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Circle diffeomorphisms

This post has been migrated from my old blog, the math-physics learning seminar.

This is the first of a series of posts based on these lecture notes on KAM theory. For now I just want to outline section 2, which is a toy model of KAM thoery.

Circle Diffeomorphisms

We consider a map (\phi: \mathbb{R} \to \mathbb{R}) defined by

[ \phi(x) = x + \rho + \eta(x) ]

where (\rho) is its rotation number and (\eta(x)) is “small”.

Define (S_\sigma) to be the strip ({ |\mathrm{Im} z|<\sigma} \subset \mathbb{C}) and let (B_\sigma) be the space of holomorphic functions bounded on (S_\sigma) with sup norm (|\cdot|_\sigma).

Goal: Show that if (|\eta|_\sigma) is sufficiently small, then there exists some diffeomorphism (H(x)) such that

[ H^{-1} \circ \phi \circ H (x) = x + \rho ]

i.e. that (\phi) is conjugate to a pure rotation.

Linearization

The idea is that if (\eta) is small, then (H) should be close to the identity, so we suppose that

[ H(x) = x + h(x) ]

where (h(x)) is small. Plugging this into the equation above and discarding higher order terms yields

[ h(x+\rho) - h(x) = \eta(x) ]

Since (\eta) is periodic, we Fourier transform both sides to obtain an explicit formula for the Fourier coefficients of (h(x)). We have to show several things:

  1. The Fourier series defining (h(x)) converges in some appropriate sense.

  2. The function (H(x) = x + h(x)) is a diffeomorphism.

  3. The composition (\tilde{\phi} = H^{-1} \circ \phi \circ H) is closer to a pure rotation than (\phi), in the sense that

[ \tilde{\phi}(x) = x + \rho + \tilde{\eta}(x) ]

where (|\tilde{\eta}| \ll |\eta|).

Newton’s Method

Carrying out the analysis, one finds that for appropriate epsilons and deltas, if (\eta \in B_\sigma) then (H \in B_{\sigma - \delta}) and that (|\tilde{\eta}|_{\sigma-\delta} \leq C |\eta|_\sigma^2). By carefully choosing the deltas, we can iterate this procedure (composing the (H)’s) to obtain a well-defined limit (H_\infty \in B_{\sigma/2}) such that

[ H_\infty^{-1} \circ \phi \circ H_\infty (x) = x + \rho, ]

as desired.

So in fact the idea of the proof is extremely simple, and all of the hard work is in proving some estimates.