Duke Mathematical Journal

Around the stability of KAM tori

L. H. Eliasson, B. Fayad, and R. Krikorian

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We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori.

We show that an analytic invariant torus T 0 with Diophantine frequency ω 0 is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at T 0 satisfies a Rüssmann transversality condition, the torus T 0 is accumulated by Kolmogorov–Arnold–Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least d + 1 that is foliated by analytic invariant tori with frequency ω 0 .

For frequency vectors ω 0 having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian H satisfies a Kolmogorov nondegeneracy condition at T 0 , then T 0 is accumulated by KAM tori of positive total measure.

In four degrees of freedom or more, we construct for any ω 0 R d , C (Gevrey) Hamiltonians H with a smooth invariant torus T 0 with frequency ω 0 that is not accumulated by a positive measure of invariant tori.

Article information

Duke Math. J., Volume 164, Number 9 (2015), 1733-1775.

Received: 15 July 2013
Revised: 12 August 2014
First available in Project Euclid: 15 June 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol d diffusion 70H08: Nearly integrable Hamiltonian systems, KAM theory
Secondary: 37J25: Stability problems 70H12: Periodic and almost periodic solutions 70H14: Stability problems

Hamiltonian systems KAM tori Birkhoff normal forms


Eliasson, L. H.; Fayad, B.; Krikorian, R. Around the stability of KAM tori. Duke Math. J. 164 (2015), no. 9, 1733--1775. doi:10.1215/00127094-3120060. https://projecteuclid.org/euclid.dmj/1434377460

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