TITLE: A new interesting pattern to $i\uparrow\uparrow n$ that looks cool (and $z\uparrow\uparrow x$ for $z\in\mathbb C,x\in\mathbb R$)
QUESTION [40 upvotes]: Many of you may recall "An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-integer values:
$$a\uparrow\uparrow b=\begin{cases}a^b,&b\in[0,1]\\a^{a\uparrow\uparrow(b-1)},&b\in(1,+\infty)\\\log_a(a\uparrow\uparrow(b+1)),&b\in(-\infty,0)\end{cases}$$
Combining this with my old question, I made the following graph of $i\uparrow\uparrow x$ for $x\in(-2,9)$, using $z=re^{i\theta},\theta\in[0,2\pi)$:

(x-axis is the real axis and y-axis is the imaginary axis)
Is there anything special to be said about this?

About how each look always connects to the previous 'branch' and then steps off the branch until it hits the next one.  Can we prove this is indeed the case?
Prove or disprove that the tangent line at each interception is equivalent for the original branch and the branch coming out that heads towards the center.
It also appears the branches connect perpendicularly.
Are these shapes similar to one another?

The pattern is rather intriguing, don't you think?

It appears $(-1)\uparrow\uparrow x$ is likewise interesting to look at:
Link to graph.
It starts off how one might expect it to start off:

It makes a loop:

Then another loop:

And then it blows up into a circular shape that reaches about 35 units away from the origin:

Closer image:

And then it goes on past $10^{30}$:

Closer image:

Medium zoom:

Another loop:


Any explanation for why this is so much more 'chaotic' than $i\uparrow\uparrow x$?  Perhaps we can define chaotic or not as follows:
\begin{align}\lim_{x\to\infty}a\uparrow\uparrow x\ \text{converges}\implies\text{stable/non-chaotic}\end{align}
\begin{align}\lim_{x\to\infty}a\uparrow\uparrow x\ \text{diverges}\implies\text{unstable/chaotic}\end{align}
Once again we see apparently perpendicular connections, though trivially all at $(-1,0)$.
There are new patterns though.  We get almost cardioids, but not quite.  We saw two interesting loop looking shapes as well.  Any idea what these are?  It appears these loops get really long and form the quasi-cardioids.
Is it the case that these loops always connect back to $(-1,0)$ from the same direction from which they came?
And is there a 4-turned patter?  The first line connecting to $(-1,0)$ came from above, the second line connecting to $(-1,0)$ came from the right, the third came from below, the fourth from the left, and if we keep graphing more of these, the fifth comes from above once again.


Can we do an analysis to these different shapes?  My graphing calculator isn't the best, and I have to do these one by one...  Particularly, what can we say about $z\uparrow\uparrow x$ for $|z|=1$ and $x\in\mathbb R$?

After quite a few graphs, I've come to the following conclusion that:
When $|z|=1,\operatorname{arg}(z)=\theta\in(-,\pi]$, then $\lim\limits_{x\to\infty}z\uparrow\uparrow x$ tends to exist for $\theta<\theta_0$ and diverges for $\theta>\theta_0$.  What is this $\theta_0$?  And is my conclusion correct?
It also appears that it may be far from trivial proving the lines in $i\uparrow\uparrow x$ actually connect.  Though they are close, I have found that $\sqrt i\uparrow\uparrow x$ clearly does not connect:
Here is the general graph.  It appears as though $\theta_0=\pi/2$.

REPLY [3 votes]: You ask:   

When $|z|=1,\operatorname{arg}(z)=\theta\in(-,\pi]$, then $\lim\limits_{x\to\infty}z\uparrow\uparrow x$ tends to exist for $\theta<\theta_0$ and diverges for $\theta>\theta_0$.  What is this $\theta_0$?  And is my conclusion correct?

For the question of fixpoints of the iteration of $x \to z^x$ the observation is relevant that we can introduce the variables $t$ and $u=\log(t)$ writing $z=t^{1/t}$ and $z=\exp(u \cdot \exp(-u))$
For this we have a proof of D. Shell & W. Thron that           

for $|u|<1$ the iteration converges to the fixpoint $t$,    
for $|u|>1$ the iteration diverges, and            
for $|u|=1$ it converges or diverges depending on the algebraic character of $\arg(u)/\pi$ 

$\qquad $Note 1: For the second case "divergence" can also mean: convergence to a set of periodic points (heuristically).
$\qquad $Note 2: there is at least one older discussion of that property of $|u|$ here on MSE; I can add the link when I've found it 
Now, if on the other hand we have by your demand $|z|=1$ and thus $z=\exp(î \theta)$, then we have $\log(z) =  u \exp(-u) = î \theta$.
There is a nice formula using the LambertW-function to find $u$ from $\log(z)$ such that we have
 $$ u = - \text{LambertW}(- \log(z)) \\
=  - \text{LambertW}(- î \theta)
$$
If we define a function $$f(\theta) =|u| = | - \text{LambertW}(- î \theta) | \qquad \qquad \text {for } 0 \le \theta \le 2 \pi $$ then we find a monoton increasing curve:

It shows that there is only one point where $f(\theta_0)=|u|=1$ and smaller $\theta$ lead to smaller $|u|$ and thus convergence in the iteration, and larger $\theta$ lead to larger $|u|$ and thus to divergence (set-of-periodic-points) in the iteration according to the mentioned categorization of Shell/Thron.
 
About $\theta_0$ such that $|z|=1$ and $|u|=1$ (third case of above list)
I found the approximation for $\theta_0$ using the solver in Pari/GP
$\qquad \qquad $ th0 = solve(th=1.9,2.0, abs(-LambertW(-I*th))-1)
getting $$\theta_0 =  1.961308846... = 0.6243039957... \cdot \pi $$
Using this $\theta_0$ I show the scatterplot of some first iterates of $z_0=1$:

The blue dots are the first 1000 iterates and by the grey connection-lines we see some very chaotic behave. To have a better grip I've also marked the first four iterates with red color.
But we can do better; the "border" of the figure seems (well:seems) to be interpolatable. And in fact we get smoother multi-section trajectories if we document only each 3rd or 11 or so dot and add an interpolating line. The following image shows the iterates in a 68-fold multi-sectioned trajectory .

In the picture I have colored the first 5 sections of the full trajectory up to some 200 steps for each section and with their dots connected. They give nicer lines, seemingly closing the figure, and are partially (nearly) overlaid.
This can be done even more dense - in fact even arbitrarily dense!         
This can be done if we use values from the convergents of the continued fraction of $\arg(u)/\pi$. Then at multi-sections accordingly with $s=68$, $s=797$, $s=5715$ ,$s=2 \cdot 15517$, $s=...$ stepwidths we get increasingly dense sectional trajectories (while of course they are always sets of discrete/disconnected complex points).        
That trajectories seem to not-to-contract, so we don't get convergence.
This is of course conjectured by visual inspection, but can be backed by the heuristic, that increasing values from the convergents of the continued fraction provides smaller distances between the steps in a sectioned trajectory. And this must then mean that we arrive arbitrarily near at $z_{-1}=0$ and then arbitrarily "near" at $z_{-2} = \log_z(0)$; so we shall always have extremely far-away dots in the picture.             
The latter consideration is only based on observation and conclusion - it would be good if one could make a proof for this....