In 1962, Moser succeeded in accomplishing the feat of proving the theorem in the space of infinitely differentiable functions [Moser, J. On invariant curves of area-preserving mappings of an annulus (1962)].
In fact, Moser used functions which are 333 times differentiable and the topology of uniform convergence on these 333 derivatives... The mere fact that it is necessary to use as many derivatives shows the difficulty of the proof. Nowadays, it is known that the theorem is true with 4 derivatives and false with 3 [Sur les courbes invariantes par les difféomorphismes de l'anneau Herman, Michael R. Astérisque, no. 144 (1986)].333 is half the number of the devil. Moser probably cut half of a deal with the devil.
The linked paper by Etienne Ghys is quite beautiful. The whole field of dynamical theory is beautiful, I got high from looking at Poincare sections just this morning. It uses all kinds of very mathematical things like Diophantine approximation, chaos theory, Farey tree, continued fractions, Fourier analysis, perturbation theory... Too bad it's not important right now.
The whole study of dynamical systems got started from the problem of stability of the solar system: is the solar system stable? The answer, it seems, is "According to KAM Theorem, almost surely yes, if we use Lebesgue measure, but almost surely now, if we use Baire meagerness."
We can therefore conclude that the set of rotation angles for which the perturbed motion is stable is of full Lebesgue measure and we should therefore be satisfied with this result since it covers most of the cases (but we should not forget that if we had preferred Baire to Lebesgue, we should have had the opposite conclusion).WAT. For more details, see the paper, section 10.
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