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.
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