In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a $C^2$ open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are $C^2$ stably ergodic if and only if they are Anosov.
This paper is an exposition of the major results of P. Eberlein's paper, "When is a geodesic flow of Anosov type? I," in the special case when the manifold $M$ is a surface. We follow Eberlein's coverage closely, adding details when helpful, and taking advantage of simplifications given by the dimension two case. The objective is to give readers a more tractable introduction to the important topic of Anosov geodesic flows, which highlights key concepts and arguments without worrying about generalizations to arbitrary dimension.
In this paper we prove that if the geodesic flow of a {compact or non-compact} complete manifold without conjugate points is of the Anosov type, then the average of the integral of the sectional curvature along the geodesic is negative and away from zero from a uniform time. Moreover, in dimension two, if the manifold has no focal points, then this condition is sufficient to obtain that the geodesic flow is of Anosov type. This sufficient condition will also be used to construct new examples of non-compact surfaces whose geodesic flow is of the Anosov type.
We introduce a generalization of the notion of Anosov representations by restricting to invariant closed geodesic subflows. Examples of such representations include many non-discrete representations with good geometric properties, such as primitive-stable representations. We give several equivalent characterizations of this type of representations and prove some properties analogous to the classical Anosov representations, such as stability, the Cartan property and regularity properties of the limit maps.
In this article, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let $M$ be a smooth connected and closed surface equipped with a $C^\infty$ Riemannian metric $g$, whose genus $\mathfrak{g} \geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tangent bundle of $M$ is ergodic with respect to the Liouville measure, under the assumption that the set of points on $M$ with negative curvature has at most finitely many connected components.
We complete the microlocal study of the geodesic X-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou and pursued by Guillarmou and the second author. We prove new stability estimates and clarify some properties of the operator $Π_m$, the generalized X-ray transform. These estimates rely on a refined version of the Livsic theorem for Anosov flows, especially on a new quantitative finite time Livsic theorem.
Anosov representations were introduced by F. Labourie [18] for fundamental groups of closed negatively curved surfaces, and generalized by O. Guichard and A. Wienhard [19] to representations of arbitrary Gromov hyperbolic groups into real semisimple Lie groups. In this paper, we focus on Anosov representations into the identity component O0(2, n) of O(2, n) for n $\ge$ 2. Our main result is that any Anosov representation with negative limit set as defined in [8] is the holonomy group of a spatially compact, globally hyperbolic maximal (abbrev. CGHM) conformally flat spacetime. The proof of the…
In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stability estimate quantifying how the marked length spectrum control the distance between the metrics. In dimension $2$ we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectr…
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 …
The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
Lichnerowicz猜想断言所有的调和流形要么是平坦的要么是秩为1的局部对称空间。这个猜想已被Z. Szabó \cite{Sz}：具有紧万有覆盖的调和流形. E. Damek和F.利玛窦\cite{DR}提供的例子表明，在非紧的情况下，猜想是错误的。然而，这样的流形不允许紧商。本文利用秩的概念研究了单连通非平坦非紧调和流形$X$上的渐近几何和测地流。在本文的第一部分中，我们证明了以下断言是等价的。体积的增长是纯指数的，$X$的秩是1，测地流是Anosov关于Sasaki度量，$X$是Gromov双曲。本文第二部分证明了若$X$是一个无焦点的非平坦调和流形，则测地流是Anosov流。在证明的过程中，我们得到了任意黎曼流形上的某些部分双曲流是Anosov流，这是调和流形以外的有趣的。结合\cite{BCG}，\cite{BFL}和\cite{FL}的刚性定理，我们对所有无焦点或含Gromov双曲基本群的紧致调和流形证实了Lichnerowicz猜想.
We prove that in a $C^1$-open and $C^k$-dense set of some classes of $C^k$ Anosov flows all Lyapunov exponents have multiplicity 1 with respect to appropriate measures. The classes are geodesic flows with equilibrium states of Hölder-continuous potentials, volume-preserving flows, and all fiber-bunched Anosov flows with equilibrium states of Hölder-continuous potentials. In the proof, we use and prove perturbative results for jets of flows to modify eigenvalues of certain Poincaré maps and, using a Markov partition, apply the simplicity criterion of Avila and Viana.
We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flo…
我们将Shnirelman、Colin de Verdière和Zelditch关于量子遍历性的经典结果推广到轨道折叠，证明了对于紧致轨道折叠X上的任何正的一阶自伴椭圆伪微分算子P，且具有正的主象征p，p的Hamilton流的遍历性蕴涵着算子P的量子遍历性。我们还证明了负的紧黎曼轨道上的测地流的遍历性。截面曲率
Let $ρ:Γ\rightarrow PSL_d(\mathbb{K})$ be a Zariski dense Borel-Anosov representation, for $\mathbb{K}$ equal to $\mathbb{R}$ or $\mathbb{C}$. Let $o$ be a form of signature $(p,d-p)$ on $\mathbb{K}^d$ (where $0<p<d)$. Let $\mathsf{S}^o$ be the corresponding geodesic copy of the Riemannian symmetric space of $PSO(o)$, inside the Riemannian symmetric space of $PSL_d(\mathbb{K})$. For certain choices of $o$ and every $t$ large enough, we show exponential bounds for the number of $γ\inΓ$ for which the distance between $\mathsf{S}^o$ and $ργ\cdot\mathsf{S}^o$ is smaller than $t$. Under an extra as…
Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. We give a sufficient condition for a nonflat billiard to be uniformly hyperbolic. As a particular case, we obtain a new criterion to show that a closed surface has an Anosov geodesic flow.
