Invariant curves near the boundary of an annulus without a twist hypothesis, following M. R. Herman

Invariant curves near the boundary of an annulus without a twist hypothesis, following M. R. Herman

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John Mather, Princeton University
Fine Hall 401

Sometime in the nineties, M. R. Herman gave a series of lectures at Columbia on KAM theory. Yasha asked me to speak on one of the results that Herman discussed in his lectures. Here is the result:Let $f$ be an area preserving infinitely differentiable diffeomorphism of a closed annulus. Suppose that the restriction of $f$ to one of the boundary components is a rotation whose rotation number satisfies a Diophantine condition. Then there exist an infinite number of rotational invariant curves in an arbitrarily small neighborhood of the given boundary component.
This result differs from earlier results in that no twist hypothesis is assumed, although then it is necessary to add the hypothesis given above about the restriction of f to the boundary component. The proof is a simple application of the formidable machinary of KAM theory. In this talk, I will state the relevant general result from KAM theory and deduce Herman's theorem from them.