Invariant Curves, Homoclinic Points, and Ergodicity in Area-Preserving Mappings

AbstractThe stability of a fixed point of an area-preserving transformation in the plane is characterized by the invariant curves which surround it. The existence of invariant curves had been extensively studied for elliptic fixed points. Here we study the similar problem for parabolic fixed points. In particular we are interested in the case where the fixed point is at infinity.

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Construction of invariant measures supported within the gaps of Aubry–Mather sets

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008713 ◽

1996 ◽

Vol 16 (1) ◽

pp. 51-86 ◽

Cited By ~ 3

Author(s):

Giovanni Forni

Keyword(s):

Variational Approach ◽

Invariant Measures ◽

Positive Entropy ◽

Almost Periodic ◽

Invariant Curves ◽

Mather Theory ◽

Entropy Measures ◽

Twist Maps ◽

Minimization Procedure ◽

Area Preserving

AbstractThis paper represents a contribution to the variational approach to the understanding of the dynamics of exact area-preserving monotone twist maps of the annulus, currently known as the Aubry–Mather theory. The method introduced by Mather to construct invariant measures of Denjoy type is extended to produce almost-periodic measures, having arbitrary rationally independent frequencies, and positive entropy measures, supported within the gaps of Aubry–Mather sets which do not lie on invariant curves. This extension is based on a generalized version of the Percival's Lagrangian and on a new minimization procedure, which also gives a simplified proof of the basic existence theorem for the Aubry–Mather sets.

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The Dynamics of a Piecewise Linear Map and its Smooth Approximation

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497000236 ◽

1997 ◽

Vol 07 (02) ◽

pp. 351-372 ◽

Cited By ~ 10

Author(s):

D. Aharonov ◽

R. L. Devaney ◽

U. Elias

Keyword(s):

Chaotic Behavior ◽

Piecewise Linear ◽

Invariant Curves ◽

Smooth Approximation ◽

Linear Map ◽

Island Structure ◽

The Real ◽

Piecewise Linear Map ◽

Real Analytic ◽

Area Preserving

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.

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Parabolic fixed points, invariant curves and action-angle variables

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700005526 ◽

1990 ◽

Vol 10 (2) ◽

pp. 231-245 ◽

Cited By ~ 2

Author(s):

Dov Aharonov ◽

Uri Elias

Keyword(s):

Fixed Point ◽

Hamiltonian System ◽

Fixed Points ◽

Phase Flow ◽

Invariant Curves ◽

Elliptic Case ◽

System A ◽

Twist Mapping ◽

Area Preserving ◽

Parabolic Fixed Point

AbstractA fixed point of an area-preserving mapping of the plane is called elliptic if the eigenvalues of its linearization are of unit modulus but not ±1; it is parabolic if both eigenvalues are 1 or −1. The elliptic case is well understood by Moser's theory. Here we study when is a parabolic fixed point surrounded by closed invariant curves. We approximate our mapping T by the phase flow of an Hamiltonian system. A pair of variables, closely related to the action-angle variables, is used to reduce T into a twist mapping. The conditions for T to have closed invariant curves are stated in terms of the Hamiltonian.

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Homoclinic Orbits in the Complex Domain

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497000182 ◽

1997 ◽

Vol 07 (02) ◽

pp. 253-274 ◽

Cited By ~ 5

Author(s):

V. F. Lazutkin ◽

C. Simó

Keyword(s):

Homoclinic Orbits ◽

Homoclinic Tangency ◽

Complex Domain ◽

Standard Map ◽

Homoclinic Points ◽

Promising Tool ◽

Theoretical Support ◽

Area Preserving Maps ◽

Area Preserving

We consider the standard map, as a paradigm of area preserving map, when the variables are taken as complex. We study how to detect the complex homoclinic points, which cannot dissappear under a homoclinic tangency. This seems a promising tool to understand the stochastic zones of area preserving maps. The paper is mainly phenomenological and includes theoretical support to the observed phenomena. Several conjectures are stated.

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On the generic existence of homoclinic points

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700004211 ◽

1987 ◽

Vol 7 (4) ◽

pp. 567-595 ◽

Cited By ~ 13

Author(s):

Fernando Oliveira

Keyword(s):

Periodic Point ◽

Invariant Manifolds ◽

Compact Manifolds ◽

Homoclinic Points ◽

Symplectic Diffeomorphisms ◽

Generic Existence ◽

Area Preserving ◽

Orientable Surfaces

AbstractThis work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.

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Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700004697 ◽

1988 ◽

Vol 8 (4) ◽

pp. 555-584 ◽

Cited By ~ 1

Author(s):

Raphaël Douady

Keyword(s):

Hamiltonian Systems ◽

Rotation Number ◽

Positive Measure ◽

Invariant Curves ◽

Rotation Numbers ◽

Twist Maps ◽

Area Preserving Maps ◽

Area Preserving

AbstractWe prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

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