Real tetration as a limit of complex tetration

[Daniel]

Daniel - good to read this.
Just a reservation: did you as well consider the periodic points (additional to the fixpoints - infinitely many "sets of p(eriodic)-points" [where p is the length of the period in one set] for each natural p)? It seems impossible to define fractional iterates along the p periodic points in a set.  
I don't know what non-degenerate areas in the complex plane are remaining for which fractional complex iteration along continuous trajectories is possible after that ...

Thanks for the input Gottfried. My immediate concern is getting a handle on real tetration for \((e^{1/e},\ldots,\infty)\).


Consider the diagram of flows from the different countable infinity fixed points. Each flow has its own value for \(^ba\). They all agree for the same value when \(b\) is a whole number. The values of \(^ba\) when \(b\) is not an whole number determine a curve that passes though the real number line. My expectation is that the further the fixed point from the real number line, the less it will drive the flow, thus "relaxing" the oscillatory curves into the real number line. The values of \(^ba\) for from distant fixed points would then approximate real tetration. Because of the symmetrical nature of the system between the upper and lower half planes, the curve intersects the real number line perpendicularly. So take the best value from the most distant fixed point it is convenient to use, and discard the imaginary component. I do need to model this to back up my intuition.

For the big picture I was inspired to read Flow proof. It discusses the importance of when there is the confluence of chaos and stability in dynamical systems.



The first image is the chaotic escape tetration Mandelbrot fractal while the second is the periodic behavior.



Julia set of \((-1)^z\)

The exponential map \(1^z\) creates a region of stability while \((-1)^z\) mixes stability and chaos. Neither is conducive to creating a flow. Maybe tetration for \(^z{(-1)}\) is a simple number representing the statistical properties of the map. As far as periodic points, I tend to think of them as a set that is invariant under the exponential map.

My main conclusion is that I need to upgrade my mathematical education. I'm retired and very bored, so I need something to do.