### 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 The latter proof is announced in [Ma5], but still unpublished.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 language | English |
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Pages (from-to) | 79-116 |

Number of pages | 38 |

Journal | Asterisque |

Issue number | 297 |

Publication status | Published - 1 Dec 2004 |

Externally published | Yes |

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### Cite this

*Asterisque*, (297), 79-116.

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*Asterisque*, no. 297, pp. 79-116.

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Instability of resonant totally elliptic points of symplectic maps in dimension 4

AU - Kaloshin, Vadim

AU - Mather, John N.

AU - Valdinoci, Enrico

PY - 2004/12/1

Y1 - 2004/12/1

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 The latter proof is announced in [Ma5], but still unpublished.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.

KW - Arnold diffusion

KW - Lyapunov stability of periodic orbits

KW - Mather theory

KW - Minimal measures

KW - Variational methods

UR - http://www.scopus.com/inward/record.url?scp=30344461845&partnerID=8YFLogxK

M3 - Article

SP - 79

EP - 116

JO - Asterisque

JF - Asterisque

SN - 0303-1179

IS - 297

ER -