Properties of the iterated exponential sequences, $z_n = e^{z_{n-1}}$

Question

Let $z_0 \in \mathbb{C}$ be a complex number, and define from it the infinite sequence $z_n = e^{z_{n-1}}$.

Question: in general, what can we say about the properties of the sequence $\{z_n\}$?

I know that if $z_0 \in \mathbb{R}$, then $\{z_n\}$ goes to $+\infty$ very rapidly.

Also, there are infinitely many $z_0$ that are fixed points of $e^z$- according to Wolfram Alpha, all values of the form $-W_n(-1)$ with $n\in \mathbb{Z}$ work.

I would guess that for all $m>1$, there are also infinitely many $z_0$ for which the sequence has a period of $m$ (though I don't know if that's true).

But all results so far were given for special values of $z_0$ (a set of measure 0). What can we say about the sequence for a general $z_0$? Does it usually diverge to $\infty$, or converge to a fixed point, or does it have weirder behavior?

I've tried to check it on Python (with $z_0 = i$ for example), and the process seems very numerically unstable, so it's hard to say what the analytic behavior is from the simulation.

Answer 1

It's a deep issue. For an overall graph see:

https://ingalidakis.com/math/expFractal.html

For the fixed points, see this question by Gottfried Helms.

The 1-period fixed points are given by $z_k=-W_k(-1)$, where $W$ is the Lambert map. Gottfried, through reverse iteration has calculated p-periodic points for $p>1$, for many $p$, in this post.

This hints towards the idea that there are infinitely many $p$-periodic points for any $p$, although this is still an open question (The periodic fixed points are shown in yellow on the first graph).

The graph for the iterates is a Cantor Bouquet type of fractal and since bouquets repeat by splitting down (bouquet "fingers") to an infinite level (depending on resolution), this suggests that there is a continuum of $p$-periodics for any $p>1$.

1-periodics will congregate around the main bouquet. 2-periodics congregate around the main sub-bouquet "fingers", 3-periodics around the sub-sub-bouquet fingers, and so on (yellow points on the graph).

To find 1-periodics, you need to solve the equation $z=\exp(z)$, whose solution is given as above, by $W$. To find 2-periodics, you need to solve the equation $z=\exp(\exp(z))$, which can only be done using numerical methods - like Gottfried's. For 3-periodics, you need to solve $z=\exp(\exp(\exp(z)))$, wtc.

If you decide to use numerics, you have to be aware that you won't be getting all of the $p$-periodic points at once, unless you iterate all branches of the inverse (complex branch $\ln_k$ in this case) - like Gottfried does.

Then, for a given period you index them, according to the $\ln$ branch, as $z_k$, $k\in\mathbb{Z}$.

To get a sense of $p$-periodics in the iterates fractal, you need to distinguish between different $p$ periods, so if you scan the plane through iterating $\exp$, you can store the iterates for given $z_0$ in an array and postprocess it to assign it a period $p$ if the iterates repeat every $p$ steps in the array. Then color them accordingly (The fractal shown in the first link does not distinguish between fixed points of different periods. It just colors yellow anything that eventually repeats).

Edit: For more info on this type of fractal - Julia Sets as iterates of $\lambda\exp(z)$ in general, see Devaney's website, for example, where he shows that it contains indecomposable continua.

Indecomposable continua are large areas which can cover the entire complex plane. Orbits of points inside those continua can be chaotic. For example, this fractal has $\lambda=1>1/e$ and as such the Julia Set has exploded through a Knaster explosion and all the regions after the main tip of the Cantor Bouquet, form an indecomposable continuum. Points from this main contiuum (exploded tip of the main feature) seem to converge spiral-like towards the two main features left and right of the main Bouquet: The two yellow spirals, left and right of the Bouquet. Other tip explosions, lead to successively deeper yellow fixed points of higher periods, inside the bouquet fingers.

Answer 2

Obligatory "not an answer but some shed light". Also, I apologize for the visually lengthy post.

Clearly, likewise with most types of recursion on the complex plain, fractals are inevitable (as Yiannis Galidakis points out).

To investigate the matter, I quickly wrote up a program to track which points on the complex plane were "hit" by an iteration (represented by luminosity), starting at an initially random value. For each of my tests, I simulated $10^7$ different starting values for $z_0$. and iterated them up to $10^3$ times. Here are the results.

scale = 40 (right most-pixel is has real-value of 40)

Not much to see here, other than $4$ attractors (seemingly on or near the real line?) being visible.

scale = 4

Here, we start to see some structure emerge. Interestingly, the right-most concentration (which I assume is an 'attractor'), can be seen to have a ring-like structure. Just eye-balling, it seems these points are at real value $\approx 0, 1, 2.5$.

scale = 2

The final image shows that there seems to be at least one or more 'ring' of luminosity around each attractor. I'm assuming that the 'incompleteness' (for lack of a better term), of each 'ring' is because I've only chosen initial points from the rendered region. I imagine that taking a larger subset for initial points would close the gap.