Intro
I am studying complex logarithms and particularly the following sequence $(z_n)$ defined by: $$z_0 \in \mathbb{C} \\ \forall n \in \mathbb{N}, z_{n+1} = \log_{\mathrm{i}} z_n$$ It involves: $ z_0 \not = {^{p}i}$ with $p \in \mathbb{N}$ i.e. $ z_0 \not = 0, 1, \mathrm{i}, \mathrm{i}^\mathrm{i}, ...$ So that we cannot have a $\log_{\mathrm{i}}(0)$ after a few iterations. We could rewrite the sequence as follows: $$ z_{n+1} = \frac{\log z_n}{\log \mathrm{\mathrm{i}}} \\ z_{n+1} = \frac{2\log z_n}{\mathrm{i}\pi}$$ Where $\log z$ is the principal value – since $\log z$ is multiple-valued – defined by: $$\log z = \log |z| + \mathrm{i}\arg z, -\pi < \arg z \leq \pi$$
Observations
So I have computed all this in Python to get more information about how this sequence behaves then plotted it.
Here is what I have noticed so far.
Sometimes the sequence simply converges (e.g. when $z_0 = -2-5\mathrm{i}, z_0 = 5-100\mathrm{i}, z_0 = 8$) to always the same value which is approximatively $-1.8617 - 0.4108\mathrm{i}$ and it looks like this when plotted:
(Note that $z_0$ is not plotted. And there is actually a million points on this graph.)
But sometimes it will diverge and after a few iterations it will alternate between 3 values which are also always the same regardless of $z_0$ (e.g. when $z_0=2+5\mathrm{i}, z_0=3\mathrm{i}$):
Some questions
How to prove that this sequence sometimes converges?
How could we determine when the sequence will converge or diverge depending on $z_0$?
Update
To expand on what @Cesareo said in response, let's assume we have : $$ \lim_{n \to +\infty} z_n = w $$ It means we should have: $$ w = \log_\mathrm{i} w \\ w = \frac{2\log w}{\mathrm{i}\pi} \\ \log w \times w^{-1} = \frac{\mathrm{i}\pi}{2} \\ \log w \times e^{-\log w} = \frac{\mathrm{i}\pi}{2} \\ - \log w \times e^{-\log w} = - \frac{\mathrm{i}\pi}{2} \\ - \log w = \mathrm{W}(- \frac{\mathrm{i}\pi}{2}) \\ w = e^{-\mathrm{W}(- \frac{\mathrm{i}\pi}{2})} \\ w = -\frac{2}{\mathrm{i}\pi}\mathrm{W}(- \frac{\mathrm{i}\pi}{2}) \\ w = \frac{2\mathrm{i}}{\pi}\mathrm{W}(- \frac{\mathrm{i}\pi}{2})$$ If we take the branch one of Lambert W function : $$w \approx -1.8617 - 0.4108\mathrm{i}$$
Guys, please. It's been eleven months.
Solving $$z=\log_i z$$ we have
$$ z=\frac{2 i W_1\left(-\frac{i \pi }{2}\right)}{\pi } = -1.8617430750131605 - 0.4107999688363923 i $$
where $W_k$ represents the Product Log $k$-branch.
NOTE
The convergence can be attained from the dotted external region as shown in the following plots.