上Alexandrov嵌入的闭磁测地线的存在性。闭合磁测地线对应于具有规定的测地曲率的闭曲线。

黎曼流形上测地线的计数与测地流的拓扑熵

Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows

设$M$是紧$C^{\infty}$黎曼流形.给定$M$和$T>0$中的$p$和$q$，将$n_{T}(p,q)$定义为连接$p$和$q$且长度为$\leq T$的测地线线段的数量。Mañé表明，指数增长率的积分$n_{T}(p,q)$超过$M \times M$是拓扑熵的测地线流的$M$。在本文中，我们展示了一个开放的度量集上的两个球的指数增长率的$n_{T}(p,q$是小于拓扑熵的测地线流的$(p,q)\in M\times M$的正测度集。这回答了Mañé提出的否定问题。

Expansiveness for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature

We study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is positive and negative separating in the sense of Gura. In this paper, we give a new proof of the expansiveness in the sense of Bowen-Walters for the geodesic flow and show that the horocycle flow is positive and negative kinematic expensive in the sense of Artigue as well as expansive in the sense of Katok-Hasselblatt but not expensive in the sense of Bowen-Walt…

Jacquet-Langlands对应在Riemann surfaces上测地流迁移算子中的应用

An application of Jacquet-Langlands correspondence to transfer operators for geodesic flows on Riemann surfaces

作为Jacquet-Langlands对应的一个新应用，本文通过比较对应的Selbergzeta函数，将不同余有限富克斯群的转移算子联系起来.

Besicovitch-Federer投影定理与Riemann surfaces上的测测地流

Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces

本文将Besicovitch-Federer投影定理推广到横截映射族。作为应用，我们证明了在一类具有常负曲率和边界的Riemann曲面上，存在在具有二维支集的测地流下不变的自然二维测度，使得它们在基流形上的投影是二维的，但投影的支集是Lebesgue可忽略的.特别地，完全测地线的并集具有豪斯多夫维数2，并且是Lebesgue可忽略的。

Markov partitions for the geodesic flow on compact Riemann surfaces of constant negative curvature

It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are very abstract and not easy to understand. To establish a more detailed understanding of Markov partitions, in this paper we consider the geodesic flow on Riemann surfaces of constant negative curvature. We provide a rigorous construction of Markov partitions for this hyperbolic flow with explicit forms of rectangles and local cross sections. The local product structure is also calculated in detail.

Prescribing the behavior of Weil-Petersson geodesics in the moduli space of Riemann surfaces

We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of geodesics is rich enough to provide for examples of closed WP geodesics in the thin part of the moduli space, as well as divergent WP geodesic rays with minimal filling ending lamination. Some ingredients of independent interest are the following: A strength version of Wolpert's Geodesic Limit Theorem proved in Sec.4. The stability of hierarchy resolution paths bet…

伪Riemannian Geometry中的Wadsley定理

Wadsley's Theorem in pseudo-Riemannian Geometry

本文用闭非类光测地线证明了叶理的Wadsley定理。作为一个应用，我们证明了每个伪黎曼和非黎曼2-流形，其所有的类时测地线或类空测地线都是封闭的，都与$S^1\times \R$同构。进一步我们证明了每一个指数为1的伪黎曼2-流形都包含一条非闭类时或类空测地线。

闭Finsler与Riemann曲面上的等分布闭测地线

Equidistributed Closed Geodesics on Closed Finsler and Riemannian Surfaces

本文利用嵌入切触同调的体积性质、局部变分构造与横截性论证，证明了闭曲面上一般Finsler、对称Finsler及Riemann度量存在非退化闭测地线的等分布序列。