# Tetration Forum

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Greetings, the Tetration Forum has been online for almost ten years now. My own tetration website tetration.org is almost thirteen years old and I am reviewing making the site more readable. I haven't closely followed the conversations here for at least several years. What are the central concepts regarding tetration that I can read about here?
(03/08/2016, 03:58 AM)Daniel Wrote: [ -> ]Greetings, the Tetration Forum has been online for almost ten years now. My own tetration website tetration.org is almost thirteen years old and I am reviewing making the site more readable. I haven't closely followed the conversations here for at least several years. What are the central concepts regarding tetration that I can read about here?

Hi Daniel,

Welcome back.

Probably the two biggest ones are that Kneser's solution has been "rediscovered", as the preferred solution for extending tetration to the real and complex numbers. In addition, Henryk Trapmann has published a uniqueness proof for Kneser's solution. I have at least two versions of pari-gp code that implement Kneser's solution, written in pari-gp. The first implements tetration for real bases greater than $\eta=\exp(1/e)$; http://math.eretrandre.org/tetrationforu...hp?tid=486; convergence is not proven, but assuming convergence, it can be shown to be equal to Kneser's solution. The second program use both fixed points to generate tetration for complex bases as well, http://math.eretrandre.org/tetrationforu...p?tid=1017; again, convergence is not proven.

This second complex tetration base program generates the Abel function on a sickle, exactly meeting the uniqueness criteria. It works by solving the problem for iterating $z \mapsto \exp(z)+k$ instead of $y \mapsto b^y$ where $k=\ln(\ln(b))+1$, so k=0 (which is parabolic; with a formal asymptotic series for the two solutions) corresponds to base $\eta=\exp(1/e)$. There is a linear transformation, so the two problems; iterating in y or z, are congruent. This is what they call the perturbed fatou coordinate in complex dynamics, which is equivalent to an slog/abel function starting by perturbing bases $>\eta$. Henryk Trapmann occasionally bugs me to publish my work, but so far, they are only here on his tetration forum.

There are other solutions for tetration as well, Kouznetsov's solution, and Andrew's slog using simultaneous equations, which appear to give the same solution as Kneser's solution, though there is no proof. Andrew's slog converges too slowly to give results of reasonable accuracy, though Jay has an accelerated solution that works up to a little better than double precision, given enough computer time and memory. There is also Walker's solution, generated from the upper solution for $g(z)=\eta^{[\circ z]}$, that is infinitely differentiable, but is conjectured to be nowhere analytic! $f(z)=\lim_{n \to \infty}\ln^{[\circ n]} g(z+n)$. I can find a link if you like; Jay and I have both done computations for this real valued solution, after "rediscovering" Walker's solution.

Feel free to ask questions, and generate discussion. There is a lot of neat stuff posted here on the forums, though it is sometimes hard to find stuff, especially at first.
- Sheldon Levenstein
The outdated things most in need to be included on wikis, but do not have citable papers are:

For bases between 0 and 1, tetration is a periodic function, damped for basis $\vspace{15}e^{-e} and very rapidly converging to a square wave function for basis $\vspace{15}0. (no proof, just numerical results).

There is a direct connection to partition numbers (number theory), in the Taylor series.

There is a direct connection to graph theory this way and this way.

Also, many equations in use of different areas of mathematics already use iterated logarithms. But to consider those tetration to negative exponents, is necessary to consider tetration a multivalued function, with multiple branches.

I threw the ball on those threads, but nobody got interested.
@marraco, you brought up a couple of issues of interest to me.

(03/08/2016, 06:58 PM)marraco Wrote: [ -> ]There is a direct connection to partition numbers (number theory), in the Taylor series.
See Combinatorics. There are several types of set partitions.

Let $f(z)$ and $g(z)$ be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula.

$D^nf(g(z))=\sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^kf)(g(z))\left(\frac{Dg(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^ng(z)}{n!}\right)^{k_n}$

A partition of $n$ is $\pi(n)$, usually denoted by $1^{k_1}2^{k_2}\cdots n^{k_n}$ with $k_1+2k_2+ \cdots nk_n=k$; where $k_i$ is the number of parts of size $i$. The partition function $p(n)$ is a decategorized version of $\pi(n)$, the function $\pi(n)$ enumerates the integer partitions of $n$, while $p(n)$ is the cardinality of the enumeration of $\pi(n)$.

Setting $g(z) = f^{t-1}(z)$ results in

$D^n f^t(z) = \sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^k f)(f^{t-1}(z))\left(\frac{Df^{t-1}(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(z)}{n!}\right)^{k_n}$
The Taylors series of $f^t(z)$ is derived by evaluating
the derivatives of the iterated function at a fixed point
$f^t(0)$ by setting $z=0$ and separating out the $k_n$
term of the summation that is dependent on $D^n f^{t-1}(0)$.

$D^n f^t(0) = \sum \frac{n!(D^k f)(0)}{k_1! \cdots k_{n-1}!} \left(\frac{Df^{t-1}(0)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(0)}{(n-1)!}\right)^{k_{n-1}} + (D f)(0) D^n f^{t-1}(0)$

The remaining $p(n)-1$ terms of the summation are only dependent on $D^k f^{t-1}(0)$, where $0.

Let me know if you have any questions.

(03/08/2016, 06:58 PM)marraco Wrote: [ -> ]Also, many equations in use of different areas of mathematics already use iterated logarithms. But to consider those tetration to negative exponents, is necessary to consider tetration a multivalued function, with multiple branches.

Yes, if logarithms are infinitely multivalued, then tetration must account for it also. All my work is based on setting arbitrary fixed points to $z=0$. That allows me to consider the dynamics of the infinite branches, because each of the branches has a fixed point for $^{-\infty}a$. Setting the entropy to being low for the exponential map can be achieved by setting $a$ close to unity in $^za$. Then the dynamics of neighboring fixed points can be computed from a fixed point.
(03/08/2016, 10:18 AM)sheldonison Wrote: [ -> ]Hi Daniel,
Welcome back.
Sheldon, thank you for your gracious welcome.

(03/08/2016, 10:18 AM)sheldonison Wrote: [ -> ]Probably the two biggest ones are that Kneser's solution has been "rediscovered", as the preferred solution for extending tetration to the real and complex numbers.
In what manner is Kneser's solution preferred? Please explain the mathematics behind it or provide links. My understanding is that use of Schroeder's equation and Abel's equation are exclusive. Abel's equation is for limit point $c$ where $f\large(c\large) = c$ and $\left|f'\large(c\large) \right|=1.$ Schroeder's equation is for $\left|f'\large(c\large) \right| \notin 0,1.$

(03/08/2016, 10:18 AM)sheldonison Wrote: [ -> ]This second complex tetration base program generates the Abel function on a sickle, exactly meeting the uniqueness criteria. It works by solving the problem for iterating $z \mapsto \exp(z)+k$ instead of $y \mapsto b^y$ where $k=\ln(\ln(b))+1$, so k=0 (which is parabolic; with a formal asymptotic series for the two solutions) corresponds to base $\eta=\exp(1/e)$. There is a linear transformation, so the two problems; iterating in y or z, are congruent. This is what they call the perturbed fatou coordinate in complex dynamics, which is equivalent to an slog/abel function starting by perturbing bases $>\eta$.
This looks interesting. I'm late to the game of looking for a tetration that maps the reals into the reals.

(03/08/2016, 10:18 AM)sheldonison Wrote: [ -> ]There are other solutions for tetration as well, Kouznetsov's solution, and Andrew's slog using simultaneous equations, which appear to give the same solution ...
I am very interested in the different solutions for tetration and how they fit together. Aldrovandi uses Bell matrices for both fractional iteration and tetration. My own work on fractional iteration is numerically consistent with Aldrovandi's work. I am somewhat familiar with Gottfried Helm's work with Bell and Carleman matrices.
(03/10/2016, 02:14 AM)Daniel Wrote: [ -> ]....
In what manner is Kneser's solution preferred? Please explain the mathematics behind it or provide links. My understanding is that use of Schroeder's equation and Abel's equation are exclusive. Abel's equation is for limit point $c$ where $f\large(c\large) = c$ and $\left|f'\large(c\large) \right|=1.$ Schroeder's equation is for $\left|f'\large(c\large) \right| \notin 0,1.$

Daniel,

Henryk has a couple of posts here. First, I confess that when I wrote my first kneser.gp pari-gp program nearly six years ago, I was merely guessing at Kneser's solution, but did not understand Kneser's Riemann mapping. Later I showed that my theta mappings are equivalent to Kneser's Riemann mapping, assuming convergence.
http://math.eretrandre.org/tetrationforu...hp?tid=213
http://math.eretrandre.org/tetrationforu...hp?tid=358

Hopefully, I don't make any typos! Now, start with the two fixed points, L,L*=0.31813 +/- 1.3372i; each fixed point leads to a Schroeder equation, and its inverse. You are familiar with the Schroeder equation, lets call the Schroeder solution $S(z)$. Call the Abel equation $\alpha(z)$, and its inverse, $\alpha^{-z}(z)$

$\alpha^{-1}(z) = S^{-1}(L^z)\;\;\;$ This is the complex valued superfunction, which you are familiar with

Now, we define two 1-cyclic $\theta(z), \; \theta*(z)$ mappings. One theta(z) mapping is defined in the upper half of the complex plane; and the other in the lower half of the complex plane.

$\theta(z)=\sum_{n=0}^{\infty} a_n
\cdot \exp(2n\cdot \pi i z)$

$\text{sexp}(z) = \alpha^{-1}_L(z+\theta(z)) \;\;\;$ we require sexp(z) be real valued at the real axis
$\text{sexp}(z) = \alpha^{-1}_{L*}(z+\theta*(z))$

Now, it turns out one can write an equation for $\theta(z)$ exactly in terms of Kneser's Riemann mapping. $\theta(z)$ has a really nasty singularity at the real axis for integer values of z, but is analytic with no other singularities in the upper half of the complex plane, and $\theta(z)$ decays to a constant as imag(z) goes to infinity.
$\lim_{z \to + i\infty}\; \theta(z)=a_0=k$

Hopefully, that's a start for you.
- Sheldon Levenstein

(03/10/2016, 03:06 AM)sheldonison Wrote: [ -> ]
(03/10/2016, 02:14 AM)Daniel Wrote: [ -> ]....
In what manner is Kneser's solution preferred? Please explain the mathematics behind it or provide links. My understanding is that use of Schroeder's equation

... Now, it turns out one can write an equation for $\theta(z)$ exactly in terms of Kneser's Riemann mapping.

Actually, I find it easiest to write an equation for Kneser's Riemann mapping in terms of my $\theta(z)$ mapping; Here are the relevant equations:

If you start with the real valued solution, sexp(z), then you can generate Kneser's Riemann mapping as follows in terms of the complex valued Abel function, $\alpha(z)$
$f(z) = \alpha(\text{sexp}(z)) = z +\theta(z)\;\;$ here, $\theta(z)$ is my theta(z) mapping with trivial algebra since
$\text{sexp}(z) = \alpha^{-1}(z+\theta(z)) \;\;\;$ this is my theta mapping for sexp(z)

Then Kneser's Riemann mapping results in $\exp(2\pi i \cdot f(z))$ where z has been wrapped around a unit circle by using the substitution $z = \frac{\log(y)}{2\pi i}$.

Then Kneser's Riemann mapping results in this unit circle function in terms of $f(z) = z +\theta(z)$

$\exp \left[ 2\pi i \cdot f\left(\frac{\log(y)}{2\pi i} \right) \right]
\;\;\;$
This is Kneser's Riemann mapping on a unit circle in terms of y and $f(z) = z +\theta(z)$

From there, Kneser works backwards to $f(z) = z+\theta(z)\;\;\;\text{sexp}(z)=\alpha^{-1}(f(z))\;\;\;$ Kneser's real valued sexp(z) function using my notation
(03/10/2016, 03:06 AM)sheldonison Wrote: [ -> ]Now, start with the two fixed points, L,L*=0.31813 +/- 1.3372i; each fixed point leads to a Schroeder equation, and its inverse.

Yes, I am familiar with the fixed points of the exponential function $e^z$

(03/10/2016, 03:06 AM)sheldonison Wrote: [ -> ]You are familiar with the Schroeder equation, lets call the Schroeder solution $S(z)$. Call the Abel equation $\alpha(z)$, and its inverse, $\alpha^{-z}(z)$

$\alpha^{-1}(z) = S^{-1}(L^z)\;\;\;$ This is the complex valued superfunction, which you are familiar with
Using the Classification of Fixed Points, there are several types of tetration. Hyperbolic tetration given by Schroeder's equation which accounts for almost all values. Parabolic tetration given by Abel's equation is only for $a=e^{1/e}$, rationally neutral tetration $a=e^{-e}$, super attracting tetration $a=1$. While I have looked for a simple way to move between Abel's and Schroeder's equations, they are not topologically conjugate.

(03/10/2016, 04:55 AM)Daniel Wrote: [ -> ]
(03/10/2016, 03:06 AM)sheldonison Wrote: [ -> ]You are familiar with the Schroeder equation, lets call the Schroeder solution $S(z)$. Call the Abel equation $\alpha(z)$, and its inverse, $\alpha^{-z}(z)$

$\alpha^{-1}(z) = S^{-1}(L^z)\;\;\;$ This is the complex valued superfunction, which you are familiar with
Using the Classification of Fixed Points, there are several types of tetration. Hyperbolic tetration given by Schroeder's equation which accounts for almost all values. Parabolic tetration given by Abel's equation is only for $a=e^{1/e}$, rationally neutral tetration $a=e^{-e}$, super attracting tetration $a=1$. While I have looked for a simple way to move between Abel's and Schroeder's equations, they are not topologically conjugate.

Kneser's would take the Schroeder function of the real valued tetration function, to get what he called the Chi-Star function; but I was actually showing my construction, and its equivalence to Kneser's; see my last post showing the Riemann mapping to a circle. As far as the topologically conjugate issue, yes the $\alpha^{-1}(z)=S^{-1}(L^z)\;\;$ complex valued superfunction is Periodic in the complex plane with period~= 4.447 + 1.058i, so it is not one to one. You have to be consistent with your branch to get the real valued sexp(z) via the theta(z) function. Knesser's Riemann mapping for that matter, must also be consistent. But the periodicity of the complex valued superfunction is not an issue. And in fact, Kneser's tetration solution is psuedo periodic with a pseudo period approaching arbitirarily closely to the period of the complex valued superfunction as imag(z) increases.
Daniel, perhaps you've not seen that, but I've composed a notice about some corrections for your tetration.org site, see in the "computation" subforum. I hope that not all my observation were useless...

Gottfried
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