Sa'ar Hersonsky, Frederic Paulin
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. We study the asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M. In the appendix, due to K. Belabas, the case of SL(2,Z) and of Bianchi groups is developed.
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Let M be a complete simply connected Riemannian manifold, with sectional curvature K bounded above by -1. Under some assumptions on the geometry of the boundary of M, which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M, and any point x_0 outside these horoballs, it is possible to shrink uniformly, by a finite amount depending only on M, these horoballs so that some geodesic ray starting from x_0 avoids the shrunk horoballs. As an application, we give a uniform upper bound on the infimum of the heights of the closed ge…
We prove that there is a true asymptotic formula for the number of one sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic structure, and it is noninteger, in contrast to counting results of Mirzakhani for simple closed curves on orientable Fuchsian surfaces.
We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.
Fix a smooth closed manifold $M$. Let $R_M$ denote the space of all pairs $(g,L)$ such that $g$ is a $C^3$ Riemannian metric on $M$ and the real number $L$ is not the length of any closed $g$-geodesics. A locally constant geodesic count function $π_M:R_M\rightarrow Z$ is constructed. For this purpose, the weight of compact open subsets of the space of closed $g$-geodesics is defined and investigated for an arbitrary Riemannian metric $g$.
We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically nite manifolds.
Counting geodesic loops on surfaces of genus at least 2 without conjugate points
本文在无共轭点的紧曲面上建立了闭测地环的渐近计数估计，并证明了更一般的扇形定理，推广了Huber、Margulis、Nicholls和Sharp等人在严格负曲率情形下的经典结果。
We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.
In this article, we consider a closed rank one Riemannian manifold $M$ without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates: \[\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\] where $h$ is the topological entropy of the geodesic flow. In the appendix, we also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.
Counting geodesics on compact symmetric spaces
本文通过焦轨道分解紧致对称空间指数映射的逆像，利用图、重数、Weyl群和格编码测地线的维数与连通分支，推广了紧致李群上的计数结果并给出已知结论的新证明。
We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish some useful properties of the measure of maximal entropy.
We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.
Counting and equidistribution of strongly reversible closed geodesics in negative curvature
本文研究含二阶挠元的受压负曲率轨形中强可逆闭测地线，利用热力学形式体系等新技术证明其渐近计数及关于 Bowen-Margulis 测度的等分布，并给出多类新例子。
Geodesic string counting invariants and arithmetic of multiplicities
该研究基于测地流的Fuller指标，推导测地弦计数的乘积型公式，结合KAM理论得出二维环面上测地弦重数的数论性质，还给出闭测地线计数及负截面曲率度量存在性的算术约束。
Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups
本文研究克莱因群强收敛下几何有限双曲流形的小特征值收敛性，利用极限流形的谱隙与测地流指数混合性质，建立了凸集间正交测地线数量的渐近一致计数公式，并应用于 Margulis 管、测地环及本原闭测地线。
Poincaré series for surfaces with boundary
本研究针对带全测地边界的负曲率曲面，证明正交测地线计数庞加莱级数及两点测地弧计数级数可亚纯延拓至全复平面，并给出其零点取值。
Prime scattering geodesic theorem
本文研究模曲面上散射测地线的分布与逗留时间，首次建立了散射测地线计数与素因子位于算术级数中的正整数之间的联系。
For positive integers $p$ and $q$ let $G:=\textrm{PSO}(p,q)$ be the projective indefinite special-orthogonal group of signature $(p,q)$. We study counting problems in the Riemannian symmetric space $X_G$ of $G$ and in the pseudo-Riemannian hyperbolic space $\mathbb{H}^{p,q-1}$. Let $S\subset X_G$ be a totally geodesic copy of $X_{\textrm{PSO}(p,q-1)}$. We look at the orbit of $S$ under the action of a projective Anosov subgroup of $G$. For certain choices of such a geodesic copy we show that the number of points in this orbit which are at distance at most $t$ from $S$ is finite and asymptotic …
Counting Salem numbers arising from arithmetic hyperbolic orbifolds
本文研究算术双曲轨道中Salem数与短测地线的关系，通过界定了Salem数比例和测地线谱乘法的指数增长下界，改进了相关理论。
Équidistribution, comptage et approximation par irrationnels quadratiques
该研究在有限体积双曲流形中证明等距超曲面的等分布性与测地弧计数渐近式，并将其应用于二次无理数在模群同余子群轨道中的计数问题。
Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ from $D^-$ to $D^+$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^-$ and $D^+$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with fin…
Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces
本文将平曲面测地线计数问题推广至 K3 曲面，分别给出了不可约特殊 Lagrangian 类数量的渐近上界，以及在一般 Bridgeland 稳定性条件下半稳定凝聚层 Mukai 向量数量渐近的精确首项。
The entropy spectrum of hyperbolic surfaces
本文研究双曲曲面的熵谱，证明其为具有限重数的逆良序多重集，在数值1附近存在可量化间隙，该间隙源于带边界曲面上曲线数量的几何估计。
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