Circle Diffeomorphisms I

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.