Subsection4.1.1Constructing the conjugation as a functional limit
The proof of Theorem 2 consists of two essential steps - (1) the construction of an approximation to the solution and (2) the proof that the construction converges as the approximation is refined. As such, the proof can be appreciated on multiple levels. From a computational perspective an understanding of the construction is essential in a couple of algorithms we'll develop for image generation.
We define \(\varphi_n(z) = \lambda^{-n}f^n(z)\text{.}\) Then,
\begin{equation*}
\varphi_n\circ f = \lambda^{-n}f^{n+1} = \lambda \varphi_{n+1}.
\end{equation*}
Now, if we assume that the sequence \(\varphi_n\) converges to \(\varphi\text{,}\) then \(\varphi\) must satisfy
\begin{equation*}
\varphi\circ f = \lambda \varphi = L\circ\varphi.
\end{equation*}
The tricky part is proving convergence. To this end, let's choose a number \(r > 0\) small enough and a positive constant \(C\) big enough so that
\begin{equation*}
\left|f(z)-\lambda z\right| \leq C|z|^2
\end{equation*}
whenever \(|z| < r\text{.}\) We can do this, of course, because the \(\lambda z\) is the first term in the Taylor series for \(f\text{.}\) Thus, an application of the reverse triangle inequality yields
\begin{equation*}
\left|f(z)\right| \leq |\lambda||z|+C|z|^2 \leq (|\lambda|+Cr)|z|.
\end{equation*}
Then, by induction,
\begin{equation*}
\left|f^n(z)\right| \leq (|\lambda| + Cr)^n|z|.
\end{equation*}
Now, if \(r\) is small enough so that
\begin{equation*}
\delta \equiv \frac{(|\lambda|+Cr)^2}{|\lambda|} < 1,
\end{equation*}
we have \begin{align*} \left|\varphi_{n+1}(z)-\varphi_n(z)\right| &= \left|\frac{f(f^n(z)) - \lambda f^n(z)}{\lambda^{n+1}}\right| \\ &\leq C\frac{\left|f^n(z)\right|^2}{|\lambda|^{n+1}} \leq C\frac{\delta^n|z|^2}{|\lambda|}. \end{align*} This implies that the sequence \(\varphi_n\) is uniformly Cauchy on the disk \(\{z:|z|<r\}\) and, therefore, uniformly convergent.