understand what’s going on, we have to better understand the derivatives of
$\tau^{n}$.

###  4.4 The Asymptotic Theorem of the third kind

The final form of our asymptotic theorems is to understand this asymptotic as
it works for derivatives and Taylor coefficients. This allows us to derive a
faux asymptotic series. I say “faux” because it will not take the form of the
usual idea of an asymptotic series; and will be a tad more involved.

Now that we know that $\tau$ is no where holomorphic on $\mathcal{B}$; we ask
how close it comes to being holomorphic. The answer, qui c’est, is as close as
it possibly can. Now, a corollary of The Asymptotic Theorem of the second kind
2.7.1; which we’ve yet to mention; is going to play a large role.

######  Corollary 4.4.1.

For all $n\in\mathbb{N}$ and all $l\in\mathbb{N}$, there exists a constant
$C_{nl}>0$; such for all $(s,\lambda)\in\mathbb{L}$ and $k>K$:

$\left|\left|\frac{d^{l}}{ds^{l}}\tau^{n}(s+k)\right|\right|_{\mathcal{N}}\leq
C_{nl}e^{-\Re\lambda k}\\\ $

For $\mathcal{N}$ a compact neighborhood of $s$ in $\mathbb{L}$.

###### Proof.

Since $\tau^{n}(s+k)\to 0$ uniformly as $k\to\infty$; then so do its
derivatives in $s$. Additionally, these terms converge like
$\mathcal{O}(e^{-\lambda k})$ because $\tau^{n}(s+k)=\mathcal{O}(e^{-\lambda
k})$. ∎

We are now at a point where we have to introduce what the Asymptotic Theorem
of the third kind is. This is a tad technical; but simple once you understand
the grand motions. There exists a sequence of functions $\sigma^{j}(s_{0})$
which are not necessarily continuous. Such that, if $P_{m}(s_{0};s)$ is a
polynomial:

$P_{m}(s_{0};s)=\sum_{j=0}^{m}\sigma^{j}(s_{0})(s-s_{0})^{j}\\\ $

Then,

$P_{m}(s_{0}+1;s)-\exp(P_{m}(s_{0};s))=\mathcal{O}(s-s_{0})^{m+1}\\\ $

This implies there’s an asymptotic series associated to
$F(s_{0})=\beta(s_{0})+\tau(s_{0})$; which doesn’t necessarily converge; but
if it doesn’t, it satisfies this functional equation. By which:

$F(s)\sim
P_{m}(s_{0},s)\,\,\text{like}\,\,(s-s_{0})^{m+1}\,\,\text{as}\,\,s\to s_{0}\\\
$

This result can be used to show; for instance, that $F_{1,1}(x)$ is infinitely
differentiable for $x\in\mathbb{R}^{+}$. But additionally, this applies to
everywhere in the weak Julia set where we converge. And if it doesn’t
converge; we are still guaranteed an asymptotic series which almost converges.

We’ve built up a lot of tools in this paper such that this result isn’t very
difficult to prove. To get a picture of how this works is a little nuanced–we
have to pay close attention to the sequence $\rho^{j}$. As $\rho^{j}(s+k)$
gets arbitrarily small as we increase $k$, and each $\rho^{j}(s_{0}+k)$ is
analytic in a neighborhood of $s_{0}\in\mathcal{B}$; the only way that $\tau$
diverges is if arbitrarily large derivatives of $\rho^{j}$ get arbitrarily
large.

Thereby, the result we are actually proving; is better phrased in a modded out
space. If we mod out by $\mathcal{O}(s-s_{0})^{m+1}$; then the algebra becomes
a bit cleaner. Let us call the spaces:

$\displaystyle\mathbb{P}_{m}$ $\displaystyle=\\{p\in\mathbb{C}[s],\,\text{the
space of polynomials with coefficients in}\,\mathbb{C}\,|\,\deg(p)=m\\}$
$\displaystyle\mathcal{H}_{s_{0}}$ $\displaystyle=\\{f\,\text{is holomorphic
in a neighborhood of}\,s_{0}\in\mathbb{C}\\}$

And the projection $\iota_{m}:\mathcal{H}_{s_{0}}\to\mathbb{P}_{m}$ which
takes:

$\displaystyle\iota_{m}(f(s))$ $\displaystyle=f(s)+\mathcal{O}(s-s_{0})^{m+1}$
$\displaystyle\iota_{m}\left(\sum_{j=0}^{\infty}a_{j}(s-s_{0})^{j}\right)$
$\displaystyle=\sum_{j=0}^{m}a_{j}(s-s_{0})^{j}$

Then the result we are really saying is that:

$\iota_{m}(\tau^{n}(s))=\iota_{m}\left(\sum_{j=1}^{n}\rho^{j}(s)\right)\to\sum_{j=0}^{m}\left(\sigma^{j}(s_{0})-\frac{\beta^{(}j)(s_{0})}{j!}\right)(s-s_{0})^{j}\\\
$

Where:

$\tau(s)=\sum_{j=0}^{m}\left(\sigma^{j}(s_{0})-\frac{\beta^{(j)}(s_{0})}{j!}\right)(s-s_{0})^{j}+\mathcal{O}(s-s_{0})^{m+1}\\\
$

Where as $m\to\infty$ we either diverge or converge; depending on whether we
are holomorphic or not. Now, to the untrained eye, this may look very close to
a proof of holomorphy. For that, it may help to remember the case $\mu=1$ and
$\lambda=1$ for $x\in\mathbb{R}^{+}$.

The function $\tau(x)$ will be smooth; and we’ll have the exact same
expansion. What’s to be remembered, this expansion will be discontinuous as we
leave the real-line. The expansion is precisely only valid for
$x_{0}\in\mathbb{R}^{+}$. Now the $\mathcal{O}(s-s_{0})^{m+1}$ term may be
complex; but it doesn’t reflect the accuracy of an expansion in the complex
plane. All we’ve done is created a polynomial $P_{m}$ such that:

$P_{m}(x_{0}+1;s)-\exp(P_{m}(x_{0};s))=\mathcal{O}(s-s_{0})^{m+1}\\\ $

Where it just so happens, that in this case we can assign these as derivatives
of $\beta(s)+\tau(s)$:

$P_{m}(x_{0};s)=\sum_{j=0}^{m}\left(\frac{d^{j}}{dx^{j}}\Big{|}_{x=x_{0}}\beta(x)+\tau(x)\right)\frac{(s-x_{0})^{j}}{j!}\\\
$

There just happens to be a path along the real-line where it is continuous.
This doesn’t necessarily happen for arbitrary points in the complex plane.
This is precisely why we’ve chosen the notation $\sigma^{j}(x_{0})$; it isn’t
necessarily the derivative in a meaningful sense of the word. It’s just an
induced coefficient of a polynomial; that can play the role of a derivative.

To over-explain; we are not always guaranteed the identity:

$\frac{d}{dx_{0}}\sigma^{j}(x_{0})=\sigma^{j+1}(x_{0})(j+1)$

But we are in the special case of $\mu=1$ and $\lambda=1$ and
$x_{0}\in\mathbb{R}^{+}$. We cannot claim this identity for
$s_{0}\in\mathbb{C}$; at least not in any sense of complex limits.

To get things rolling, we’ll begin by reducing our cases to large $k>K$ in the
orbit.

######  Lemma 4.4.2.

If $P_{m}(s_{0}+1;s)$ exists and additionally $\sigma^{0}(s_{0}+1)\neq 0$,
then there exists $P_{m}(s_{0};s)$.

###### Proof.

There exists a polynomial $P_{m}(s_{0};s)$ such that:

$\exp P_{m}(s_{0};s)-P_{m}(s_{0}+1;s)=\mathcal{O}(s-s_{0})^{m+1}\\\ $

So long as the first term of $P_{m}(s_{0}+1;s)$ is non-zero (there’s no
logarithmic singularity). By way, there is no polynomial $p$ in $s$ such that
$\exp p=\mathcal{O}(s)$; but there’s always a polynomial $p$ in $s$ such that
$\exp p=B+\mathcal{O}(s)$ for $B\neq 0$. ∎

From here we have to discuss our supremum norms. We’ll begin by writing, if
$a(s)\in\mathcal{H}_{s_{0}}$; is holomorphic about $s_{0}$:

$||a(s)||_{\mathcal{N},m}=||\iota_{m}(a(s))||_{\mathcal{N}}\\\ $

Which means the supremum norm on the Taylor polynomial up to $m$ terms of $a$
about a point $s_{0}$. The give all end all of The Asymptotic Theorem of the
third kind, is that we converge uniformly under this norm for fixed $m$. A
second norm that is going to be very important, is a derivative of this norm.
We’ll call this slightly differently:

$||a(s)||_{m}=\sum_{j=0}^{m}|a^{(j)}(s)|\\\ $

By which, these two norms are related as:

$1/\delta||a(s_{0})||_{m}\leq||a(s)||_{\mathcal{N},m}\leq||a(s_{0})||_{m}\\\ $

For some $\delta>0$. This is derived from bounding terms of the sum as:

$||a(s)||_{\mathcal{N},m}=\left|\left|\sum_{j=0}^{m}a^{(j)}(s_{0})\frac{(s-s_{0})^{j}}{j!}\right|\right|_{\mathcal{N}}\leq\sum_{j=0}^{m}|a^{(j)}(s_{0})|\\\
$

Which is true so long as $|s-s_{0}|\leq 1$; as we are concerned with
arbitrarily small neighborhoods $\mathcal{N}$ of $s_{0}$; it’s perfectly valid
to make this comparison. Bounding from below is just as straight forward.

Within the above norms we want to derive convergence of the $\tau$ process.
We’re going to follow a similar procedure as done in The Inverse Abel Theorem
3.1.5; but we’re going to have to be a tad more careful. But the steps are
similar.

We are given two different modes of convergence; which we’ve detailed fairly
well before. But, now, we have to talk about point wise norms, as opposed to
norms on compact sets. By which, we have to create a new distinction. The use
of these norms is to avoid much of the trouble of the weak Julia set.

######  Lemma 4.4.3.

Let $m\in\mathbb{N}$ and let $\iota_{m}$ be centered about $s_{0}+k$; such
that $\beta(s)$ is holomorphic in a neighborhood of $s_{0}$. Then,

$\lim_{k\to\infty}\left|\left|\frac{1}{\beta(s+k)}\right|\right|_{\mathcal{N},m}<\infty\\\
$

######  Remark 4.4.4.

It’s very important to notice that we are using the projection $\iota_{m}$
about $s_{0}+k$; not about $s_{0}$. So we are changing the projection/norm as
we move $k$. This means that:

$||a(s+k)||_{\mathcal{N},m}\leq\sum_{j=0}^{m}|a^{(j)}(s_{0}+k)|\\\ $

This allows us to talk fluidly about the orbits under this norm.

###### Proof.

If $s_{0}\in\mathcal{P}$, the weak Fatou set; then $\beta(s+k)\to\omega\neq
0$; as detailed in the first half of Theorem 3.3.1. Thereby, for
$s_{0}\in\mathcal{P}$:

$\lim_{k\to\infty}\left|\left|\frac{1}{\beta(s+k)}\right|\right|_{\mathcal{N},m}=\frac{1}{|\omega|}\\\
$

Because all the derivatives of $\beta(s+k)\to 0$ as $k\to\infty$.

The more difficult case is when $s_{0}\in\mathcal{B}$, the weak Julia set. We
essentially just have to show that the norm about $s_{0}$ diverges. We’ll
start with $m=0$; and, note that:

$\lim_{k\to\infty}\frac{1}{|\beta(s_{0}+k)|}\to 0\\\ $

This can be done by the process of elimination. It cannot tend to a nonzero
constant, as that would mean $s_{0}$ is in the weak Fatou set. If this tended
to infinity, it would imply there’s a fixed point $\exp 0=0$; which is clearly
impossible.

The difficult part is to prove that this limit exists. But once it exists it
must be $0$. But every orbit $\beta(s+k)$ of a neighborhood $\mathcal{N}$
about $s_{0}$ of the Julia set is dense in the Julia set; and hence each point
gets arbitrarily close to an orbit sending to $\infty$. Hence, eventually
$\beta(s_{0}+k)$ gets arbitrarily close to an orbit which sends to infinity.
This causes a subsequence to converge. But no subsequence can converge to $0$
or a constant; which means $|\beta(s_{0}+k)|\to\infty$. The Riemann Sphere is
compact; and therefore bounded; and therefore every subsequence must tend to
$\infty$, and therefore the sequence tends to $\infty$. Therefore, so do its
derivatives in $s_{0}$. Therefore:

$\lim_{k\to\infty}\left|\left|\frac{1}{\beta(s+k)}\right|\right|_{\mathcal{N},m}=0\\\
$

∎

This gives us a normality theorem which is very valuable to us. Under the norm
$||...||_{\mathcal{N},m}$ the process $\tau^{n}$ is bounded. We are performing
a very similar procedure now; as we did at the start of Section 3.1. We are
going to reduce to the linear case.

We can now invoke the constant:

$A_{\mu}^{m}=\lim_{k\to\infty}\left|\left|\frac{1}{\mu\beta(s+k)}\right|\right|_{\mathcal{N},m}<\infty\\\
$

Which will play the exact role that $A_{\mu}$ played; but for a different
norm. From Corollary 4.4.1 we know that:

$\lim_{k\to\infty}\left|\left|\tau^{n}(s+k)\right|\right|_{\mathcal{N},m}=0\\\
$

Because all of the derivatives of $\tau$ in $s_{0}$ converge to zero. We can
now play the game of linearizing the iteration in this norm. Each:

$\left|\left|\tau^{n+1}(s+k)\right|\right|_{\mathcal{N},m}\leq\left|\left|\log\left(1+\frac{\tau^{n}(s+k+1)}{\beta(s+k+1)}\right)-\log(1+e^{-\lambda(s+k)})\right|\right|_{\mathcal{N},m}\\\
$

Where,

$\left|\left|\log\left(1+\frac{\tau^{n}(s+k+1)}{\beta(s+k+1)}\right)\right|\right|_{\mathcal{N},m}\leq(A_{\mu}^{m}+\delta)\left|\left|\tau^{n}(s+k+1)\right|\right|_{\mathcal{N},m}\\\
$

So that we can reduce to the linearization:

$\displaystyle\left|\left|\tau^{n+1}(s+k)\right|\right|_{\mathcal{N},m}$
$\displaystyle\leq(A_{\mu}^{m}+\delta)\left|\left|\tau^{n}(s+k+1)\right|\right|_{\mathcal{N},m}+(1+\delta)\left|\left|e^{-\lambda(s+k)}\right|\right|_{\mathcal{N},m}$
$\displaystyle\vdots$
$\displaystyle\leq(1+\delta)\left|\left|e^{-\lambda(s+k)}\right|\right|_{\mathcal{N},m}\sum_{j=0}^{n}(A_{\mu}^{m}+\delta)^{j}e^{-\lambda
j}$

And we’re given the following lemma:

######  Lemma 4.4.5.

For all $(s_{0},\lambda)\in\mathbb{L}$ such that $\Re\lambda>\log
A^{m}_{\mu}$; for all $n$ and all $k>K$, for some large $K$:

$||\tau^{n}(s+k)||_{\mathcal{N},m}\leq M\\\ $

For some $M\in\mathbb{R}^{+}$ and $\mathcal{N}$ an arbitrarily small
neighborhood about $s_{0}$.

From this lemma, the result appears more obvious. We are still given the
contraction mapping:

$\left|\left|\log(1+a(s))-\log(1+b(s))\right|\right|_{\mathcal{N},m}\leq\frac{1+\delta}{|\mu|}||a(s)-b(s)||_{\mathcal{N},m}\\\
$

For $a,b\approx 0$. And from this we can state the Asymptotic Theorem of the
third kind.

######  Theorem 4.4.6 (The Asymptotic Theorem of the third kind).

For all $m\in\mathbb{N}$ and all $(s_{0},\lambda)\in\mathbb{L}$–setting
$\iota_{m}$ about $s_{0}+k$. For all $\Re\lambda>\log A_{\mu}^{m}$, there
exists $K$ such for $k>K$:

$\lim_{n\to\infty}\tau^{n}(s+k)=\tau(s+k)\,\,\text{under
the}\,\,||...||_{\mathcal{N},m}\,\,\text{norm}\\\ $

Where $\mathcal{N}$ is a neighborhood about $s_{0}$; and:

$\lim_{k\to\infty}\left|\left|\frac{1}{\mu\beta(s+k)}\right|\right|_{\mathcal{N},m}=A_{\mu}^{m}\\\
$

###### Proof.

This theorem is a weakening of The Convergence Theorem 3.1.3 for
$A_{\mu}^{m}\neq 0$; therefore we only need to prove the case of
$A_{\mu}^{m}=0$. Beginning with the inequality:

$\displaystyle||\tau^{n+1}(s+k)-\tau^{n}(s+k)||_{\mathcal{N},m}$
$\displaystyle=\left|\left|\log\left(1+\frac{\tau^{n}(s+k+1)}{\beta(s+k+1)}\right)-\log\left(1+\frac{\tau^{n-1}(s+k+1)}{\beta(s+k+1)}\right)\right|\right|_{\mathcal{N},m}$
$\displaystyle\leq(A_{\mu}^{m}+\delta)||\tau^{n}(s+k+1)-\tau^{n-1}(s+k+1)||_{\mathcal{N},m}$
$\displaystyle\vdots$
$\displaystyle\leq(A_{\mu}^{m}+\delta)^{n}||\log(1+e^{-\lambda(s+k+n)})||_{\mathcal{N},m}$

We have to justify this inequality; and the result follows. What is meant by
this inequality is a bit of a word salad. Each of these norms are about a
Taylor polynomial around $s_{0}+k$. Each of the terms are multiplied by
$e^{-\lambda n}$; and that’s all we care about. Giving us:

$||\log(1+e^{-\lambda(s+k+n)})||_{\mathcal{N},m}\leq Ce^{-\Re\lambda n}\\\ $

Now, $\delta>0$ can be made as small as possible by increasing $k>K$ as large
as we want. Choose a $K$ such that $\Re\lambda>\log(A_{\mu}^{m}+\delta)$
(which always exists); then set $0<r=(A_{\mu}^{m}+\delta)e^{-\Re\lambda}<1$:

$||\tau^{n+1}(s+k)-\tau^{n}(s+k)||_{\mathcal{N},m}\leq Cr^{n}\\\ $

Thereby:

$||\tau^{l}(s+k)-\tau^{n}(s+k)||_{\mathcal{N},m}\leq\sum_{j=n}^{l-1}Cr^{j}\\\
$

Setting:

$\sum_{j=n}^{l-1}Cr^{j}<\epsilon\\\ $

Arbitrarily small for $n,l>N$; completes the proof. ∎

This theorem provides the quick corollary:

######  Corollary 4.4.7.

For $\mu>1/e$ and $\lambda\in\mathbb{R}^{+}$ the function:

$F(x)=\beta(x)+\tau(x)\\\ $

For

$e^{\mu F(x)}=\exp F(x)=F(x+1)\\\ $

Is infinitely differentiable for $x>x_{0}$ where $F(x_{0}+1)=0$; but it’s
nowhere holomorphic.

###### Proof.

Since the norms $||a||_{m}$ and $||a||_{\mathcal{N},m}$ are equivalent;
convergence under either is equivalent. Therefore all the derivatives up to
$m$ of $\tau^{n}(x)$ converge; since $m$ is arbitrary, all derivatives
converge. Since $F$ is real valued and monotone and divergent as $x\to\infty$,
there’s a point $x_{0}+N$ such that $F(x_{0}+N)=\exp^{\circ N-1}(0)$
(recalling in this case $\exp^{\circ N-1}(0)$ gets arbitrarily large); which
means $F(x_{0}+1)=0$; where $F(x_{0})=\log(0)=-\infty$. This function is
nowhere holomorphic because every $x>x_{0}$ is in the weak Julia set. ∎

##  Chapter 5 The Elliptic View of Iterated Exponentials

###  5.1 A fundamental domain of $\beta$

######  Remark 5.1.1.

This is where we are going to play gloves off. Much of this work will be far
more complicated than earlier work. This section requires a strong
understanding of complex analysis, and additionally, complex dynamics. We will
try to be as forward and explanatory as possible; but we aren’t going to hold
your hand. Additionally; this section serves as supplementary commentary on
our discussion. For this reason, we will not argue too extensively; instead we
will be focused on fleshing out deeper avenues of research.

The function $\beta$ has a much tighter relationship to iterated exponentials;
and its dynamics; than we’ve discussed. But first, it’s helpful to reduce the
size of the domain we’re working with. Such, it is possible to find a single
simply connected domain, in which $\beta_{\lambda,\mu}$ is fully determined.
The function $\beta$ has a fundamental domain of:

$\mathcal{H}=\\{s\in\mathbb{C}\,|\,0\leq\Re(s)<1,\,0\leq\Im(s)<|\Im(2\pi
i/\lambda)|\\}\\\ $

Such the entire complex plane is constructed from iterations of the following
identities:

$\displaystyle\beta(s\pm 2\pi i/\lambda)$ $\displaystyle=\beta(s)$
$\displaystyle\beta(s+1)$ $\displaystyle=\frac{\exp\beta(s)}{1+e^{-\lambda
s}}=\frac{e^{\mu\beta(s)}}{1+e^{-\lambda s}}$ $\displaystyle\beta(s-1)$
$\displaystyle=\log(\beta(s))+\log(1+e^{-\lambda(s-1)})$

We have described how this object converges; but we’ve yet to really comment
on its geometric structure when related to $e^{\mu z}$. Alors, to study the
$\beta$ function, we need only look at
$\beta(s):\mathcal{H}\to\mathbb{C}/\\{0\\}$. But this fundamental domain is
not the domain we are interested in exactly. We would like to shear this
domain by $\lambda$.

We will begin by describing the best fundamental domain possible. This begins
by taking the points: $1+\pi i/\lambda$, $1-\pi i/\lambda$, $-\pi i/\lambda$,
$\pi i/\lambda$, and draw the parallelogram connecting them. We will call this
domain $\mathcal{Q}$.

Now under the generation of the above iterative formulas
$\mathcal{Q}=\mathcal{H}$; of which, both of these domains are fundamental
domains. And every point in the orbit of $\mathcal{Q}$ equals every point in
the orbit of $\mathcal{H}$, and both cover the orbit of $\mathbb{C}$.

We can refer to the tetrations these functions induce based solely on the
domain $\mathcal{Q}$. The weak Julia set and weak Fatou sets can be reduced to
their behaviour on $\mathcal{Q}$. Such, $\tau$ is holomorphic or asymptotic on
domains in $\mathcal{Q}$.

We are going to look at the tetration function induced on $\mathcal{Q}$, and
we are going to construct a function on the torus from it. To do this, we’re
going to have to look at Kneser’s tetration briefly; and look at regular
iteration of tetration. This produces what is usually called the
$\theta$-mapping.

###  5.2 The $\theta$-mapping in general situations

Let us fix a base of the exponential $e^{\mu z}$; and let’s assume there
exists two tetration functions $F_{1}$ and $F_{2}$, such that $F_{1}\neq
F_{2}$. Let’s additionally assume that we can construct an inverse function to
$F_{1}$, and let’s call it $S_{1}$. Then, what is known as a theta mapping is:

$\theta(s)=S_{1}(F_{2}(s))-s\\\ $

Which is a periodic function, satisfying:

$\theta(s+1)=\theta(s)\\\ $

This mapping is used to measure the difference between various tetrations; and
is considered a manner of measuring how close two tetrations are to each
other. This can be quite laborious, considering that $S_{1}$ can be very
misbehaved. But what is helpful, is that we need only invert a single
tetration to construct a $\theta$ mapping.

This can be done rather easily if we stick to using regular iterations. For
that reason, our discussion of $\theta$ terms will take two forms. The first
form being when $b=e^{\mu}$ is in the interior of the Shell-Thron region; and
when $b>e^{1/e}$. We won’t talk about the other possible Abel functions that
can be used to create our $\theta$ mapping. Though, in truth, a $\theta$
mapping is a mapping between two tetrations, we will focus only on the Abel
function induced from regular iteration.

For this conversation, we will need Koenig’s Linearization Theorem. A proof
can be found in Milnor [8].

######  Theorem 5.2.1 (Koenig’s Linearization Theorem).

Let $f$ be a holomorphic function with a fixed point $z_{0}$ such
$\gamma=f^{\prime}(z_{0})$ and:

$0<|\gamma|<1\\\ $

Then there exists a unique function $\phi:\mathcal{A}\to\mathbb{C}$ such
$\phi^{\prime}(z_{0})=1$ and:

$\phi(f(z))=\gamma\phi(z)\\\ $

Where $\mathcal{A}$ is the immediate basin of attraction of the fixed point
$z_{0}$.

Some quick notes on this theorem are needed; specifically, a definition of the
immediate basin of attraction is in order. This is simply defined as:

######  Definition 5.2.2.

The immediate basin of attraction $\mathcal{A}$ of $z_{0}$ is the maximal
connected domain about the fixed point $z_{0}$ such:

$\lim_{n\to\infty}f^{\circ n}(z_{0})=z_{0}\\\ $

We can point to Devaney [3] for a proof that this domain is simply connected;
despite being rather chaotic on the boundary. This domain will play the role
of the domain of our Abel function. From this function, we can construct an
Abel function.

If we take:

$\alpha(z)=\ln\phi(z)/\ln(\gamma)$

Then,

$\alpha(f(z))=\alpha(z)+1\\\ $

This function will be holomorphic almost everywhere on
$\mathcal{A}/\\{z_{0}\\}$. To be precise, it will be holomorphic on
$\mathcal{A}$ excluding a branch cut spawned from the point $\phi(z_{0})=0$.
This won’t matter much to us; and is largely a sigh of relief because it’s a
much easier branch cut then what we’re used to so far in this paper. And as
our function will avoid the fixed point excepting at infinity; it will be
largely negligible.

Now, if we assume that $\alpha$ is an Abel function for $f(z)=e^{\mu z}$; then
any tetration function $F(s)$ constructs a $\theta$ function:

$\theta(s)=\alpha(F(s))-s\\\ $

What’s particularly interesting is when $F$ has a period. By which, let us
assume that $F(s+2\pi i/\lambda)=F(s)$. Then, we must have $\theta$
satisfying:

$\displaystyle\theta(s+1)=\theta(s)$ $\displaystyle\theta(s+2\pi
i/\lambda)=\theta(s)-2\pi i/\lambda$

Which implies it is periodic, and “nearly” linear. This instantly tells us
there must be either singularities, or branch cuts, or domains cut from its
fundamental domain. That fundamental domain is again $\mathcal{Q}$.
Understanding how $\theta$ behaves, gives us a very quick manner of
understanding how $F$ is behaved.

###  5.3 Treating $\theta$ mappings up to elliptic mappings

The point of this section is to talk about the permissible periods of
$F_{\lambda}(s)$ for $e^{\mu}$ in the Shell-Thron region; and comparing how
unique they are. As we’ve found from The Inverse Abel Function on the Shell-
Thron region 3.3.1; we know that a holomorphic function can be constructed on
the weak Fatou set, so long as $\Re\lambda>-\ln|\mu\omega|$; $\omega$ is the
attracting fixed point. We know that $0<|\mu\omega|<1$, so the values of
$\lambda$ where our construction fails are a significant vertical strip.

To begin the foray, we can create the fundamental domain
$\mathcal{Q}^{*}\subset\mathcal{Q}$ such that, $F_{\lambda}$ is holomorphic on
$\mathcal{Q}^{*}$ up to branch cuts. This domain is precisely the weak Fatou
set $\mathcal{P}$ and its intersection with $\mathcal{Q}$. Thereby,
$\mathcal{Q}^{*}=\mathcal{P}\cap\mathcal{Q}$. Thereby,

$\theta(s)=\alpha(F_{\lambda}(s))-s\\\ $

Is holomorphic on $\mathcal{Q}^{*}$; but since this is $1$-periodic, and
eventually the further right we go the branch cuts of $F$ disappear, we must
have that $\theta$ is holomorphic on $\mathcal{Q}^{*}$; and there are no
branch cuts induced by $F$. We recall that $\alpha$ is the Abel function of
$e^{\mu}$ constructed through the regular iteration method.

We wish to show two things in this section. The first being; a natural
boundary appears about the point $\lambda\to-\ln(\mu\omega)$; which invokes a
period $2\pi i/\lambda=-2\pi i/\ln(\mu\omega)$. This is the period of the
regular iteration. Thereupon the goal is; it’s impossible to stretch the width
of our fundamental domain further from this.

This means the regular iteration serves as a sort of boundary to how much the
period can be stretched. It’s helpful to think of this with the case
$\mu=\ln(2)/2$, $b=\sqrt{2}$. To get a look of the regular iteration; we turn
to Sheldon Levenstein’s Fatou.gp [7]. A graph of the regular iteration is
given in Figure 5.1.

We’ve also provided a graph of $F_{\lambda}$ that is very close to the regular
iteration. The value of the period of the regular iteration is $-2\pi
i/\ln\ln(2)=17.143148i$. The value of the period we will display in the next
figure, is $16i$. This is the shrinking of the fundamental domain; which
causes some fractal errors; but retains much of the same picture. This is seen
in Figure 5.2.

Figure 5.1: This is a graph of the regular iteration for $b=\sqrt{2}$. This
means it’s the function
$\text{tet}_{\sqrt{2}}(s)=\phi^{-1}(e^{\ln\ln(2)(s-s_{0})})$; with
$\text{tet}_{\sqrt{2}}(0)=1$ and
$\text{tet}_{\sqrt{2}}(s+1)=\sqrt{2}^{\text{tet}_{\sqrt{2}}(s)}$; while
$\text{tet}_{\sqrt{2}}$ is real-valued and $2\pi i/\ln(\ln(2))$ periodic. This
function is graphed over the domain $|\Re(s)|,|\Im(s)|<10$. This was drawn
using Levenstein’s Fatou.gp [7]; but I tweaked the protocols a bit. Figure
5.2: This is a graph of the beta method
$F_{\lambda}(s)=\beta_{\lambda}(s)+\tau(s)$ for $\mu=\log(2)/2$. We have set
$\lambda=0.392699082$ which is just a bit away from
$-\ln\ln(2)=0.366512920$–the natural border. We have normalized this function,
so $F_{\lambda}(0)=1$, and is graphed over the domain $|\Re(s)|,|\Im(s)|<10$

The regular iteration for this base, has a period of $-2\pi i/\ln\ln 2$. Then
what we are saying is that the width of our fundamental domain cannot exceed
the width of this fundamental domain. Where the width is simply the imaginary
part of the period (in this case it’s purely imaginary). By which, we can
think of the regular iteration as the limit of the various $F_{\lambda}$.

This is a very difficult concept to cognize; but it is a surprisingly simple
argument when you know what you are doing. By, of course, if we relinquish
ourselves to $\theta$ mappings; it’s a statement on the maximum width of our
parallelogram $\mathcal{Q}^{*}$. Thereso, the manner this statement works is,
$\Im\left(2\pi i/\lambda\right)=\Im\left(2\pi i/\ln(\mu\omega)\right)$ serves
as a natural boundary to all elliptic $\theta$ mappings.

To begin, we provide an alternative representation of the $\theta$ mapping.

######  Theorem 5.3.1.

The $\theta$-mapping holomorphic on $\mathcal{Q}^{*}$ can be alternatively
represented as:

$\theta(s)=\lim_{n\to\infty}\alpha(\beta(s+n))-s-n\\\ $

###### Proof.

The function $\theta$ is $1$-periodic and initially described as:

$\theta(s)=\alpha(F(s))-s\\\ $

Où,

$\displaystyle\theta(s+1)$ $\displaystyle=\alpha(F(s+1))-s-1$
$\displaystyle=\alpha\left(e^{\mu F(s)}\right)-s-1$
$\displaystyle=\alpha(F(s))+1-s-1$ $\displaystyle=\alpha(F(s))-s=\theta(s)$

And thereby:

$\theta(s)=\alpha(F(s+n))-s-n\\\ $

But $F(s+n)=\beta(s+n)+\tau(s+n)$. The function $\tau(s+n)\to 0$; and does so
geometrically like $\mathcal{O}(e^{-\lambda n})$ on compact subsets. Upon,
nous voyons,

$||\alpha(F(s+n))-\alpha(\beta(s+n))||_{\mathcal{N}}\leq C|e^{-\lambda n}|\\\
$

For any compact neighborhood $\mathcal{N}\subset\mathcal{Q}^{*}$. The function
$\alpha(s)$ grows at worst like $\mathcal{O}(1/(s-\omega))$ as $s\to\omega$
(we reference Milnor for this tidbit [8]; but it isn’t a difficult result to
intuit). Any Abel function grows at worst linearly; this is comparable to the
statement $\log(z)=\mathcal{O}(z)$. ∎

Because of this, we’ve shown an alternative representation for the $\beta$
tetrations. We summarize in the next theorem.

######  Theorem 5.3.2.

Let $F_{\lambda}(s)$ be a tetration induced from $\beta$, for
$\Re\lambda>-\log|\mu\omega|$ and $b=e^{\mu}\in\mathfrak{S}^{o}$ (the interior
of the Shell-Thron region). Let $\text{tet}_{b}(s)$ be the regular tetration
base $b$, and let $\alpha$ be its Abel function. Then, defining:

$\theta(s)=\lim_{n\to\infty}\alpha(\beta(s+n))-s-n\\\ $

We must have:

$F_{\lambda}(s)=\text{tet}_{b}(s+\theta(s))\\\ $

This means we can fully construct $F_{\lambda}$ with pretty much no mention of
$\tau$ or $\rho$; we just have to construct $\theta$ mappings. This begs the
question, how unique is our theta mapping? The answer is surprisingly simple;
they are equivalent up to the addition of an elliptic function.

Take two $\theta$ and $\widetilde{\theta}$ such that:

$\displaystyle\theta(s+1)=\theta(s)\,\,\,$
$\displaystyle\text{and}\,\,\,\widetilde{\theta}(s+1)=\widetilde{\theta}(s)$
$\displaystyle\theta(s+2\pi i/\lambda)=\theta(s)-2\pi i/\lambda\,\,\,$
$\displaystyle\text{and}\,\,\,\widetilde{\theta}(s+2\pi
i/\lambda)=\widetilde{\theta}(s)-2\pi i/\lambda$

Therefore, if we take the difference of these two functions:

$p(s)=\theta(s)-\widetilde{\theta}(s)\\\ $

We must have that:

$\displaystyle p(s+1)$ $\displaystyle=p(s)$ $\displaystyle p(s+2\pi
i/\lambda)$ $\displaystyle=p(s)$

There are two cases here, where $p$ is what we typically call an elliptic
function; and the case that $p$ is only holomorphic on the parallelogram
$\mathcal{Q}$ with some deleted domains. Both cases are equally interesting.
We will leave this discussion for the next section; but we’ve reduced the
uniqueness of our $\theta$ mapping into a question of elliptic functions.

###  5.4 Arguing for the uniqueness of the $\beta$ method

We won’t be using any deep details of elliptic functions excepting some basic
properties. We will point to the introductory text by Ahlfors which handles
everything we will use in this section [1]. The two cases we have to handle in
this section are when $p$ is your run of the mill elliptic function; and when
$p$ is not your run of the mill elliptic function; meaning it has domains
deleted from its parallelogram of holomorphy (It’s holomorphic on
$\mathcal{Q}^{*}\subset\mathcal{Q}$ as opposed to being meromorphic on
$\mathcal{Q}$).

The first instance, when $p$ is an elliptic function in the traditional sense,
is very easy to handle. To explain this, we will stay with the base case of
$F_{\lambda}$ induced by the beta method; and ask what happens if we perturb
the $\theta$ mapping by an elliptic function. This means that:

$\displaystyle\widetilde{\theta}(s)$ $\displaystyle=\theta(s)+p(s)$
$\displaystyle\widetilde{F}_{\lambda}(s)$
$\displaystyle=\text{tet}_{b}(s+\widetilde{\theta}(s))$
$\displaystyle\widetilde{F}_{\lambda}(s)$
$\displaystyle=\text{tet}_{b}(s+\theta(s)+p(s))$

Now, without loss of generality, we can assume that $p(s)$ has all its
singularities contained within the parallelogram $\mathcal{Q}$ (if not, just
shift the fundamental parallelogram). This means that $\widetilde{F}$ will
have “more” singularities than $F$. But it’s difficult to quantize this
intuition.

We want to be able to determine whether a $\theta$ mapping has a elliptic
function attached to it, or not; while just looking at $\theta$. The idea is
fairly simple; since $p$ is meromorphic, if it has a singularity at $s^{*}$,
then $\widetilde{F}(s^{*})$ has an essential singularity. Since it has an
essential singularity, it attains every point in the complex plane near it;
but then we must have a whole lot of branching; as $\text{tet}_{b}(s)$ has a
big branch cut along $(-\infty,-2]$. So, by direct comparison $\widetilde{F}$
is holomorphic on a smaller domain than $F$! If $\widetilde{F}$ has an
essential singularity, it must have a non-constant elliptic function attached
to its $\theta$ mapping.

So, if we have a tetration function $F$ induced by $\beta$; any tetration
created by perturbing the $\theta$ mapping by an elliptic mapping $p$ must
necessarily be holomorphic on a smaller domain than $F$. And, this tetration
function must have an essential singularity.

Now, if $p$ is not a traditional elliptic mapping, things are a tad more
complicated, as we need not have singularities; or $p$ itself may have
essential singularities. If it is defined on the whole parallelogram but with
essential singularities (it isn’t meromorphic), the above argument still
applies. Thus, it suffices to consider $p$ defined on the parallelogram with
deleted points/domains.

We are going to assume a result here and save the proof for the next section.
We will assume something using Lebesgue’s theory of integration. Let
$\mathcal{Q}$ be the parallelogram, and let $\mathcal{Q}^{*}$ be the
parallelogram that the beta methods $\theta$ mapping is holomorphic on. Then,

$\int_{\mathcal{Q}^{*}/\mathcal{Q}}\,dA=0\\\ $

This is equivalent to the statement that:

$\int_{\mathcal{B}}\,dA=0\\\ $

Which, is the statement that the weak Julia set is measure zero under an area
mapping. This is where it is absolutely pivotal that we assume that
$b=e^{\mu}\in\mathfrak{S}^{o}$. This is a result which only holds for the
Shell-Thron region. Therefore, we can begin by only considering
$\widetilde{\theta}$ holomorphic almost everywhere on $\mathcal{Q}$ (up to
branch cuts). So what we are qualifying with adding an elliptic mapping, is
moving the branch cuts around.

This is where it becomes very difficult to determine why the beta method is
any better than any other tetration method which is equally holomorphic on
$\mathcal{Q}$ almost everywhere. We have very little data in ways of
identifying the beta method from any other method which produces a periodic
tetration of the same period. Thus, enters the key.

The function $\theta$ induced by the beta method, has its singularities
exactly at the four corners of $\mathcal{Q}$. If we can qualify the behaviour
of these singularities, we have the uniqueness of the beta method in terms of
$\theta$ mappings.

The last thing we have to do, before completing this proof; is proving that
any tetration function $F$ with a different period than the regular iteration,
cannot be holomorphic in the right half-plane. I.e: there must be branch-cuts
or singularities. This result will take us away from everything we’ve done in
this paper, and is actually very simple to show with the right tools. We will
also put this in coming sections; as it’s a tad technical.

To summarize the goal of the next section:

If $\widetilde{\theta}(s)$ is $\theta$ upto a traditionally non-constant
elliptic function we will have singularities outside of the singularities at
the four corners of the parallelogram. If $\widetilde{\theta}(s)$ has branch
cuts on the parallelogram and singularities outside of the singularities at
the four corners, we will have more singularities than $\theta$. If the only
singularities of $\widetilde{\theta}$ are at the four corners of
$\mathcal{Q}$; and the singularities are “no worse” than the singularities of
$\theta$, then $\widetilde{\theta}=\theta$.

###  5.5 Comparing the $\beta$ method to a $\widetilde{\beta}$ method

We are going to start by putting a wall of equations and relations; and how
our functions interact:

$\displaystyle F(s)$ $\displaystyle=\text{tet}_{b}(s+\theta(s))$
$\displaystyle\widetilde{F}(s)$
$\displaystyle=\text{tet}_{b}(s+\widetilde{\theta}(s))$
$\displaystyle\theta(s)$ $\displaystyle=\widetilde{\theta}(s)+p(s)$
$\displaystyle p\,\,$ is doubly periodic (almost elliptic) $\displaystyle
p(0)$ $\displaystyle=0$ $\displaystyle\theta(s)$
$\displaystyle=\lim_{n\to\infty}\alpha(\beta(s+n))-s-n$
$\displaystyle\widetilde{\theta}(s)$
$\displaystyle=\lim_{n\to\infty}\alpha(\beta(s+n))-s-n-p(s)$
$\displaystyle=\lim_{n\to\infty}\alpha(\widetilde{\beta}(s+n))-s-n$
$\displaystyle F(s)$ $\displaystyle=\beta(s)+\tau(s)$
$\displaystyle\widetilde{F}(s)$
$\displaystyle=\widetilde{\beta}(s)+\widetilde{\tau}(s)$
$\displaystyle\beta(s)$ $\displaystyle=\beta(s+2\pi i/\lambda)$
$\displaystyle\widetilde{\beta}(s)$ $\displaystyle=\widetilde{\beta}(s+2\pi
i/\lambda)$

This gives us access to a $\widetilde{\beta}$ function from the function
$\widetilde{\theta}$. If we can create a uniqueness criterion for $\beta$; it
will transfer over as a uniqueness condition for $\theta$. The thing we’re
looking for is; $\beta$ is holomorphic for $\Re(\lambda s)<1$, and has simple
poles at $s=1+(2k+1)\pi i/\lambda$ with residue
$\frac{1}{\lambda}e^{\mu\beta(\pi i/\lambda)}$. This defines a unique function
by mapping $\Re(\lambda s)<1.5$ to the unit disk, and treating the zeroes of
$\frac{1}{\beta(s)}$; then, any function with the same poles
$\widetilde{\beta}(s)=\beta(s)e^{g(s)}$. But then, the residues are not the
same; they don’t equal $e^{\mu\widetilde{\beta}(\pi i/\lambda)}/\lambda$. This
uniquely characterizes the $\beta$ function.

###  5.6 Tying up the loose ends

There are two results we need to solidify before we can state that the
$\theta$ mapping induced by $\beta$ is unique. The first result is that every
periodic solution to the tetration equation which is not the period of the
regular iteration must have singularities in the right half plane other than
the singularities of the regular iteration. Let us assume everything is
normalized, and then:

$\theta(s)=\alpha(F(s))-s\\\ $

Where we assume that $F$ has no singularities for large $\Re(s)$. Where
additionally, as we’ve assumed normalization, $\theta(0)=\theta(1)=0$. The
function $\theta(s)=\mathcal{O}(s)$; and therefore applying Ramanujan’s master
theorem $\theta(s)=0$. This implies that $F$ is identically the regular
iteration. For our purposes, we need only reference Ramanujan’s theorem as, if
$f(s)=\mathcal{O}(s)$ and is holomorphic in the right half plane; if
$f\big{|}_{\mathbb{N}}=0$, then $f=0$. Typically this is done with functions
of exponential type, but that’s a slight overkill for us. We reference an
introductory journal article for this [2].

The second thing we need to clear up is that the weak Julia set $\mathcal{B}$
when $e^{\mu}=b\in\mathfrak{S}^{o}$ is measure zero under an area mapping. For
this, we detract from an actual proof and reference Devaney [3]; and his
analysis of the exponential function for $0<\mu<1/e$; which extends exactly
for everywhere in the Shell-Thron region. This is never exactly stated in
Devaney; but the Julia set of $e^{\mu z}$ for the Shell-Thron region is all
the preimages of the Devaney hairs which tend to infinity.

For example, in the case of $b=\sqrt{2}$; if we take all the preimages of
$(4,\infty)$, we have the Julia set. Since this is a measure zero set; so are
a countable union of preimages which are measure zero. So the Julia set is
measure zero. Since the weak Julia set is a remapping of the Julia set where
$\beta\in\mathcal{J}$; the actual weak Julia set is measure zero under an area
mapping.

###  5.7 Collecting Results

From here, we know that the $\theta$ mapping is unique. Collecting how and why
it’s unique, is difficult. But, if the reader has been paying attention this
chapter, it’s pretty straight forward. The $\beta$ function induces a $\theta$
mapping. If the $\theta$ mapping is moved, it induces a different $\beta$
function. The $\beta$ function is unique if we ask that:

$\displaystyle\beta$ $\displaystyle\,\,\text{is holomorphic
for}\,\,\Re(\lambda s)<1$ $\displaystyle\beta$ $\displaystyle\,\,\text{has a
period}\,\,2\pi i/\lambda$ $\displaystyle\beta$ $\displaystyle\,\,\text{has
simple poles at}\,\,1+(2k+1)\pi i/\lambda\,\,\text{for}\,\,k\in\mathbb{Z}$
$\displaystyle\beta(s)$ $\displaystyle=\frac{e^{\mu\beta(\pi
i/\lambda)}/\lambda}{s-1-(2k+1)\pi i/\lambda}+h(s)\,\,\,\,h\,\text{locally
holomorphic}$ $\displaystyle\log\beta(s+1)-\beta(s)$
$\displaystyle=\mathcal{O}(e^{-\lambda s})$

And the $\theta$ function is unique if we ask that its singularities are at
the corners of the parallelogram $\mathcal{Q}$; and that the $\beta$ function
induced from $\theta$ is as above. This settles the Shell-Thron region (or at
least its interior). The very important complex analytic fact, which is
implicit in this solution:

$\beta(s)=\sum_{k\in\mathbb{Z}}\left(\frac{e^{\mu\beta(\pi
i/\lambda)}/\lambda}{s-1-(2k+1)\pi i/\lambda}-p_{k}(s)\right)+H(s)\\\ $

For a holomorphic function $H$ for $\Re(s)<2$; and polynomials
$p_{k}=\mathcal{O}(1/k)$. This is largely what insures $\beta$ is unique. We
can reduce this; by doing some manipulations. Firstly $s\mapsto\pi
is/\lambda+1$:

$\beta(\pi is/\lambda+1)=\frac{e^{\mu\beta(\pi i/\lambda)}}{\pi
i}\sum_{k\in\mathbb{Z}}\left(\frac{1}{s-2k-1}-b_{k}(s)\right)+B(s)\\\ $

Now, we can map $s\mapsto s+1$ and:

$\beta(\pi i(s+1)/\lambda+1)=\frac{e^{\mu\beta(\pi i/\lambda)}}{\pi
i}\sum_{k\in\mathbb{Z}}\left(\frac{2}{\frac{s}{2}-k}+\frac{2}{k}\right)+A(s)\\\
$

Using the expansion for $\cot(s)$:

$\frac{1}{\pi}\cot(\pi
s)=\frac{1}{s}+\sum_{k\in\mathbb{Z}/\\{0\\}}\frac{1}{s-k}+\frac{1}{k}\\\ $

This reduces exactly to:

$\beta(\pi i(s+1)/\lambda+1)=-2ie^{\mu\beta(\pi i/\lambda)}\cot(\pi s/2)+A(s)$

Where, notably $A(s+2)=A(s)$. The uniqueness condition we are imploring is
that $\beta$ satisfies this equation; and it is the only function to do so if
we require $\log\beta(s+1)-\beta(s)$ tends to zero geometrically; which is
required for any $\beta$ function to induce a $\theta$ mapping. It’s important
to recognize that this is a recursive equation. We are requiring the function
to satisfy a residue equation with a different value of the function. This is
the real heavy lifting of our theorem. We are checking whether $\beta$ induced
by $\theta$ has this expansion. If not, it’s not the correct $\theta$.

Producing this residue is rather simple too; for any contour $\mathcal{C}$
about the singularity $1+(2k+1)\pi i/\lambda$:

$\int_{\mathcal{C}}\beta(s)\,ds=\int_{\mathcal{C}}\frac{e^{\mu\beta(s-1)}}{1+e^{-\lambda(s-1)}}\,ds=\frac{2\pi
i}{\lambda}e^{\mu\beta(\pi i/\lambda)}\\\ $

Due to the simple pole of the denominator with residue $1/\lambda$; and the
holomorphy of the numerator. If this isn’t self-explaining; I don’t know what
to say. The result then, is this uniquely defines $\beta$.

As an author, I have not chosen to solidify these results into a theorem. The
result of these discussions are intended as a blue print; and if they are not
produced perfectly; they still display the general shape of the $\theta$
mappings of $F_{\lambda}$. This ends our discussion of $\theta$ mappings for
the $\beta$ method on the Shell-Thron region.

###  5.8 Closing with the $b>e^{1/e}$ case

We are beginning to close this report with a very different subject. This
subject is much deeper at home to Levenstein’s work on tetration–and the
$\theta$ mapping philosophy of the tetration forum [12]. Much of this work was
epitomized in Levenstein’s Fatou.gp program. The mathematical work hasn’t been
written as of yet; but it is in the process of being written by him and I.
There exists an exact dichotomy between this case, and the cases handled for
the most of this paper. It’s precisely the question, how can we make a
holomorphic inverse Abel function on the Julia set?

As we’ve described over and over again in this paper; the $\beta$ method is
suited exactly for Fatou sets; and is at best an asymptotic expansion for
Julia sets. The trouble is, for $b>e^{1/e}$; the Julia set is the entire
complex plane. This can be found in Devaney [3]; but is a well known result;
which is largely why it makes sense to just refer to base $b=e$; and call it a
day. Although the exponentials $b^{z}$ are not conjugate similar to $e^{z}$;
when it comes to iteration, the mantra is everything you can do with
$e^{z}$–you can do with $b^{z}$ when $b>e^{1/e}$.

With that being said, the central philosophy of constructing a tetration from
$e^{z}$ is to use a $1$-periodic $\theta$ mapping. As Levenstein likes to say,
the $\theta$ mappings I just talked about are very different from these
$\theta$ mappings. These $\theta$ mappings need an application of Schwarz’s
reflection principle to ever have a hope of being a full tetration.

Whereas; when $b=\sqrt{2}$, for example:

$\displaystyle\theta(s)$ $\displaystyle=\sum_{k=-\infty}^{\infty}c_{k}e^{-2\pi
iks}$ $\displaystyle\overline{c_{k}}$ $\displaystyle=c_{-k}$
$\displaystyle\theta$ $\displaystyle:\mathbb{R}\to\mathbb{R}$

And then, using regular iteration $F(s)=\text{tet}_{\sqrt{2}}(s+\theta(s))$. A
very different result happens for base $b=e$. Since there is no fixed point in
the orbit of $\exp$ about $0$; we have to choose a different iteration base.
This is done primarily by focusing on $L\approx 0.31813+1.33723i$–which is the
fixed point of $\exp$ with minimal imaginary part. This fixed point is
repelling, and consequently there does exist an iteration $\Psi(e^{Ls})$ which
satisfies the tetration equation. En plus, $\Psi(0)=L$ and
$\Psi^{\prime}(0)=1$.

The trouble is, this will never be real valued. The only way it can be real
valued is if you can find a real valued path and map it to the real line. This
can be done for the upper half plane; with a boundary of the real line.
Additionally it will be unique. This produces a very different kind of
$\theta$ mapping:

$\displaystyle\theta(s)$ $\displaystyle=\sum_{k=0}^{\infty}d_{k}e^{2\pi iks}$
$\displaystyle\theta\,\,$ $\displaystyle\text{is holomorphic for}\,\,\Im(s)>0$
$\displaystyle\lim_{\Im(s)\to\infty}\theta(s)$ $\displaystyle=d_{0}$
$\displaystyle\lim_{\Im(s)\to 0}\Psi(e^{L(s+\theta(s))})$
$\displaystyle\in\mathbb{R}\,\,\text{when}\,\,\Re(s)>-2$

Then, to extend to the lower half plane, we can use Schwarz’s reflection
principle. Which is equivalent to doing the exact same thing I’ve just done
here, but with the fixed point $\overline{L}\approx 0.31813-1.33723i$. Then,
thank the heavens, we are still analytic on $\mathbb{R}$; and we are pasting
two analytic functions together.

It is well conjectured this is the only real valued tetration we get from
$e^{z}$. Thereby, because these $\theta$ mappings are inherently different
from the elliptic-like $\theta$ mappings, there is no way to “correct” the
$\beta$ function (which is periodic) into Kneser’s method. This underscores
the deepness of The Asymptotic Theorem of the third kind 4.4.6. We are trying
to force a periodic solution to this iteration; which makes an elliptic-like
$\theta$ mapping; and this is impossible.

This leads to the only way that the $\beta$ method can become holomorphic for
$b>e^{1/e}$. We must remove the elliptic aspect; there can be no period. The
way the author refers to this, is as a double limit. If, as we iterate
$\tau^{n}$, we additionally let $\lambda\to 0$; then $2\pi
i/\lambda\to\infty$; and the period disappears. Upon which, the only hope in
hell we have of making the $\beta$ method work for $b>e^{1/e}$ is to work
precisely with the limit:

$\lim_{n\to\infty,\,\lambda\to 0}\log^{\circ n}\beta(s+n)\\\ $

It is beyond the scope of this paper, but the author has much of the jigsaw
discovered. The hidden trick is to ensure that
$\lambda=\mathcal{O}(n^{-\delta})$ for $0<\delta<1$. Which should directly
produce Kneser’s tetration. The difficulty in even approaching this, is that
any program calculating $\beta$ for $\Re\lambda<0.001$ is doomed to fail.
Again, to even attempt a numerical solution would require a supercomputer. But
the author is confident in this result; despite the inability to empirically
justify it.

This is largely why the author has avoided talking about the case $b>e^{1/e}$
in a holomorphic respect. The code is expected to fail; but the result is
still prominent. With that statement, we nod towards what is so incredible
about Levenstein’s Fatou.gp [7]; it uses the bare amount of data points to
produce Kneser’s tetration. The author feels in no manner that the $\beta$
method is feasible (at least computationally) for these cases. Regardless, the
$\beta$ method is still a pretty good approximation. It can be made to work;
but it is not as natural, or in anyway as convenient as Levenstein’s approach.
Why use the language of all these $\beta$’s when we can just find $\theta$ and
leave it at that.

## Appendix

### A

This portion of the appendix is intended to give a rough explanation of the
program beta.gp included with this report. All the phase-plot graphs in this
paper were produced using this program (excepting one graph using
fatou.gp–Figure 5.2). The program is based off of a recursive philosophy;
where much of the entire program is written using recursion. It is, as one
would say, a formal program that runs perfectly; but is limited by the
confines of the computer which runs it. Where its greatest enemy is memory
overflow errors. This is something that is unavoidable, as tetration is
ingrained in the study of large numbers (numbers too large for a desktop
computer’s memory).

This program is programmed using the language PARI; and executed through the
shell PARI-GP; or the source can be compiled into a C program, which runs
noticeably faster. The author chose PARI for its handling of large numbers,
that far surpasses any competitor. This is the language of the tetration forum
for the most part, because it allows for values of the order exp(1E6), if you
have the memory capacity. It’s an odd fit; as PARI is more at home amongst
p-adic analysis; but the ideas go hand in hand. PARI can hold an obscene
amount of Taylor series/polynomial data that is perfect for tetration–let
alone the size and detail of a number it can hold.

Built into the program beta.gp is a rudimentary graphing protocol MakeGraph;
which accounts for all the phase-plot graphs we have included in this paper.
This function protocol was written by Mike3 of the tetration forum; and is the
only piece of code not written by Levenstein, or myself, in this program. Most
of the code in beta.gp was written by myself; but Levenstein offered a couple
of optimization techniques which are invaluable.

This program allows us to analyse: the $\beta$ function, the inverse Abel from
$\beta$, the $\tau$ function, the $\rho$ functions, and a bunch of weak
Fatou/weak Julia set protocols. Half heartedly coded in are some normalization
protocols, but they aren’t needed. Additionally we have added alternative
initializations which allow us to analyse the $\mu=1E100$ or higher cases. The
analysis of these cases are out of the scope of this paper; but their
construction still exists. The code of beta.gp requires different
initializations for such extremes, though (and be prepared to eat memory like
the cookie monster).

One is able to grab Taylor series from any function involved; so calculating
beta(1+z), or tau(1+z), will produce a polynomial (to the desired series
precision) of $\beta$, or $\tau$, about $1$. Equally, we are constrained by
the digit precision declared at the beginning of the program. This can be used
to grab the asymptotic series as well; and works identically in either case.

This program is not without its flaws, but it is quite literally an exact
translation of the math within this paper. Where things like
$\operatorname*{{{\Omega}}}$ are translated directly into a recursive
protocol; and $\tau^{n}$ similarly. The major difference between beta.gp and a
formal algorithm of the math with in this paper; there is a training wheels
protocol, which initiates a breaking sequence. The training wheels protocol is
done solely if we are about to hit the capacity of PARI; otherwise this is
just a formal program where 90% of the time it will fail because $\beta$ can
grow too large sporadically and we’ll have a memory error. So there is a
predictive if statement which catches before we’ll even get close to having an
error; this is the only unnatural part of our code. By unnatural, I mean it’s
not a perfect recursive program. Nonetheless, it is modeled exactly after a
perfect recursive algorithm.

The catch statement is precisely if x>exp(1E6), quit. This is all we’ve added
as an artificial construction to save face within PARI.

I have included a large amount of documentation with the program beta.gp. It
explains more clearly the exact function calls, how to initialize values, how
to understand what each variable/function means. I have included a readme.txt;
which runs through the purpose of each function. I have also included a
bloated amount of commentary within beta.gp itself.

The program beta.gp is included alongside this paper to demonstrate how
everything we’ve talked about is reproducible. This is the scientific end of
this paper; we want everything to be clear, but reproducible. For more
detailed use of this program (creating more dynamic types of graph); it is
possible to attach this program to SageMath–which has a plethora of graphing
protocols not used in this paper, but helpful for visualizations.

### B

This portion of the appendix is a break down of Sheldon Levenstein’s analysis
of the case $\mu=1$ and $\lambda=1$. This case is particularly interesting, as
the beta function has a base $b=e$ and a period $2\pi i$. This means if $\tau$
were holomorphic, we’d construct a $2\pi i$ periodic real valued tetration
base $e$. As we know, this doesn’t happen; $\tau$ is not holomorphic. The
author had originally thought he’d proven that $\tau$ was holomorphic.
Levenstein, however, provided a large amount of counter evidence, which
eventually led towards The Asymptotic Theorem of the third kind 4.4.6. Where
we miss holomorphy by a hair’s breadth. Levenstein employed some genius
observations.

These comments begin as such:

> The $\beta$ function appears very well behaved in the complex plane, and
> matches to precision a Taylor series generated by sampling a unit circle,
> and also matches the function’s iterative definition. The results are
> accurate to 112 decimal digits with 240 sample points. The $\tau$ function
> does not match its Taylor series generated by sampling around a unit circle
> with 32 sample points, except at those 32 sample points.

What Levenstein is saying here, in so many words, Cauchy’s Integral Formula is
failing. Sampling along the boundary of a disk, is causing, using the
generated taylor series of $\tau$ at 32 points, a divergence when we try to
use these points to evaluate on the interior of said disk. As we increase
sample points; we yet again see further and further divergence; where, if
$\tau$ were holomorphic we’d see some semblance of convergence.

This is essentially the basis of a Cauchy type argument. As we express $\tau$
on the border of the disk (using as many sample points, which are really
sample polynomials), they don’t reflect the behaviour of $\tau$ within the
disk in a holomorphic manner. Where Cauchy’s integral formula would guarantee
this process to work. As so well it does work with $\beta$ itself.

Levenstein then goes on to single out the points $1+e^{2.3826i}$ and
$1+e^{2.3827i}$; where along the arc connecting them $\tau$ has a singularity.
He posts a graph describing this phenomena in Figure 5.3. This throws a wrench
in the gears, because further iterations only go to sharpen these spikes, and
additionally create more distortions in the nice looking waveform we have.
This is the germ of the idea that $\tau$ converges point wise; is absolutely
great as an asymptotic (like $\tau^{9}$ is in this graph); but as we iterate
further, we experience more jagged discontinuous behaviour. The resolution of
this problem, is the eventual Asymptotic Theorem of the third kind.

Figure 5.3: Levenstein’s graph displaying the singularity which occurs for
$\tau^{n}$ for $n\approx 9$.

Levenstein then points his attention to the real line. This begins by
introducing the $\rho$ terms; which we dubbed Levenstein’s form of the
recursion:

$\displaystyle\tau^{n}(s)$ $\displaystyle=\sum_{j=1}^{n}\rho^{j}(s)$
$\displaystyle\rho^{n+1}(s)$
$\displaystyle=\ln\left(1+\frac{\rho^{n}(s+1)}{\beta(s+1)+\sum_{j=1}^{n-1}\rho^{j}(s+1)}\right)$

The real line becomes incredibly difficult because $\beta(6)$ is about where
Pari-gp overflows $\beta(6)\approx 3.49302404\times 10^{48220669901186881}$.
This means we are limited when talking about $\rho$. For $0\leq\Re(s)\leq 1$
we can only reach $\rho^{5}$. So it is very difficult to draw a conclusion on
an infinite amount of $\rho$ terms from a sample size of $5$.

The value $\rho^{0}=0$, and we start our chain of equations with
$\rho^{1}=-\ln(1+e^{-s})$. Levenstein observed, that as we add $\rho^{2}$,
$\rho^{3}$, $\rho^{4}$, and $\rho^{5}$–we converge ridiculously fast
pointwise. But something very anomalous happens with the Taylor coefficients.
The odd thing being, they happen very very far out. To notice this we can look
at $\beta(s)+\rho^{1}(s)$ and see that it hits zero; and that a clustering of
zeroes start to happen near zero. This is displayed in Figure 5.4

Figure 5.4: A zoom in of the zeroes which start to appear.

This hints directly that our Taylor series on $\mathbb{R}$ is going to be
tricky to prove convergence. And our gut should say that it isn’t going to
converge at this point. As soon as we hit $\rho^{3}$ we are going to feel the
effect of these zeroes. In Figure 5.4, on the border of the white and pale
green petals are a countable number of singularities which cluster towards
$\mathbb{R}$.

Levenstein, then refers to the Taylor expansions in a neighborhood of the
interval $[-1,1]$. Specifically trying to test; using polynomials:

$\displaystyle\widetilde{\tau}^{5}(x)$
$\displaystyle=\sum_{k=0}^{1000}\tau^{5,(k)}(0)\frac{x^{k}}{k!}\,\,\text{for}\,\,-1\leq
x\leq 1$ $\displaystyle\beta(x)+\widetilde{\tau}^{5}(x)$
$\displaystyle\approx\exp(\beta(x-1)+\widetilde{\tau}^{5}(x-1))\,\,\text{for}\,\,0\leq
x\leq 1$ $\displaystyle\beta(x+1)+\widetilde{\tau}^{5}(x+1)$
$\displaystyle\approx\exp(\beta(x)+\widetilde{\tau}^{5}(x))\,\,\text{for}\,\,-1\leq
x\leq 0$

Where, if the Taylor series converges, it must be that these objects are
virtually identical with enough terms. It doesn’t converge, though. So, if we
take the Taylor polynomial of $\tau^{5}$ about $0$, call it
$\widetilde{\tau}^{5}$, and test $\tau^{5}(0.5)$ against
$\widetilde{\tau}^{5}(0.5)$; the polynomials begin to disagree rapidly and
rabidly as we take further and further terms. Which is to say
$\exp(\beta(-0.5)+\widetilde{\tau}(-0.5))\neq\beta(0.5)+\widetilde{\tau}(0.5)$.
Despite the fact, $\exp(\beta(-0.5)+\tau(-0.5))=\beta(0.5)+\tau(0.5)$. But
furthermore, we can map how it gets worse with each iteration.

Figure 5.5: A graph of $\tau^{3}$, drawn across an arc around a circle of
radius $0.489$, about the point $x_{0}$ such that $\beta(x_{0})=1$. A clear
display of singularity-like behaviour arises.

In Figure 5.5 we begin to see a chaos in the polynomials describing $\tau$
(albeit with $3$ iterations instead of $5$). This means quite clearly, if:

$\displaystyle\widetilde{\tau}^{n}_{m}(x)$
$\displaystyle=\sum_{k=0}^{m}\tau^{n,(k)}(0)\frac{x^{k}}{k!}$
$\displaystyle\lim_{m,n\to\infty}\widetilde{\tau}^{n}_{m}(x)-\tau(x)$
$\displaystyle\neq 0$

The Asymptotic Theorem of the third kind, ensures that:

$\lim_{n\to\infty}\widetilde{\tau}^{n}_{m}(x)-\tau_{m}(x)\,\,\text{converges}\\\
$

It’s as we explore further terms of the Taylor polynomials that we start to
see more spikes. These spikes cause the series to diverge; despite all the
terms converging pointwise. From this, Levenstein made an astute calculation
as to where we start to see spikes in the derivatives of $\tau$; they are far
off in the $50,000$ terms range. Levenstein came up with a very ingenious way
of isolating only far out terms without calculating the lower terms.

In Figure 5.6; we can see how very large terms of the Taylor series start to
grow faster and faster. First with $\rho^{3}$, growing in an unbounded way;
but $\rho_{3}$ itself is holomorphic, so this is okay. The function $\rho^{4}$
is just as holomorphic; but we can see the further out Taylor coefficients
start growing faster than $\rho^{3}$–it’s going to be holomorphic on a smaller
domain. As we add $\rho^{5}$, we experience the same phenomena–the further out
series coefficients start growing unreasonably. When we add up all the terms
$\tau=\sum_{j=1}^{\infty}\rho^{j}$, we will see this phenomena causes the far
out terms to be unreasonably large, and the power series will not converge.

Figure 5.6: A discrete graph of $\ln\frac{d^{k}}{dx^{k}}\rho_{3}(0)$ (red) and
$\ln\frac{d^{k}}{dx^{k}}\rho_{4}(0)$ (blue) from $0\leq k\leq 1000000$

Levenstein then does the unthinkable, and produces an unreasonably correct
approximation. Let
$\ell\rho^{1}(s)=\ln(\ln(1+e^{-s}))=\ln\left(-\rho^{1}(s)\right)$; described
by the recursion:

$\displaystyle\ell\rho^{n}(s)$
$\displaystyle=\ln\left(-\ln(1-e^{\ell\rho^{n-1}(s+1)-F_{n-1}(s)})\right)$
$\displaystyle F_{n}(s)$
$\displaystyle=\beta(s)+\tau^{n}(s)=\beta(s)+\sum_{j=1}^{n}\rho^{n}(s)$

This causes us to get:

$\displaystyle\ell\rho^{n}(s)$
$\displaystyle\approx\ell\rho^{n-1}(s+1)-F_{n-1}(s)$
$\displaystyle\ell\rho^{n}(s)$
$\displaystyle\approx\ln\ln\left(1+e^{-s-n}\right)-\sum_{j=1}^{n}F_{n-1}(s+n-j)$

This serves as an awesome approximation, and helps us pull out a faster
formula for the further out series terms. From here, we can see that higher
$\rho$ terms cause the higher terms of the series to blow up; and not only
that–helps us estimate their eventual growth. All in all, this shows that
$\tau$ cannot be analytic anywhere on $\mathbb{R}$. This is not shown, per se,
as we’ve only analyzed $\rho^{1},\rho^{2},\rho^{3},\rho^{4},\rho^{5}$; but it
provides a lot of solid evidence.

As to this, I abstracted much of this to derive Levenstein’s Theorem 4.2.1,
and using this knowledge, abstracted it to The Generalized Levenstein Theorem
4.3.3. We named these theorems for Sheldon Levenstein as they are truly his
work. He just didn’t know how to compile the results into a proof. Again, as
he comes from an engineering background, he’d prefer to hold the calculator in
his hand; and can make the calculator. But the proof may escape his train of
thought. For that, we call these results as his–especially because he found a
hole in my original idea that this can’t happen. This entire results hinges on
the fact that $||\frac{1}{\beta(s+n)}||_{\mathcal{N}}\not\to 0$ for any
compact set $\mathcal{N}\subset\mathbb{C}$, despite the fact
$\frac{1}{\beta(s+n)}\to 0$ for each element of $\mathcal{N}$.

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