## Duke Mathematical Journal

### Around the stability of KAM tori

#### Abstract

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 ${\mathcal{T}}_{0}$ with Diophantine frequency $\omega _{0}$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at ${\mathcal{T}}_{0}$ satisfies a Rüssmann transversality condition, the torus ${\mathcal{T}}_{0}$ 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 $\omega _{0}$.

For frequency vectors $\omega _{0}$ 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 ${\mathcal{T}}_{0}$, then ${\mathcal{T}}_{0}$ is accumulated by KAM tori of positive total measure.

In four degrees of freedom or more, we construct for any $\omega _{0}\in{\mathbb{R}}^{d}$, $C^{\infty}$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus ${\mathcal{T}}_{0}$ with frequency $\omega _{0}$ that is not accumulated by a positive measure of invariant tori.

#### Article information

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

Dates
Revised: 12 August 2014
First available in Project Euclid: 15 June 2015

https://projecteuclid.org/euclid.dmj/1434377460

Digital Object Identifier
doi:10.1215/00127094-3120060

Mathematical Reviews number (MathSciNet)
MR3357183

Zentralblatt MATH identifier
1366.37126

#### Citation

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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