I am interested in the long term behaviour of the sequence $a_{n+1}=a_0^{a_n}$ where $a_0\in\mathbb{C}$. I am aware there are many sources discussing convergence of this sequence(for example) but I am interested in the long-term oscillatory behaviour of the sequence which I cannot find any discussion on. I will mention I am using the principle branch of the logarithm.
Above is a plot I have made displaying what period the sequences approach up to period $25$(any higher period is plotted as divergent). I have noticed several properties of these regions of "convergence" although have no idea how I would go about proving them.
Firstly it appears that if the sequence approaches a cycle of $m$ numbers $\{\xi_1,\xi_2\cdots,\xi_m\}$ we can find $i,j$ such that the sequences defined by $a_0=\xi_i,a_0=\xi_j$ converge and have period-$2$. For example in the picture the blue point defines a sequence which approaches period $4$ and there two numbers in this cycle which specify convergent and period-$2$ sequences. One can make a more general statement about $m$ cycles containing numbers specifying sequences with cycles up to $m-1$ or similar however I have not been able to check this well and so am less sure about the truth of this claim.
Secondly, I would like to know if one can give details about certain regions of low period. For example one can easily show if $\Re(a_0)<0$ as $|a_0|\to\infty$ then the sequence approaches the cycle $\{a_0,0,1\}$ as if $a_0=Re^{i\theta}, a_1=Re^{i\theta R(\cos(\theta)+i\sin(\theta))}=Re^{i\theta\cos(\theta)}e^{-R\theta\sin(\theta)}\approx0$. I was wondering if this can be strengthened to say that if $|a_0|$ is sufficiently large then you can guarantee that it approaches a period-$3$ cycle. I would also be interested if one can specify the region of convergence and region around $0$ which approaches a period-$2$ cycle in terms of $a_0$. One would maybe guess that for $|a_0|<e^{-e}$ the sequence approaches a period-$2$ cycle which appears true although it appears that this does not full specify the region.
The only other thing I noticed is that for all $n$ one appears to be able to find and $a_0$ such that the sequence approaches a cycle of period $n$.
I apologise if any of the above has already been asked and would appreciate answers to any of them and any other observations I missed are greatly welcomed.
