Weak stability of the geodesic flow and Preissmann's theorem

2000 ◽  
Vol 20 (4) ◽  
pp. 1231-1251
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Geodesic Flow ◽  
Abelian Subgroup ◽  
Conjugate Points ◽  
Weak Stability ◽  
Shadowing Property

Let $(M,g)$ be a compact, differentiable Riemannian manifold without conjugate points and bounded asymptote. We show that, if the geodesic flow of $(M,g)$ is either topologically stable, or satisfies the $\epsilon$-shadowing property for some appropriate $\epsilon > 0$, then every abelian subgroup of the fundamental group of $M$ is infinite cyclic. The proof is based on the existence of homoclinic geodesics in perturbations of $(M,g)$, whenever there is a subgroup of the fundamental group of $M$ isomorphic to $\mathbb{Z}\times \mathbb{Z}$.

Topological stability and Gromov hyperbolicity

1999 ◽  
Vol 19 (1) ◽  
pp. 143-154 ◽  
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Hyperbolic Space ◽  
Geodesic Flow ◽  
Positive Curvature ◽  
Universal Covering ◽  
Conjugate Points ◽  
Gromov Hyperbolicity ◽  
Topological Stability ◽  
Shadowing Property ◽  
Gromov Hyperbolic

We show that if the geodesic flow of a compact analytic Riemannian manifold $M$ of non-positive curvature is either $C^{k}$-topologically stable or satisfies the $\epsilon$-$C^{k}$-shadowing property for some $k > 0$ then the universal covering of $M$ is a Gromov hyperbolic space. The same holds for compact surfaces without conjugate points.

On a conjecture about expansive geodesic flows

1996 ◽  
Vol 16 (3) ◽  
pp. 545-553 ◽  
Author(s):  
Rafael Oswaldo Ruggierot
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Conjugate Points ◽  
Geodesic Flows

AbstractWe show that near the geodesic flow of a compact Riemannian manifold with no conjugate points which is expansive, every expansive geodesic flow has no conjugate points. We also prove that in the above hypotheses the geodesic flow istopologically stable.

The accessibility property of expansive geodesic flows without conjugate points

2008 ◽  
Vol 28 (1) ◽  
pp. 229-244
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Conjugate Points ◽  
Stable Set ◽  
Geodesic Flows ◽  
Continuous Path ◽  
Two Dimensional ◽  
Unstable Set ◽  
Unit Tangent Bundle ◽  
Continuous Curves

AbstractLet (M,g) be a compact, smooth Riemannian manifold without conjugate points whose geodesic flow is expansive. We show that the geodesic flow of (M,g) has the accessibility property, namely, given two pointsθ1,θ2in the unit tangent bundle, there exists a continuous path joiningθ1,θ2formed by the union of a finite number of continuous curves, each of which is contained either in a strong stable set or in a strong unstable set of the dynamics. Since expansive geodesic flows of compact surfaces have no conjugate points, the accessibility property holds for every two-dimensional expansive geodesic flow.

Expansive geodesic flows in manifolds with no conjugate points

1997 ◽  
Vol 17 (1) ◽  
pp. 211-225 ◽  
Author(s):  
RAFAEL O. RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Product Structure ◽  
Conjugate Points ◽  
Geodesic Flows ◽  
Closed Geodesics ◽  
Local Product ◽  
Unit Tangent Bundle

Let $M$ be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. We show that there exists a local product structure in the unit tangent bundle of the manifold which is invariant under the geodesic flow. In particular, we have that the set of closed geodesics is dense and that the flow is topologically transitive.

Expansive geodesic flows on surfaces

1993 ◽  
Vol 13 (1) ◽  
pp. 153-165 ◽  
Author(s):  
Miguel Paternain
Keyword(s):  
Geodesic Flow ◽  
Compact Surface ◽  
Conjugate Points ◽  
Riemannian Surface ◽  
Geodesic Flows ◽  
Flows On Surfaces

AbstractWe prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

Rigidities of asymptotically Euclidean manifolds

1999 ◽  
Vol 60 (3) ◽  
pp. 521-528 ◽  
Author(s):  
Seong-Hun Paeng
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Group ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Covering Space ◽  
Universal Covering Space

Let M be an n-dimensional compact Riemannian manifold. We study the fundamental group of M when the universal covering space of M, M is close to some Euclidean space ℝs asymptotically.

scholarly journals Dynamics of the geodesic flow of a foliation

1988 ◽  
Vol 8 (4) ◽  
pp. 637-650 ◽  
Author(s):  
Paweł G. Walczak
Keyword(s):  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Geodesic Flow ◽  
Smooth Measure ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Foliated Riemannian Manifold

AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

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

1997 ◽  
Vol 17 (5) ◽  
pp. 1043-1059 ◽  
Author(s):  
KEITH BURNS ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Positive Measure ◽  
Geodesic Flows ◽  
Open Set ◽  
Negative Questions

Let $M$ be a compact $C^{\infty}$ Riemannian manifold. Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number of geodesic segments joining $p$ and $q$ with length $\leq T$. Mañé showed in [7] that \[ \lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq = h_{\rm top}, \] where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of $M$.In this paper we exhibit an open set of metrics on the two-sphere for which \[ \limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top}, \] for a positive measure set of $(p,q)\in M\times M$. This answers in the negative questions raised by Mañé in [7].

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