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We write down an asymptotic expression for action coordinates in anintegrable Hamiltonian system with a focus-focus equilibrium. From thesingularity in the actions we deduce that the Arnol'd determinant growsinfinitely large near the pinched torus. Moreover, we prove that it is possibleto globally parametrise the Liouville tori by their frequencies. If oneperturbs this integrable system, then the KAM tori form a Whitney smoothfamily: they can be smoothly interpolated by a torus bundle that isdiffeomorphic to the bundle of Liouville tori of the unperturbed integrablesystem. As is well-known, this bundle of Liouville tori is not trivial. Ourresult implies that the KAM tori have monodromy. In semi-classical quantummechanics, quantisation rules select sequences of KAM tori that correspond toquantum levels. Hence a global labeling of quantum levels by two quantumnumbers is not possible.Comment: 11 pages, 2 figure

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