Intereting Posts

Example of a finitely generated module with submodules that are not finitely generated
Fekete's lemma for real functions
Important applications of the Uniform Boundedness Principle
Number of sigma algebras for set with 4 elements
Strict cyclic order
A book for abstract algebra with high school level
Questions related to nilpotent and idempotent matrices
Infinitely many systems of $23$ consecutive integers
What is the meaning of $\exp(\,\cdot\,)$?
Does a section that vanishes at every point vanish?
Motivation for Ramanujan's mysterious $\pi$ formula
How to find all roots of the quintic using the Bring radical
How do I understand $e^i$ which is so common?
If $z_n \to z$ then $(1+z_n/n)^n \to e^z$
Connectedness of a regular graph and the multiplicity of its eigenvalue

Let $z \in \mathbb{C}$ and let $W$ be the Lambert $W$ function. In this post it is shown that if $|W(-\ln z)| > 1$ then the infinite power tower $z^{z^{z^{z^…}}}$ does not converge, that is $|W(-\ln z)| \leq 1$ is a necessary condition for the convergence of $z^{z^{z^{z^…}}}$.

Here I would like to show that $|W(-\ln z)| < 1$ is a sufficient condition, that is if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent.

- polynomial approximation in Hardy space $H^\infty$
- Triangle inequality for integrals of complex functions of real variable
- Residue of two functions
- How do I find out the symmetry of a function?
- Limit to infinity and infinite logarithms?
- Complex integral help involving $\sin^{2n}(x)$

- Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$
- Radius of convergence of Taylor series of holomorphic function
- Finding value of exponential sum
- Is $z^{-1}(e^z-1)$ surjective?
- geometrical interpretation of a line integral issue
- Proving an identity relating to the complex modulus: $z\bar{a}+\bar{z}a \leq 2|a||z|$
- Equation of line in form of determinant
- Bijective holomorphic map with a fixed point
- Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
- How many roots have modulus less than $1$?

The key concept is here the “Shell-Thron-region”. In articles in the previous century initially W. Thron and later D. Shell based on Thron’s work proved that if you have a complex base, say $b$ such that $b=t^{1/t}$ or, with $u=\log(t)$, such that $b=\exp(u \exp(-u))$ then the infinite power tower converges if $|u| \lt 1$ and the point of convergence is $t$. (See my earlier picture in MSE where I’ve related those 3 variables with each other)

The numerical values given in Yiannis Galidakis’ answer have $|u|=1-\varepsilon$ so the iteration should converge although very slowly. I found, that a nice picture occurs if you separate the trajectory in *4*, or even better: *72* subtrajectories. With Pari/GP and *800* digits precision you get a nice shape which has some “fractal-like” or “snowflake-like” border. I’ve done the iterations from $z_0=1$ to up to *80 x 72* iterations so each partial curve has *80* points, nearly neighboured with each other – and each pair of neighboured points of the same color has distance of *72* iterations; for a real good image one should proceed to at least *72^3 x 72* points to get a valid impression that this strangely shaped curve really contracts. See a q&d picture made with Excel using values made with Pari/GP, *800* digits precision:

One recognizes *4* segments which together do about one round. These are the first four segments of *72* segments, (where the *5* ‘th would nearly overlap the first, the *6* ‘th nearly the second and so on, but are not shown here to keep the picture clean). The brown segment is the *32* ‘th and its small additional excess from the blue one shows that the expected contraction is -at least- not smooth.

I’ve no nerve to increase the number of points at the moment (it’s night here), possibly my hints give enough ideas to proceed on your own.

** [update]** I couldn’t stop to try to discern the convergence. It appears, that not only in steps of 72 the iterates are tight together, but that it needs 322 of such 72-steps to fill one round of the curve. So I took an arbitrary initial value from my existing list, $\small y_0=-0.5602531521 – 0.6868631844 I$, then iterated 322*72=23184 times to arrive at $ \small y_{23184} \approx -0.5602563718 – 0.6868510240 I$ and proceeded

```
real y_k imag y_k | y_k - t| =distance to fixpoint
-0.5602563718 -0.6868510240 0.8863698615
-0.5602654611 -0.6868245642 0.8863551032
-0.5602477332 -0.6867936307 0.8863199274
-0.5602391855 -0.6867763400 0.8863011262
-0.5602265922 -0.6867593629 0.8862800106
-0.5602215274 -0.6867495245 0.8862691855
-0.5602178010 -0.6867358203 0.8862562109
-0.5602148750 -0.6867278280 0.8862481684
-0.5602175553 -0.6867173579 0.8862417497
-0.5602183774 -0.6867059453 0.8862334262
-0.5602266751 -0.6866972799 0.8862319570
-0.5602460790 -0.6866801357 0.8862309393
-0.5602492634 -0.6866452713 0.8862059387
-0.5602499101 -0.6866233340 0.8861893503
-0.5602465541 -0.6865997931 0.8861689891
-0.5602452598 -0.6865858545 0.8861573713
-0.5602473046 -0.6865693793 0.8861458993
-0.5602478202 -0.6865574553 0.8861369869
-0.5602548561 -0.6865457844 0.8861323930
-0.5602616629 -0.6865325382 0.8861264340
```

*[end update]*

That is different with values *b* where the according value of *u* is $|u|=1$ and thus lie on the boundary of the complex unit disk. An example has been given in the comment at Yiannis Galidakis’ answer with *u* as some complex unit root. Then we have no convergence and the curve (with roughly same shape as the beginning of the shown curve) does *not* contract but has its trajectory “stationary” – I called this, when I’ve seen it first time, “equator” because it reminds me to the meridians on a globus – not disappearing away from and also not contracting towards the fixpoint, but of course there are established technical terms for such -thanks to Yiannis pointing me to this in a comment last night.

P.s.: to improve numerical stability and computing speed when many iterations are needed, use the following conjugacy relation:

the original iteration demands

- use some $\small z_0$,

compute $\small z_{k+1} = b^{z_k}$ and iterate

until some $\small z_n$ .

You can do the following replacement:

- use the same $\small z_0$,

compute $\small y_0=(z_0/t)-1$ ,

compute $\small y_{k+1}=\exp(u \cdot y_k) – 1$ for as many iterations as before

to get $\small y_n$,

then compute $\small z_n = (y_n+1) \cdot t$

$ \qquad \qquad $ with $\small t=\exp(-W(-\ln(b)))$ and $\small u=\ln(t)=-W(-\ln(b))$

Here I would like to show that $|W(−\ln(z))|\le 1$ is also a sufficient condition, that is if $|W(−\ln(z))|\le 1$ then $z^{z^{z^{\ldots}}}$ is convergent.

It’s not true. Take $c=2.043759690+0.9345225945i$. Then (with some Maple code:)

```
restart;
with(plots);
F := proc (z, n)#power tower recursively defined
option remember;
if n = 1 then z else z^F(z, n-1)
end if
end proc;
W := LambertW;
c := 2.043759690+.9345225945*I;
evalf(abs(W(-ln(c))));
```

0.99999999

```
L := [seq(evalf(F(c, n)), n = 1 .. 100)];
complexplot(L, style = point);
```

Here’s the list of values $c,c^c,c^{c^c},\ldots$ plotted against the Complex plane:

- How to show determinant of a specific matrix is nonnegative
- Let $K$ be a field and $f(x)\in K$. Prove that $K/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K$.
- Isomorphism between complex numbers minus zero and unit circle
- Show that every group $G$ of order $175$ is abelian and list all isomorphism types of these groups
- Brouwer's fixed-point theorem and the intermediate value theorem?
- Prove $\int_0^1 \frac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(9\Gamma(3/4)^2{}_4F_3(\cdots)+\Gamma(5/4)^2{}_4F_3(\cdots)\right)$
- Connection on a restricted bundle
- In a ring $a*a=0$ then $a+a=0$
- Why there is a unique empty set?
- Calculating conditional entropy given two random variables
- A improper integral with complex parameter
- How do i prove how $S_5$ is generated by a two cycle and a five cycle?
- Asymptotic upper bound of Bisecting trees
- How to prove that $e^x=x$ has no real solution?
- If $ \cos(x) \cos(2x) \cos(3x) = \frac{4}{7} $ find $ \frac{1}{\cos^2{x}}+\frac{1}{\cos^2{2x}} + \frac{1}{\cos^2{3x}} $