new results from complex dynamics

[sheldonison]

...So far it seems these pertubed Fatou coordinates only give real analytic functions for conjugated complex fixed point pairs. To give it a distinguished name (and as "perturbured Fatou coordinates" does not really hit the point) I will call it from here bipolar iteration in mnemonic that the Abel function is defined on a sickel between *two* fixed points, in opposite to the regular iteration which is defined in a neighborhood of *one* fixed point.
.....
Second, the bipolar Fatou coordinates may exist between two real fixed points (though even that is not completely clear to me and not backed by the theory) but it may not be real analytic.

So the question is whether there is at all a real tetration on (1,oo) which is particularly analytic in the base at e^(1/e) ...

Any updates on whether the theory supports bipolar Fatou solutions between two real fixed points? This would be analogous to the tetration solution, in my complex base solution post, where I propose the bipolar solution continues along a circle past the Shell Thron boundary, where one of the fixed points switches from repelling to attracting.

...start with a parabolic fixed point \( z_0 \) of a holomorphic function \( f \), i.e. \( f(z_0)=z_0 \) and \( f'(z_0)=1 \).
If we slightly perturb this function by a complex \( \epsilon \), \( g(z)=f(z)+\epsilon \), then the one fixed point splits up into several fixed points (at least 2). But the perturbed function still behaves quite similar to the original function.

It would seem that the bipolar Fatou solutions should have encountered this exact question. I very briefly experimented with some of these ideas applied to a simple function like \( f(z)=z+z^2+\epsilon \), which has a parabolic fixed point at z=0, for epsilon=0, and it should have a real valued superfunction if \( \epsilon \) is a postive real number, which behaves similar to tetration for real bases \( >\eta \) in many ways.

...If we slightly perturb this function by adding \( \epsilon>0 \) ... then the one parabolic fixed point splits up into two conjugate repelling fixed points. If we slightly perturb with \( \epsilon<0 \) then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling).

So, just as in the tetration solution, the perturbed Fatou method generates a bipolar superfunction if both fixed points are repelling. And, like tetration, if epsilon starts out as a positive real number, then as epsilon goes around in a circle around 0 counterclockwise, very soon one of the two repelling fixed points switches to attracting. Also, of course, there is the case where epsilon is exactly on the transition, with one repelling fixed point, and the other fixed point rationally or irrationally indifferent, which would be analogous to the Shell Thron boundary. As \( \epsilon \) is varied, \( f(z)=z+z^2+\epsilon \) has such a boundary, which seems very similar to the Shell Thron boundary for tetration.

It would be interesting to know what the theory has to say about bipolar solutions on that boundary, and bipolar solutions inside the boundary. Also, mathematicians interested in perturbed Fatou coordinates, and parabolic implosion should have a natural interest in tetration, as an example of a bipolar Fatou solution.
- Sheldon