Instability of resonant totally elliptic points of symplectic maps in dimension 4

Vadim Kaloshin, John N. Mather, Enrico Valdinoci

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

A well known Moser stability theorem states that a generic elliptic fixed point of an area-preserving mapping is Lyapunov stable. We investigate the question of Lyapunov stability for 4-dimensional resonant totally elliptic fixed points of symplectic maps. We show that gcnerically a convex, resonant, totally elliptic point of a symplectic map is Lyapunov unstable. The proof heavily relies on a proof of J. Mather of existence of Arnold diffusion for convex Hamiltonians in 2.5 degrees of freedom. The latter proof is announced in [Ma5], but still unpublished.

Original languageEnglish
Pages (from-to)79-116
Number of pages38
JournalAsterisque
Issue number297
Publication statusPublished - 1 Dec 2004
Externally publishedYes

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Lyapunov
Fixed point
Arnold Diffusion
Lyapunov Stability
Stability Theorem
Unstable
Degree of freedom

Cite this

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Instability of resonant totally elliptic points of symplectic maps in dimension 4. / Kaloshin, Vadim; Mather, John N.; Valdinoci, Enrico.

In: Asterisque, No. 297, 01.12.2004, p. 79-116.

Research output: Contribution to journalArticle

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AU - Kaloshin, Vadim

AU - Mather, John N.

AU - Valdinoci, Enrico

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N2 - A well known Moser stability theorem states that a generic elliptic fixed point of an area-preserving mapping is Lyapunov stable. We investigate the question of Lyapunov stability for 4-dimensional resonant totally elliptic fixed points of symplectic maps. We show that gcnerically a convex, resonant, totally elliptic point of a symplectic map is Lyapunov unstable. The proof heavily relies on a proof of J. Mather of existence of Arnold diffusion for convex Hamiltonians in 2.5 degrees of freedom. The latter proof is announced in [Ma5], but still unpublished.

AB - A well known Moser stability theorem states that a generic elliptic fixed point of an area-preserving mapping is Lyapunov stable. We investigate the question of Lyapunov stability for 4-dimensional resonant totally elliptic fixed points of symplectic maps. We show that gcnerically a convex, resonant, totally elliptic point of a symplectic map is Lyapunov unstable. The proof heavily relies on a proof of J. Mather of existence of Arnold diffusion for convex Hamiltonians in 2.5 degrees of freedom. The latter proof is announced in [Ma5], but still unpublished.

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KW - Lyapunov stability of periodic orbits

KW - Mather theory

KW - Minimal measures

KW - Variational methods

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