Federico Rodriguez Hertz
Let $S$ be a surface of nonpositive curvature of genus bigger than 1 (i.e. not the torus). We prove that any flat strip in the surface is in fact a flat cylinder. Moreover we prove that the number of homotopy classes of such flat cylinders is bounded.
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For any rank 1 nonpositively curved surface $M$, it was proved by Burns-Climenhaga-Fisher-Thompson that for any $q<1$, there exists a unique equilibrium state $μ_q$ for $qφ^u$, where $φ^u$ is the geometric potential. We show that as $q\to 1-$, the weak$^*$ limit of $μ_q$ is the restriction of the Liouville measure to the regular set.
Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed "flat" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed "flat" geodesics.
We study the one parameter family of potential functions $qφ^u$ associated with the geometric potential $φ^u$ for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For $q<1$ it is known that there is a unique equilibrium state associated with $qφ^u$, and it has full support. For $q > 1$ it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value $q=1$ and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure, or measures supported on …
We study geodesic flows over compact rank 1 manifolds and prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not carry full pressure. In dimension 2, this proves uniqueness for scalar multiples of the geometric potential on the interval $(-\infty,1)$, which is optimal. In higher dimensions, we obtain the same result on a neighborhood of 0, and give examples where uniqueness holds on all of $\mathbb{R}$. For general potential functions $φ$, we prove that the pressure gap holds whenever $φ$ is locally constant on a neighborhood of the sing…
We show that the metric of nonpositively curved graph manifolds is determined by its geodesic flow. More precisely we show that if the geodesic flows of two nonpositively curved graph manifolds are $C^0$ conjugate then the spaces are isometric.
Superdiffusive central limit theorems for geodesic flows on nonpositively curved surfaces
本文研究了非正曲率表面测地流的超扩散中心极限定理，证明了非标准中心极限定理和弱不变性原理，并改进了稳定/不稳定叶的正则性结果。
We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.
Polynomial decay of correlations for nonpositively curved surfaces
本文证明了一类零曲率仅沿一条闭测地线出现的非正曲率曲面上测地流的多项式相关性衰减，并证明其满足包括中心极限定理在内的多种统计极限定律。
Surfaces with nonpositive magnetic curvature
该研究针对具有非正磁曲率的负曲率曲面的弱双曲磁流，借助万有覆盖大尺度几何开展拓扑动力学与遍历理论相关研究。
We study the ergodic properties of horospheres on rank 1 manifolds with non-positive curvature. We prove that the horospheres are equidistributed under the action of the geodesic flow towards the Bowen-Margulis measure, on a large class of manifolds. In the case of surfaces, we define a parametrization of the horocyclic flow on the set of horocycles containing a rank 1 vector that is recurrent under the action of the geodesic flow. We prove that the horocyclic flow in restriction to this set is uniquely ergodic. The results are valid for large classes of manifolds, including the compact ones.
We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. Finally, we give examples of complete rank one surfaces for which the non wandering set of the geodesic flow is connected, the periodic orbits are dense in that set, yet the geodesic flow is not transitive in restriction to its non wandering set.
Patterson-Sullivan construction of equilibrium states and weighted counting in nonpositive curvature
本文研究非正曲率闭rank one流形上的测地流，通过引入核心极限集等概念并构造Patterson-Sullivan测度，显式构建了唯一的平衡态，证明了其Bernoulli性质，并用于加权自由同伦类计数。
We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set. In the case of a compact surface, we get the following sharp result: ergod- icity is a generic property in the space of all invariant measures defined on the uni…
On the unit tangent bundle of a nonflat compact nonpositively curved surface, we prove that there is a unique probability Borel measure invariant by a horocyclic flow which gives full measure to the set of rank $1$ vectors recurrent by the geodesic flow. If we assume in addition that the surface has no flat strips, we show that the horocyclic flow is uniquely ergodic. These results are valid for any parametrization of the horocyclic flow.
On the singular set of the geodesic flow on nonpositively curved surfaces
该研究证明非正曲率曲面上测地流奇异轨迹的特征函数属于特定几何各向异性 Sobolev 空间，由此推导测地流非遍历的条件，证明依赖半经典分析与抛物流新径向估计。
Topology of horocycles on geometrically finite nonpositively curved surfaces
该研究针对有限生成基本群的秩1非正曲率曲面，探究horocycle的拓扑性质，完成半horocycle分类、确定horocycle流非游荡集并刻画存在极小集的曲面，推进了相关动力系统与几何拓扑研究。
The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature
本文研究非正曲率秩一流形上测地流的遍历性，通过引入由曲率消失定义的子集刻画奇异集，证明其零体积时测地流关于刘维尔测度遍历，并推广至实解析度量。
In this article, we consider a closed rank one $C^\infty$ Riemannian manifold $M$ of nonpositive curvature and its universal cover $X$. Let $b_t(x)$ be the Riemannian volume of the ball of radius $t>0$ around $x\in X$, and $h$ the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates \[\lim_{t\to \infty}b_t(x)/\frac{e^{ht}}{h}=c(x)\] for some continuous function $c: X\to \mathbb{R}$. We prove that the Margulis function $c(x)$ is in fact $C^1$. If $M$ is a surface of nonpositive curvature without flat strips, we show that $c(x)$ is constant if and …
This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-Λ^2 < K < -λ^2$ where $λ/Λ> >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity t…
We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass phenomenon and the measure theoretic entropy on finite volume nonpositively curved manifolds satisfying the Visibility axiom. We provide a general criterion for the same relation to hold between the escape of mass and the measure theoretic entropy. This gives a criterion for the existence of measures of maximal entropy for the geodesic flow on some nonpositively c…
Local product structure for equilibrium states of geodesic flows and applications
本文证明紧致曲面（带锥点）及秩1非正曲率流形上测地流平衡态的局部乘积结构，并由此建立伯努利性质。
We study the non-persistence of horizontal invariant circles for geodesically convex perturbations of the geodesic circular billiard on surfaces of constant curvature and show that the result obtained by Ramírez-Ros for the planar case, remains true for billiards on surfaces with constant curvature.
In this article, we consider the geodesic flow on a compact rank $1$ Riemannian manifold $M$ without focal points, whose universal cover is denoted by $X$. On the ideal boundary $X(\infty)$ of $X$, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on $M$ has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in $X$ and the growth rate of the number of closed geodesics on $M$. These r…
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