Convex Hamiltonians without conjugate points

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

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On the Group of Almost Periodic Diffeomorphisms and its Exponential Map

International Mathematics Research Notices ◽

10.1093/imrn/rnz155 ◽

2019 ◽

Author(s):

Xu Sun ◽

Peter Topalov

Keyword(s):

Lie Group ◽

Riemannian Metric ◽

Conjugate Points ◽

Exponential Map ◽

Almost Periodic ◽

Exponential Maps ◽

Fluid Equations ◽

Periodic Diffeomorphisms

Abstract We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.

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Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism

Journal of Dynamical and Control Systems ◽

10.1007/s10883-008-9054-8 ◽

2009 ◽

Vol 15 (1) ◽

pp. 133-156 ◽

Cited By ~ 3

Author(s):

Marcos M. Diniz ◽

José M. M. Veloso

Keyword(s):

Riemannian Manifolds ◽

Exponential Map ◽

Local Diffeomorphism

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Almost existence from the feral perspective and some questions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.20 ◽

2021 ◽

pp. 1-43

Author(s):

JOEL W. FISH ◽

HELMUT H. W. HOFER

Keyword(s):

Periodic Orbit ◽

Energy Level ◽

Existence Result ◽

Symplectic Manifolds ◽

Extended Version ◽

Pseudoholomorphic Curves ◽

Energy Surfaces

Abstract We use feral pseudoholomorphic curves and adiabatic degeneration to prove an extended version of the so-called ‘almost existence result’ for regular compact Hamiltonian energy surfaces. That is, that for a variety of symplectic manifolds equipped with a Hamiltonian, almost every (non-empty) compact energy level has a periodic orbit.

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Exponential Functions in Cartesian Differential Categories

Applied Categorical Structures ◽

10.1007/s10485-020-09610-0 ◽

2020 ◽

Author(s):

Jean-Simon Pacaud Lemay

Keyword(s):

Differential Equations ◽

Exponential Function ◽

Differential Calculus ◽

Exponential Map ◽

Dual Numbers ◽

Exponential Functions ◽

Smooth Functions ◽

Exponential Maps ◽

Differential Categories ◽

Complex Exponential

Abstract In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $$e^x$$ e x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $$e^0 = 1$$ e 0 = 1 , $$e^{x+y} = e^x e^y$$ e x + y = e x e y , and $$\frac{\partial e^x}{\partial x} = e^x$$ ∂ e x ∂ x = e x all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.

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The Exponential Map Near Conjugate Points In 2D Hydrodynamics

Arnold Mathematical Journal ◽

10.1007/s40598-015-0019-1 ◽

2015 ◽

Vol 1 (3) ◽

pp. 243-251 ◽

Cited By ~ 3

Author(s):

Gerard Misiołek

Keyword(s):

Conjugate Points ◽

Exponential Map

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The orbit of Pluto over a long interval of time

Symposium - International Astronomical Union ◽

10.1017/s0074180900105509 ◽

1966 ◽

Vol 25 ◽

pp. 227-229 ◽

Cited By ~ 1

Author(s):

D. Brouwer

Keyword(s):

Periodic Orbit ◽

Numerical Integration ◽

Restricted Problem ◽

Three Dimensional ◽

The Third ◽

Circular Restricted Problem ◽

Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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