Real tetration as a limit of complex tetration

[Daniel]

I have delivered the first release of my Mathematica software as a function for performing fractional iteration. I plan on using this as a basis for my future research. But before I model tetration through software, I need to model it mentally so that I have some expectations of what I will see.

The type of complex tetration I do is based on fixed points. There seems to be general agreement as to the value of expressions like \(^zb\) when evaluated with \(b\) within the Shell-Thron region. Since my work is based on fixed points and there are a countable infinity of fixed points, I argue that complex tetration should be considered as an infinite ensemble with one tetration function per fixed point.

What I expect to see is an infinite set of tetration functions that agree in the tetration of whole numbers, but vary from one another for fractional values. The fixed points imaginary values grows linearly while their real values grow much slower. Considered in polar coordinates from the fixed point, as the radius grows, the arc length from one whole number tetration to the next becomes smaller. I'm expecting to see real tetration as a limit of complex tetration as the fixed points approach imaginary infinity. I expect to see the variance of imaginary values to be inverse to the distance of the fixed point from the real numbers.

[Gottfried]

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 ...

[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.

[tommy1729]

I agree with Daniel.

I had similar ideas in the past.
I made a post somewhere I am sure of that.


The thing is this :

If the derivative at the fixpoint has small imaginary value the angular rotation is small.

So if the fixpoint is also far away , combined with bounded angular rotation , we get almost no rotation.

The rotation converges to zero if the distance to the fixpoint converges to infinity and we pick a subset that has small angular rotation ( aka imag part of the derivative ).

So the question is ; is it justified to take that limit ?? 

Does that converge properly to a unique and analytic function ?

I am not sure any of this or a variant is even justified but it is worth investigating.



regards

tommy1729

[Daniel]

I agree with Daniel.

I had similar ideas in the past.
I made a post somewhere I am sure of that.


The thing is this :

If the derivative at the fixpoint has small imaginary value the angular rotation is small.

So if the fixpoint is also far away , combined with bounded angular rotation , we get almost no rotation.

The rotation converges to zero if the distance to the fixpoint converges to infinity and we pick a subset that has small angular rotation ( aka imag part of the derivative ).

So the question is ; is it justified to take that limit ?? 

Does that converge properly to a unique and analytic function ?

I am not sure any of this or a variant is even justified but it is worth investigating.



regards

tommy1729

I'm first trying to develop a mental picture of the problem so that I have something to look for when I model the problem. Plus we now have Paulsen's JavaScript calculator, which I believe gives valid results for expressions like \(^{.5}2\). Next I need to analyze the values of an expression like \(^{.5}2\) that are associated to the different fixed points.

[tommy1729]

I want to add something important I forgot to mention.

I think we need to use fixpoint pairs being a fixpoint and its conjugate.

We know this fails for any finite fixpoint pair but taking the limit to infinity maybe helps ...


regards

tommy1729

[leon]

Complex tetration can be as simple as a Riemann sum and a Schroeder equations so limiting it  with standard tetration is like the step before pentation the more you do it the more you get it


Using the two together is like tetration that is more operative on open variables that 2 specific divergent tetrations so fast you can split the variables down the middle like a Gauss string my thread on a similar subject to help explain how it's basically always better to use just tetration even if you can do tetration and something else at the same time, or two (even complex) tetrations at the same time or two and something else like two tetrations and a pentation. That will fix the leap to pentation and I can explain two pentation at the same time with a good hyper-infinite perspective

https://tetrationforum.org/showthread.php?tid=1775