Geodesic
On the measure of the KAM-tori in a~neighbourhood of a~separatrix
Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 755-774

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Consider a Liouville-integrable Hamiltonian system with $n$ degrees of freedom. Assume that the foliation of the phase space by invariant Lagrangian $n$-tori is degenerate on a $(2n-1)$-dimensional singular manifold $\mathbb{W}$ formed by the asymptotic manifolds of hyperbolic $(n-1)$-tori. The system usually ceases to be integrable after a small perturbation of order $\varepsilon$, but in accordance with the KAM-theory most invariant $n$-tori persist. The dynamics on the complement $C$ to this toric set is commonly associated with chaos. The measure of the set of points obtained as the intersection of a neighbourhood of $\mathbb{W}$ with $C$ is considered. Under natural assumptions it has the order of $\sqrt \varepsilon$. This results generalizes and complements the estimates for the measure of $C$ away from $\mathbb{W}$ due to Svanidze, Neishtadt and Pöschel. Bibliography: 14 titles.
Keywords: KAM-theory, separatrices, systems with small parameter, measure of the invariant tori, perturbation theory.
Mots-clés : chaos 
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     title = {On the measure of the {KAM-tori} in a~neighbourhood of a~separatrix},
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A. G. Medvedev. On the measure of the KAM-tori in a~neighbourhood of a~separatrix. Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 755-774. http://geodesic.mathdoc.fr/item/SM_2024_215_6_a2/