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

[sheldonison]

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

Hi Sheldon -

in the meantime I've aggregated more data for plots based on c=silver constant, and I've to recompute accordingly data for the other types of c - just to apply the new insights to the small paper that I'd linked to, and hopefully extend that small paper towards becoming a better "catalogue".            

However, I need a break for a couple of days and surely cannot be much productive in this matter. A question which is coming up at the moment is how to characterize the interior of the \( z_0=1 \) fractal shape, say the set of curves produced by  \( 0.5 < z_0 < 0.99 \), somehow like a gradient-field, displaying little arrows instead of dots, perhaps including directions of fractional iteration-height (as far as this might be meaningfully applicable). That's just the desire to embed the observation into known phenomena in other areas, hopefully of the physical world.           

Moreover, I think I've to meditate first a bit about that Yoccoz-work and Siegel-discs which you have directed me to and what this shall give me for the understanding of the whole phenomen.

Cordially -  
Gottfried

Gottfried,
I just read your equater.pdf; very nice.  There you also cover the dynamics for the rational case as well.  Since the Schroeder function doesn't exist when the multiplier at the fixed point is 1, or a rational root of 1, then one can use Ecalle's solution for the Abel function.  Considering what happens as the continued fraction becomes less well behaved is more difficult, as the irrational number gets closer and closer to behaving like a rational number.  Most cases with an irrational multiplier still have a Schroeder function so there is still an infinite number of copies where \( z_n \) gets arbitrarily close to zero and there is a logarithmic singularity at zero so \( z_{n-1} \) gets arbitrarily large so the fractal should still be unbounded, but it might take an uncountable number of iterations to show that behavior...  The nice thing about using a multiplier of the golden ratio is that one can actually compute the Schroeder function and get good numerical results for the orbits for  \( 0.5 < z_0 < 0.99 \), which gives the same gradient curves as iterating \( z\mapsto b^z \).  

The Schroder function has a 1-1 mapping from the unbounded fractal to a circle.  One can also study the Julia set for these iteration mappings, but I haven't done it.  There are an infinite number of other pre-images of the fractal since the \( b^z \) exponential function has a period of \( \frac{\2\pi i}{\ln b} \).  

Thanks for posting a delightful topic, both on MSE, and here.