The "cheta" function

[bo198214]

Hmm, that sparked a memory of a long-forgotten converstation we had:
http://math.eretrandre.org/tetrationforu...ght=entire

It's amazing how much more sense all of that makes now (and clear from my posts where my misunderstandings at the time were, as well as my gaps in understanding complex analysis). And yes, looking at just the descriptions in that old discussion, it would appear that cheta is nothing new.

Yaya, things are also much clearer for me now. As I already suggested the book of Milnor deals exactly with (the dynamics of) that cases, a must read.

E.g. non-parabolic fixed points \( |f'(0)|\neq 0,1 \) have always a neighborhood that is either attractive |f'(0)|<1 or repellent |f'(0)|>1. The regular iteration powerseries has non-zero convergence radius.

Parabolic fixed points however have no such neighborhood but alternating attractive and repelling petals (the Leau-Fatou-flower). If the powerseries is of the form
\( z+a_m z^m + a_{m+1} z^m + \dots \), \( a_m\neq 0 \), one says it is of multiplicity \( m \).
E.g. exp(z)-1 at 0 or \( \eta^z \) at \( e \) is of multiplicity 2.
Now the Leau-Fatou-flower has m-1 attractive and m-1 repellent petals.
In our case e^x-1 the one repelling petal covers \( x>0 \) and the attractive petal covers \( x<0 \).
Each petal has an associated (regular) Abel function (and hence regular iteration).
The regular iterations of any petal have the same asymptotic powerseries in the fixed point. But this asymptotic powerseries has mostly convergence radius 0 (non-integers iterates).

I hope I could make the matter more clear for our forum members.

The bright side is, this saves me the work of having to rigorously prove various properties, as they apparently have been proven for nearly 20 years. I need only work on getting good numerical approximations (several thousands of bits of accuracy), to use in my change-of-base formula.

Haha dont dare! You have to show that your algorithm indeed reproduces the regular iteration!
And whether the corresponding superfunction for base \( b>\eta \) is holomorphic is also not solved yet, we only know about infinite differentiability by Walker.
If he uses your change of base at all, but to find out is your task now.