[MSE] Shape of orbit of iterations with base b on Shell-Thron-region

[sheldonison]

In MSE I've started a discussion providing also some pictures on the properties of shapes of the orbits 0 \to 1 \to b \to b^b \to ... and in which way there occurs "divergence". Such "divergence" has been proven for classes of bases b on boundary of the Shell-Thron-region by I.N.Baker & Rippon in 1983 and according to Sheldonison has been thoroughly        investigated by J.C.Yoccoz .                      

Hey Gottfried,

The unbounded values in the orbits of \( \exp_b^{[\circ n]}(0) \) for these values of b in your MSE post is really cool.
\( \phi=\frac{\sqrt{5}+1}{2};\;\;\;\;\lambda=\exp\left(\frac{2\pi i}{\phi}\right);\;\;\;b=\exp\left(\lambda\exp(-\lambda)\right);\;\;\;l=\exp(\lambda);\;\;\;l=b^L \)

Just a clarification on \( \lambda=\frac{2\pi i}{c} \) multiplies with c real, and/or rational.  I think Yoccoz's work primarily involved iterating \( z \mapsto \lambda z + z^2 \) and his proof of the sharp convergence; non-convergence of the Schroeder \( \Psi(f(z))=\lambda\Psi(z) \) when c is an irrational number.  Yoccoz proved that if c has a continued fraction that doesn't misbehave super badly, the the \( \Psi(z) \) series converges, and that \( \Psi(z) \) doesn't converge if the continued fraction misbehaves, see https://en.wikipedia.org/wiki/Brjuno_number  I don't know if Yoccoz's proof has been extended to proof it also applies to iterating exponentials, but the conjecture would be that it applies.  So when you find Siegel disc's pictures, they typically like to use a value of \( c=\phi=\frac{\sqrt{5}+1}{2} \) since the golden ratio has the most ideally behaved continued fraction so it tends to show nice easily computable fractal behavior.

I think \( \Psi^{-1}(z) \) is the ideal tool to understand the behavior, but perhaps Yoccoz's work isn't quite as directly applicable as the general work on https://en.wikipedia.org/wiki/Siegel_disc