A Cantor set of tori with monodromy near a focus-focus singularity

Abstract

We write down an asymptotic expression for action coordinates in an integrable Hamiltonian system with a focus-focus equilibrium. From the singularity in the actions we deduce that the Arnol'd determinant grows infinitely large near the pinched torus. Moreover, we prove that it is possible to globally parametrise the Liouville tori by their frequencies. If one perturbs this integrable system, then the KAM tori form a Whitney smooth family: they can be smoothly interpolated by a torus bundle that is diffeomorphic to the bundle of Liouville tori of the unperturbed integrable system. As is well-known, this bundle of Liouville tori is not trivial. Our result implies that the KAM tori have monodromy. In semi-classical quantum mechanics, quantisation rules select sequences of KAM tori that correspond to quantum levels. Hence a global labeling of quantum levels by two quantum numbers is not possible.

arXiv comment11 pages, 2 figures

References in corpus (2)

• Direction reversing travelling waves in the Fermi-Pasta-Ulam chain
• Geometry of KAM tori for nearly integrable Hamiltonian systems

Cited by in corpus (7)

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• Defect in the Joint Spectrum of Hydrogen due to Monodromy
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• Hamiltonian monodromy and Morse theory
• Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance
• Nearly-integrable perturbations of the Lagrange top: applications of KAM-theory
• Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems