# Does infinite tetration of negative numbers converge for any value other than -1?

Okay, so I know that for positive values, $^{\infty}x$ converges to $-\frac{W(-\ln x)}{\ln x}$ for $e^{-e}\le x \le e^{\frac1e}$. Above that, it diverges. For positive values less than $e^{-e}$, any attempt to evaluate the infinite tetration oscillates between two values. $^{\infty}0$ is undefined, but $^{\infty}(-1) = -1$. By way of experimentation, other negative values of $^{\infty}x$ seem to quickly become nonconvergent complex values. Are there any other negative values for which infinite tetration converges? If not, is there a proof that no other vaues do?

• Quote "For positive values less than $e^{-e}$... oscillates between two values". I see the same thing for negative values of x, where if x<=-3.25, then it oscillates between three values. Might be able to look at the derivitive at the fixed point for negative values... using the lambertw function. May 29, 2014 at 14:19
• And for other negative values of $x$, you can get periods different from 3. Maybe you should check out the "Tetration Forum" math.eretrandre.org/tetrationforum/index.php May 29, 2014 at 14:22
• Moreoever, the closer you get to $-1$, the bigger the period seems to be. I seems to be a fairly complex problem to analyze. May 29, 2014 at 14:24

Are there any other negative values for which infinite tetration converges?

Not really. Without going into too much detail, the potential map for the infinite exponential is:

$$\phi(z)=\exp(z/\exp(z))$$

If you plug in the above the parametrized unit circle $\exp(i\cdot\theta)$ for $0\le\theta\le 2\pi$, you'll get what is known as the Shell-Thron region boundary on the Complex plane. In Maple for example:

phi:=z->exp(z/exp(z));

complexplot(phi(exp(I*theta)), theta = 0 .. 2*Pi, scaling = constrained);

Shell (of Shell-sort fame) in his Ph.D. thesis proved that convergence occurs only for $c$ inside this region on the complex plane. Alternatively, if you don't want to check against this region on the complex plane, you can check that the inverse potential map (multiplier) sends you inside the unit circle (which is crucial for convergence). That is, given $c$, check that:

$$t=|\phi^{-1}(c)|=|-W(-\log(c))|< 1$$

where $W$ is the principal branch of the Lambert function.

Baker and Rippon later made the result stronger, by proving that convergence occurs only either if $t<1$ or $t=1$ and $t^n=1$.

You can see the Shell-Thron region here (red nephroid).

The value (-1) is sort of a rogue in the complex plane and is the only value outside this region for which the infinite exponential trivially converges to itself, since ${^n}(-1)=-1$ for all $n\in\mathbb{N}$.

• Daniel; thanks for the great graph! Daniel's graph on the right is color coded by period. Red is converging, the other colors are periodic... May 30, 2014 at 3:17
• Another graph of the Shell-Thron-region is in the tetration-forum's wiki-page: math.eretrandre.org/hyperops_wiki/… Jun 6, 2014 at 16:32
• @GottfriedHelms The images in your link appear to be broken May 30, 2017 at 14:08
• I can provide a link to my own picture from that small entry so far, see go.helms-net.de/math/tetdocs/_equator/ShellThron.png May 30, 2017 at 14:28
• Dear Yiannis, late addendum: please see the update of my answer. I give 4 initial bases in the near of $-1$ which converge to sets of periodic-points. These are not fixed-points but they also do not diverge to infinity by this function-definition. Oct 29, 2020 at 19:33

Surely there are no other fixpoints in that area. However, one should observe the difference between divergence and convergence-to-cyclic-points . In the left side (negative real part) I was surprised that we have mostly convergence-to-cyclic-points , while divergence (of absolute values) was only observed in the right side (positive real part) - which I think is a better characterization of the properties presented in the question.

A relatively raw attempt to (compute and) illustrate the cycle-lengthes of sets of attracting cyclic points gives the following picture (coordinates $$-5..+5, -5i..+5i$$) .

In the middle we see that black nephroid - this is the region where the complex numbers with definition $$z_0=z$$ and then $$z_{k+1}=z^{z_k}$$ iterate to the fixpoints (cycle-lengthes 1), known as "Shell-Thron-region".

But also mainly on the left side (negative real part) the complex bases $$z$$ have attracting cyclic points, most of cyclelength $$3$$ - the dark red color. Attracting cyclic points with higher cyclelenghtes got lighter red color.

(Mainly) on the rights side (positive real part of $$z$$) it seems that we have the "point at infinity" as fixpoint - the iterations diverge quickly or less quick; the fastest diverging bases are at the right side, the dark teal color. The lighter teal/blue colors indicate bases which diverge slower.

As I wrote above, this is based on a somehow crude procedure - the characteristics of the divergent bases are not yet well analyzed due to the exorbitant values which occur- sometimes alternating with extremely small absolutes values; those are all the blue and green and reaining grey colored points.

Late addition: I've looked for some specific values. Assume a base $$b$$ with negative real and positive imaginary part. Compute $$c=\log(b)$$, do everything with larger decimal precision (for instance I took 200 decimal digits). Assume $$z_0=0$$. Iterate $$z_{k+1}=b^z$$ . The following three nearly randomly chosen values became periodic by simple fixpoint-iteration:

• b = -3 + 2*I a 3-periodic cycle, nearest value to b:-2.99636967476 + 1.99878169538*I
• b = -1.1+0.5*I a 6-periodic cycle, nearest value to b:-1.09989523565+0.500062582963*I
• b = -1.01+0.5*I a 72-periodic cycle, nearest value to b:-1.00999999804+0.499999999221*I
• b =-1.001+0.2*I a 7-periodic cycle, nearest value to b:-1.00100000000+0.200000000000*I

Done with 200 dec digits precision; because I nearly could believe the last of the four examples I re-checked with 3200 digits precision, getting the same results. In that orbit we get the markant numbers
-285.781368475 + 290.236031597*I
2.02035401807 E-374 + 3.73161384866 E-375*I very near zero!
1.00000000000 + 5.95638505604 E-374*I very near 1
-1.00100000000 + 0.200000000000*I very near to the base
and then
-0.532977390140 - 0.107137563189*I
-0.000949870602674 - 1.35593373706*I
54.1580223026 - 1.66264875036
-285.781368475 + 290.236031597*I (... cycling)

If you re-check this using your favorite programming language make sure you have sufficient internal precision! (I use Pari/GP)