F(f(x))=exp(exp(x)).

[JmsNxn]

A quick way to remedy this problem is stick to a fixed point.

Call \(G(x) = \exp^{\circ n}(x)\), and we want to know how many \(n\) roots there are about the fixed point. The general rule is that there are \(n\) \(n\)-roots about a fixed point.

So if:

\[
\Psi(G(z)) = \lambda \Psi(z)\\
\]

There are precisely \(n\) functions \(g_i\) such that:

\[
\Psi(g_i(z)) = \sqrt[n]{\lambda} \Psi(z)\\
\]

These are given as:

\[
g_i(z) = \Psi^{-1}\left(\sqrt[n]{|\lambda|}\zeta_i\Psi(z)\right)\\
\]

Where \(\zeta_i\) is an \(n\)'th root of unity.