Does the hyperbolicity of periodic orbits of a geodesic flow without conjugate points imply the Anosov property?

2021 ◽  
pp. 1
Author(s):  
Rafael O. Ruggiero
Keyword(s):  
Periodic Orbits ◽  
Geodesic Flow ◽  
Conjugate Points

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.

On the entropy of the geodesic flow in manifolds without conjugate points

1982 ◽  
Vol 69 (3) ◽  
pp. 375-392 ◽  
Author(s):  
A. Freire ◽  
R. Ma��
Keyword(s):  
Geodesic Flow ◽  
Conjugate Points

scholarly journals Coding of geodesics and Lorenz-like templates for some geodesic flows

2016 ◽  
Vol 38 (3) ◽  
pp. 940-960
Author(s):  
PIERRE DEHORNOY ◽  
TALI PINSKY
Keyword(s):  
Periodic Orbits ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Hyperbolic Metric ◽  
Geodesic Flows ◽  
Unit Tangent Bundle

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

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.

scholarly journals On the Conley conjecture for Reeb flows

2015 ◽  
Vol 26 (07) ◽  
pp. 1550047 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Başak Z. Gürel ◽  
Leonardo Macarini
Keyword(s):  
Magnetic Field ◽  
Energy Level ◽  
Periodic Orbits ◽  
Geodesic Flow ◽  
Low Energy ◽  
Contact Manifolds ◽  
Reeb Flows ◽  
Circle Bundles ◽  
Conley Conjecture

In this paper, we prove the existence of infinitely many closed Reeb orbits for a certain class of contact manifolds. This result can be viewed as a contact analogue of the Hamiltonian Conley conjecture. The manifolds for which the contact Conley conjecture is established are the pre-quantization circle bundles with aspherical base. As an application, we prove that for a surface of genus at least two with a non-vanishing magnetic field, the twisted geodesic flow has infinitely many periodic orbits on every low energy level.

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.

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 $\mathbb{Z}\times \mathbb{Z}$.

scholarly journals Equicontinuous geodesic flows

2009 ◽  
Vol 29 (6) ◽  
pp. 1951-1963
Author(s):  
CHRISTIAN PRIES
Keyword(s):  
Geodesic Flow ◽  
Chaotic Behavior ◽  
Conjugate Points ◽  
Necessary Condition ◽  
Riemannian Surface ◽  
Geodesic Flows ◽  
Higher Dimensional ◽  
The Stability ◽  
Flows On Surfaces ◽  
Sufficient And Necessary Condition

AbstractThis article is about the interplay between topological dynamics and differential geometry. One could ask how much information about the geometry is carried in the dynamics of the geodesic flow. It was proved in Paternain [Expansive geodesic flows on surfaces. Ergod. Th. & Dynam. Sys.13 (1993), 153–165] that an expansive geodesic flow on a surface implies that there exist no conjugate points. Instead of considering concepts that relate to chaotic behavior (such as expansiveness), we focus on notions for describing the stability of orbits in dynamical systems, specifically, equicontinuity and distality. In this paper we give a new sufficient and necessary condition for a compact Riemannian surface to have all geodesics closed; this is the idea of a P-manifold: (M,g) is a P-manifold if and only if the geodesic flow SM×ℝ→SM is equicontinuous. We also prove a weaker theorem for flows on manifolds of dimension three. Finally, we discuss some properties of equicontinuous geodesic flows on non-compact surfaces and on higher-dimensional manifolds.

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