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15 June 2015 Around the stability of KAM tori
L. H. Eliasson, B. Fayad, R. Krikorian
Duke Math. J. 164(9): 1733-1775 (15 June 2015). DOI: 10.1215/00127094-3120060


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.


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L. H. Eliasson. B. Fayad. R. Krikorian. "Around the stability of KAM tori." Duke Math. J. 164 (9) 1733 - 1775, 15 June 2015.


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

zbMATH: 1366.37126
MathSciNet: MR3357183
Digital Object Identifier: 10.1215/00127094-3120060

Primary: 37J40 , 70H08
Secondary: 37J25 , 70H12 , 70H14

Keywords: Birkhoff normal forms , Hamiltonian systems , KAM tori

Rights: Copyright © 2015 Duke University Press


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Vol.164 • No. 9 • 15 June 2015
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