Central limit theorems for counting measures in coarse negative curvature

无共轭点的亏格至少为2的曲面上的测地环计数

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。

Counting common perpendicular arcs in negative curvature

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 essential minimal surfaces in closed negatively curved n-manifolds

For closed odd-dimensional manifolds with sectional curvature less or equal than -1, we define the minimal surface entropy that counts the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic. Moreover, we compute the entropy associated with other locally symmetric spaces and cusped hyperbolic 3-manifolds.

Counting for some convergent groups

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincaré series converges at the critical exponent $δ_Γ$. We obtain an explicit asymptotic for their orbital growth function. Namely, for any $α\in ]1, 2[ $ and any slowly varying function $L : \mathbb R\to (0, +\infty)$, we construct $N$-dimensional Hadamard manifolds $(X, g)$ of negative and pinched curvature, whose group of oriented isometries admits convergent geometrically finite subgroups $Γ$ such that, as $R\to +\infty$, $$ …

共轭类中的轨道计数

Orbital Counting in Conjugacy Classes

本文研究了离散群在合适空间上的作用下的受限轨道计数问题，重点分析了在单个共轭类内点的计数渐进行为，探讨了凸余紧作用在CAT(-1)空间上的一般结果。

Counting closed geodesics on rank one manifolds

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.

无共轭点粗双曲流形上测地流最大熵测度的唯一性

Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

本文在无共轭点且基本群为Gromov双曲、剩余有限的闭黎曼流形上，通过对扩张集施加适当条件，证明了测地流最大熵测度的唯一性，从而推广了此前仅在负截面曲率背景度量下成立的相关结果。

Equilibrium states in negative curvature

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications, to the Variational Principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on…

Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces

Let $X$ be a proper, geodesically complete Hadamard space, and $\ Γ<\mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed point of a rank one isometry of $X$ in its infinite limit set. In this paper we prove that if $Γ$ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of $Γ$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit coun…