Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems

1994 ◽  
Vol 14 (4) ◽  
pp. 757-785 ◽  
Author(s):  
Anatole Katok ◽  
Keith Burns
Keyword(s):  
Lyapunov Exponents ◽  
Geodesic Flow ◽  
Lyapunov Functions ◽  
Riemannian Metric ◽  
Dimensional Manifold ◽  
Invariant Cone ◽  
Ergodic Component ◽  
3 Dimensional ◽  
Stochastic Properties ◽  
Volume Preserving

AbstractWe establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C∞ Riemannian metric whose geodesic flow is Bernoulli.

Correlation based swarm trackers for 3-dimensional manifold mesh formation

2009 ◽  
Author(s):  
Charles Casey ◽  
Laurence G. Hassebrook ◽  
Priyanka Chaudhary
Keyword(s):  
Dimensional Manifold ◽  
3 Dimensional

scholarly journals The Long-Time Behavior of the Ricci Tensor under the Ricci Flow

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Christian Hilaire
Keyword(s):  
Ricci Flow ◽  
Ricci Tensor ◽  
Closed Manifold ◽  
Time Behavior ◽  
Dimensional Manifold ◽  
Long Time Behavior ◽  
Bounded Curvature ◽  
3 Dimensional ◽  
Long Time ◽  
Uniformly Bounded

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as . We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the curvature and the square of the diameter is uniformly bounded, then this solution must be of type III.

Three Conditions Lyapunov Exponents Should Satisfy

2003 ◽  
Author(s):  
Jingjun Lou ◽  
Shijian Zhu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Chaotic Motion ◽  
The Other ◽  
Cubic Nonlinearity ◽  
3 Dimensional ◽  
Duffing System ◽  
Dissipative Dynamical System

In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1-dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1-dimensional and 3-dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity.

scholarly journals A characterization of harmonic foliations by the volume preserving property of the normal geodesic flow

2002 ◽  
Vol 29 (10) ◽  
pp. 573-577
Author(s):  
Hobum Kim
Keyword(s):  
Geodesic Flow ◽  
Riemannian Foliation ◽  
Volume Form ◽  
Normal Connection ◽  
Riemannian Volume ◽  
Canonical Metric ◽  
Normal Geodesic ◽  
Harmonic Foliations ◽  
Volume Preserving

We prove that a Riemannian foliation with the flat normal connection on a Riemannian manifold is harmonic if and only if the geodesic flow on the normal bundle preserves the Riemannian volume form of the canonical metric defined by the adapted connection.

THE ADAPTED COMPLEXIFICATION OF AN ELLIPSOID

2007 ◽  
Vol 18 (01) ◽  
pp. 43-68
Author(s):  
RAUL M. AGUILAR
Keyword(s):  
Geodesic Flow ◽  
Riemannian Metric ◽  
Classical Theorem ◽  
Round Sphere

We show that an n-dimensional real ellipsoid in ℝn+1 with the induced Riemannian metric does not admit an unbounded adapted complexification in the sense of Lempert/Szőke and Guillemin/Stenzel, unless it is a round sphere. In other words, an ellipsoid whose (maximal) Grauert tube has infinite radius must be a round sphere. For the proof we take advantage of the integrability of the geodesic flow and use a classical theorem on umbilic geodesics. We carry out an extension of this result to Liouville metrics elsewhere.

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.

scholarly journals Lyapunov exponents in Hilbert geometry

2012 ◽  
Vol 34 (2) ◽  
pp. 501-533 ◽  
Author(s):  
MICKAËL CRAMPON
Keyword(s):  
Lyapunov Exponents ◽  
Convex Functions ◽  
Geodesic Flow ◽  
Regularity Property ◽  
Hilbert Geometry

AbstractWe study the Lyapunov exponents of the geodesic flow of a Hilbert geometry. We prove that all of the information is contained in the shape of the boundary at the endpoint of the chosen orbit. We have to introduce a regularity property of convex functions to make this link precise. As a consequence, Lyapunov manifolds tangent to the Lyapunov splitting appear very easily. All of this work can be seen as a consequence of convexity and the flatness of Hilbert geometries.

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