On the stability conjecture for geodesic flows of manifolds without conjugate points

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.

Expansive geodesic flows on surfaces

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007264 ◽

1993 ◽

Vol 13 (1) ◽

pp. 153-165 ◽

Cited By ~ 9

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.

The Coadjoint Operator, Conjugate Points, and the Stability of Ideal Fluids

Arnold Mathematical Journal ◽

10.1007/s40598-016-0043-9 ◽

2016 ◽

Vol 2 (2) ◽

pp. 249-266 ◽

Cited By ~ 2

Author(s):

James Benn

Keyword(s):

Conjugate Points ◽

Ideal Fluids ◽

The Stability

A stability index for travelling waves in activator-inhibitor systems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.92 ◽

2019 ◽

Vol 150 (1) ◽

pp. 517-548

Author(s):

Paul Cornwell ◽

Christopher K. R. T. Jones

Keyword(s):

Eigenvalue Problem ◽

Symplectic Form ◽

Travelling Waves ◽

Stability Index ◽

Maslov Index ◽

Evans Function ◽

Conjugate Points ◽

Spatial Evolution ◽

The Stability ◽

Activator Inhibitor

AbstractWe consider the stability of nonlinear travelling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. We formulate the Evans function for the eigenvalue problem and show that the parity of the Maslov index determines the sign of the derivative of the Evans function at the origin. The connection between the Evans function and the Maslov index is established by a ‘detection form,’ which identifies conjugate points for the curve of Lagrangian planes.

Uniqueness of central foliations of geodesic flows for compact surfaces without conjugate points

Nonlinearity ◽

10.1088/0951-7715/20/2/011 ◽

2007 ◽

Vol 20 (2) ◽

pp. 497-515 ◽

Cited By ~ 2

Author(s):

J Barbosa Gomes ◽

Rafael O Ruggiero

Keyword(s):

Conjugate Points ◽

Geodesic Flows ◽

Compact Surfaces

On a conjecture about expansive geodesic flows

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008968 ◽

1996 ◽

Vol 16 (3) ◽

pp. 545-553 ◽

Cited By ~ 1

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.

On focal stability in dimension two

Anais da Academia Brasileira de Ciências ◽

10.1590/s0001-37652007000100001 ◽

2007 ◽

Vol 79 (1) ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Mauricio M. Peixoto ◽

Charles C. Pugh

Keyword(s):

Conjugate Points ◽

Stability Conjecture ◽

Riemann Structure

In Kupka et al. 2006 appears the Focal Stability Conjecture: the focal decomposition of the generic Riemann structure on a manifold M is stable under perturbations of the Riemann structure. In this paper, we prove the conjecture when M has dimension two, and there are no conjugate points.

STABILITY CONJECTURE IN THE THEORY OF TENSEGRITY STRUCTURES

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455403000744 ◽

2003 ◽

Vol 03 (01) ◽

pp. 1-16 ◽

Cited By ~ 2

Author(s):

K. Y. VOLOKH

Keyword(s):

General Problem ◽

Tensegrity Structures ◽

Stability Conjecture ◽

The Stability

The general problem of the stability of tensegrity structures comprising struts and cables is formulated. It is conjectured that any tensegrity system with totally tensioned cables is stable independently of its topology, geometry and specific magnitudes of member forces.

Hyperbolicity versus weak periodic orbits inside homoclinic classes

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.122 ◽

2017 ◽

Vol 38 (6) ◽

pp. 2345-2400 ◽

Cited By ~ 5

Author(s):

XIAODONG WANG

Keyword(s):

Lyapunov Exponents ◽

Periodic Orbits ◽

Homoclinic Class ◽

Intrinsic Nature ◽

Significant Difference ◽

Stability Conjecture ◽

The Stability ◽

Classical Proof ◽

Homoclinic Classes

We prove that, for$C^{1}$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class$H(p)$have all their Lyapunov exponents bounded away from zero, then$H(p)$must be (uniformly) hyperbolic. This is in the spirit of the works on the stability conjecture, but with a significant difference that the homoclinic class$H(p)$is not known isolated in advance, hence the ‘weak’ periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an ‘intrinsic’ nature, and the classical proof of the stability conjecture does not work. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.

The accessibility property of expansive geodesic flows without conjugate points

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385707000399 ◽

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.