08/08/2007, 04:44 PM

This category being empty is a perfect opportunity to think about the proper way to categorize this subject. If you go to Math Atlas you may find bits and pieces about iteration and tetration in Functional Equations, in Dynamical Systems, or Foundations or higher-level Statistics even. These are not the subjects of iteration/tetration. I believe that there are 4 sub-subjects within the subject that most people are interested in (whatever its called - you know - the field tetration researchers are in) and I think that these 4 sub-subjects are a natural separation of tetration-related research:

* Generalized Hyper-operators

* Generalized Iteration

* Iterated Exponentials

* Nested Exponentials

All four naturally have a lot to do with iteration, and as such there are 4 commonly encountered function types which people tend to perform iteration upon (Bennet actually classifies around 15 function types in his "Analytic Iteration"):

* All Functions

* Analytic Functions

* Analytic Functions with a Hyperbolic Fixed-point (ex: f(0) = 0 and f'(0) < 1)

* Analytic Functions with a Parabolic Fixed-point (ex: f(0) = 0 and f'(0) = 1)

Any post talking about iteration should explicitly specify which kind of functions they apply to, as it would facilitate searching, and cataloging the content posted. This can be done with keywords, like "parabolic" or "hyperbolic iteration" or something to that effect.

I have also noticed some commonalities between various methods of approaching generalized iteration of analytic functions:

* Functional Equations (Abel, Schroeder, Boettcher, Julia, ref: Szekeres)

* Double-Binomial Expansions (Woon's, Trappmann's)

* Iterate-Derivative Matrices (ref: Bell's "Iterated Exponential Integers")

* Power-Derivative Matrices (Bell matrix, Carleman matrix, refs:Koch,Jabotinsky)

I hesitate to call these separate methods, because they all seem to produce identical results for (parabolic FP) analytic functions, and some may apply to (hyperbolic FP) analytic functions as well. The only method I have encountered so far that doesn't produce results consistent with the 4 types of methods above is Iga's method which is not an exact "analytic iteration" method but a set-of-all-possibilities "continuous iteration" method, and as such, one member of Iga's set MAY be "analytic iteration", but he gives no clues as to how to find it.

Also, in http://en.wikipedia.org/wiki/Dynamical_s...inition%29 a wonderful notation for iteration is given, in three parts:

* As a 1-variable function Phi^t(x), (sometimes called an _Iterate_ or the "t-fold iterate of f")

* As a 1-variable function Phi_x(t), (sometimes called an _Orbit_ or the "orbit of f from x")

* As a 2-variable function Phi(t, x), (sometimes called a _Flow_)

But I am less interested in the notation, and more in the terminology. As far as I have seen, the three terms (iterate, orbit, flow) are not very common, but they are VERY useful in talking about iteration. Three analogous terms have been used for exponentiation for quite some time: powers, exponentials, and exponentiation respectively. I also think that the analogous terms for tetration are also emerging, and if we are going to talk at length about tetration we should begin to use them accordingly (notice how "power-tower" is nowhere in the list, I personally discourage the term "power-tower"):

* Hyper-power functions (hpr_n(x) == x^^n) where n is constant

* Tetrational functions (tet_x(y) == x^^y) where x is constant

* Tetration (hyper4(x, y) == x^^y)

After extensive meditation, I realized that hyper-operators are based on orbits, not iterates, so naturally, a better definition of tetration would be: "tetration is the orbit of an exponential from 1".

Lastly, although Galidakis gives a nearly exhaustive list of references, I think there should be place set aside for links and references that other people can add to if possible. For example, for many of the integer sequences I have been researching, I have found a variant of them on OEIS, and most of the time I find they were posted by Vladeta Jovovic. Although I have yet to contact Jovovic, if i didn't know better, I would guess that Jovovic is researching tetration as well! Perhaps if this forum takes off, people like Jovovic can post here directly, and not have to distill their research into an integer sequence suitable for OEIS.

Andrew Robbins

* Generalized Hyper-operators

* Generalized Iteration

* Iterated Exponentials

* Nested Exponentials

All four naturally have a lot to do with iteration, and as such there are 4 commonly encountered function types which people tend to perform iteration upon (Bennet actually classifies around 15 function types in his "Analytic Iteration"):

* All Functions

* Analytic Functions

* Analytic Functions with a Hyperbolic Fixed-point (ex: f(0) = 0 and f'(0) < 1)

* Analytic Functions with a Parabolic Fixed-point (ex: f(0) = 0 and f'(0) = 1)

Any post talking about iteration should explicitly specify which kind of functions they apply to, as it would facilitate searching, and cataloging the content posted. This can be done with keywords, like "parabolic" or "hyperbolic iteration" or something to that effect.

I have also noticed some commonalities between various methods of approaching generalized iteration of analytic functions:

* Functional Equations (Abel, Schroeder, Boettcher, Julia, ref: Szekeres)

* Double-Binomial Expansions (Woon's, Trappmann's)

* Iterate-Derivative Matrices (ref: Bell's "Iterated Exponential Integers")

* Power-Derivative Matrices (Bell matrix, Carleman matrix, refs:Koch,Jabotinsky)

I hesitate to call these separate methods, because they all seem to produce identical results for (parabolic FP) analytic functions, and some may apply to (hyperbolic FP) analytic functions as well. The only method I have encountered so far that doesn't produce results consistent with the 4 types of methods above is Iga's method which is not an exact "analytic iteration" method but a set-of-all-possibilities "continuous iteration" method, and as such, one member of Iga's set MAY be "analytic iteration", but he gives no clues as to how to find it.

Also, in http://en.wikipedia.org/wiki/Dynamical_s...inition%29 a wonderful notation for iteration is given, in three parts:

* As a 1-variable function Phi^t(x), (sometimes called an _Iterate_ or the "t-fold iterate of f")

* As a 1-variable function Phi_x(t), (sometimes called an _Orbit_ or the "orbit of f from x")

* As a 2-variable function Phi(t, x), (sometimes called a _Flow_)

But I am less interested in the notation, and more in the terminology. As far as I have seen, the three terms (iterate, orbit, flow) are not very common, but they are VERY useful in talking about iteration. Three analogous terms have been used for exponentiation for quite some time: powers, exponentials, and exponentiation respectively. I also think that the analogous terms for tetration are also emerging, and if we are going to talk at length about tetration we should begin to use them accordingly (notice how "power-tower" is nowhere in the list, I personally discourage the term "power-tower"):

* Hyper-power functions (hpr_n(x) == x^^n) where n is constant

* Tetrational functions (tet_x(y) == x^^y) where x is constant

* Tetration (hyper4(x, y) == x^^y)

After extensive meditation, I realized that hyper-operators are based on orbits, not iterates, so naturally, a better definition of tetration would be: "tetration is the orbit of an exponential from 1".

Lastly, although Galidakis gives a nearly exhaustive list of references, I think there should be place set aside for links and references that other people can add to if possible. For example, for many of the integer sequences I have been researching, I have found a variant of them on OEIS, and most of the time I find they were posted by Vladeta Jovovic. Although I have yet to contact Jovovic, if i didn't know better, I would guess that Jovovic is researching tetration as well! Perhaps if this forum takes off, people like Jovovic can post here directly, and not have to distill their research into an integer sequence suitable for OEIS.

Andrew Robbins