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07/20/2019, 03:39 PM
(This post was last modified: 07/20/2019, 03:41 PM by Daniel.)
Members of the Tetration Forum have been interested in extending the domain and range of tetration to the real numbers. The problem is that the results of Schroeder's functional equation are exponential functions whose domain is the complex numbers. But the results of Abel's functional equation's allows for polynomial functions such that the domain and range are real numbers.

Now the ugly part. According to the Classification of Fixed Points while setting the fixed point to zero, Abel's functional equation only works when

. Can you get a paper published with this mistake? Well, people are doing it. To my knowledge, the only published paper handling Schroeder's functional equation is by Aldrovandi and Freitas. My research from the Nineties produces the same results that Aldrovandi and Freitas obtain.

R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps,
J. Math. Phys. 39, 5324 (199

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If you have a Schröder solution

then you also have an Abel solution

, by

. Can't remember where the Schröder solution was introduced (am on travel - limited resources).

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The Abel functional equation and the Schroeder functional equation represent different symmetries. Saying that there can be a mathematical connection between the Abel and Schroeder functional equations doesn't mean that they coexist in the same system. Let's consider complex functions. Generally speaking, we will be looking at a system with an infinite set of hyperbolic fixed points, although complex conjugates can give pairs of parabolic fixed points along with infinite hyperbolic fixed points. The same for parabolically neutral fixed points. So while some folks like the properties of Abel's functional equation, the mathematical existence of these systems are rare.

I am striving for the ultimate generalization of my work (we probably all are). That means working with iterated functions in Banach or Frechet space. Representation theory ties symmetries and matrices together. Currently, the Tetration Forum focuses on real numbers as iterators while I'm looking at how to use the General Linear group.

At Wolfram's request, I'm working on a Mathematica application that can be migrated to the new Mathematica Notebook interface. Having this application will allow folks to see tetration based phenomena that is core to my work and that I have never seen a reference to. I hope this will be a picture that is worth a thousand words.

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07/21/2019, 08:54 PM
(This post was last modified: 07/22/2019, 02:32 PM by sheldonison.
*Edit Reason: Typo*
)
(07/21/2019, 01:29 AM)Daniel Wrote: The Abel functional equation and the Schroeder functional equation represent different symmetries. Saying that there can be a mathematical connection between the Abel and Schroeder functional equations doesn't mean that they coexist in the same system. Let's consider complex functions. Generally speaking, we will be looking at a system with an infinite set of hyperbolic fixed points, although complex conjugates can give pairs of parabolic fixed points along with infinite hyperbolic fixed points. The same for parabolically neutral fixed points. So while some folks like the properties of Abel's functional equation, the mathematical existence of these systems are rare.

I'm not that fluent in group theory, which is where symmetries would be studied, but I certainly get that the set of Tetration bases with an attracting fixed point is bound by our familiar Shell-Thron boundary curve.

All of the bases inside this curve have an attracting fixed point with a well defined Schroeder equation from which the standard equation can be used to generate the Abel function, and the slog. and this family of slogs is an analytic family of slogs. And for bases outside this region, the function

is a singularity.

is the derivative at the attracting fixed point.

One interesting question is does this limit exist, and is it equal to the slog generated from the Ecalle's parabolic Abel function at

?

And then another interesting question, for the Kneser's family of slogs, is the limit the same as the base approaches eta (from>eta)? The Kneser family of slogs can be extended to the inside of the Shell-Thron region, and would be unequal to the Schroeder family of slogs everywhere else other than possibly b=eta.

- Sheldon

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Just to reiterate my position (har har har), I study dynamics. So when I discuss tetration, it is solely as an example of an iterated function. I could just as easily be discussing pentation or inverse hexation. You want to take one matrix tetrated to a second matrix? Sure. Pentation of quaternions. Yup. A tetration superlog is just a pentation to -1.

My work provides a straight forward algebraic derivation of the entire Classification of Fixed Points including the Abel and Schroeder functional equations and their range. The one limitation of this work is it provides no insight into the Diaphontine irregular neutral fixed points on the Shell-Thron boundary.

Want to hear something cool about the Shell-Thron boundary curve? All systems in physics are measure-preserving dynamical systems. So any physical tetration system will have a base on the Shell-Thron boundary curve. Now strap yourselves in. The complex Shell-Thron boundary curve is associated with multipliers of the roots of unity. Now imagine a multi-dimensional Shell-Thron boundary curve where the multipliers are matrixes, maybe quaternions. The multi-dimensional Shell-Thron boundary is the loci for physically realizable systems.

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(07/22/2019, 12:24 AM)Daniel Wrote: Just to reiterate my position (har har har), I study dynamics. So when I discuss tetration, it is solely as an example of an iterated function....

All systems in physics are measure-preserving dynamical systems. So any physical tetration system will have a base on the Shell-Thron boundary curve. Now strap yourselves in. The complex Shell-Thron boundary curve is associated with multipliers of the roots of unity...

No matter what approach you take, studying tetration involves mastering a lot of complex dynamics. Your approach seems like a personal taste to focus strictly on the Schroder equation and on Ecalle's Abel function solution for f^[n] for rational roots of unity with denominator of "n", for bases on the Shell Thron boundary.

I'm not fluent in the math behind graduate level Physics, but the focus on neutral fixed points seems really cool.

- Sheldon