Infinite tetration of $-2.5$

Question

Let $a_n$ be the sequence $z, z^z, z^{z^z} ...$ for $z \in \mathbb{C}$. This is sometimes called the iterated exponential with base $z$. I am investigating the above sequence for $z = -2.5$. After 6 terms it is on the order of $10^{26649}$. My question is whether the sequence is eventually sent very close to $0$ or if the entire sequence diverges to $\infty$.

I have tried manipulating the sequence $a_n$ in various ways, most of which involve the natural log. These include computing $\ln a_{n+1} = a_n \ln z$ as well as the sequence $b_n = \ln a_n$ using $b_0 = \ln z$ and $b_{n+1} = e^{b_n}\ln z$. For other values of $z$ I am able to conclude that some term $a_n \sim 0$ because $b_n$ has a negative real part. But this is not the case for $z = -2.5$. I have also found it is extremely awkward to evaluate more than a few terms in these situations; the numbers involved tend to get much too large to manipulate directly, even with a system that supports arbitrary precision arithmetic.

Edit: What I have tried so far is essentially asymptotic analysis. To $10$ digits $a_6 = 1.048867589\cdot10^{26649}-5.4257156893\cdot10^{26648}i$. If this, or some later term, were of the form $-\infty+\infty i$ I could stop there since we would have $a_n \sim 0\cdot0 =0$. Otherwise, I need to explicitly compute at least $1$ more term, because we would have $a_n \sim \infty\cdot\infty = \infty$ or $a_n \sim 0\cdot\infty$. Alternatively, I have tried computing $b_n = \ln(a_n)$ until $\Re(b_n) < 0$. I have also thought about using other iterative formulas, such as for $ c_n = \ln(\ln(a_n)), d_n = \ln(\ln(\ln(a_n)))$, etc, but I have not had much luck with this.

Answer 1

The following is not trying to give an answer how to overcome the numerical problem, but is thought to help the intuition.
From the wikipedia-entry for tetration the remark about the Shell-Thron region implies, that the base $b=-2.5$ should give a divergent orbit for $z_{k+1}=b^{z_k}$, but unfortunately, the orbit beginning at $0,1,b,...$ runs early in numerically such extreme values that we cannot continue looking heuristically for more than a couple of steps (as the OP has put it already).

I've tried to give an intuition for the general shape of the iteration using an interpolation based on log-polar values of the (conjugate) iteration of $y_{k+1} = t^{y_k}-1$ where $t = \exp(-W(-\log(b)))$ (or expressed differently: $b = t^{1/t} $). Here $t$ is also the attracting fixpoint for the inverse operation $z_k = \log_b(z_{k+1})$ - so we can start at a point, say $z_{-120}$ in the near in the fixpoint and iterate a couple of times more to make the tendency better visible.
Moreover, there is a "poor-man's" approximation of the Schröder-mechanism for the interpolation to a continuous flow, using the log-polar representation of the values $y_k$ and interpolating linearly, quadratic or by some higher polynomial. (The orbit $y_k$ are the values of the conjugate iteration $y_{k+1}=t^{y_k}-1$ where $y_k = z_k/t-1$ or vice versa $ z_k = (y_k+1)\cdot t$ .)
It gives for some (random) initial value $y_{-120} $ near zero the following contour, which spirals out (showing the basic divergence of the orbit), arriving at the near of $y_0 $ near $-1$ from where then the numerical problems explode after few more iterations.

The big orange points are the integer-iterates (the orbit of $y_k$) where the problems begins after the $y_k$ arrive in the near of $-1$ (which means $z_k=(y_k+1)t$ is near zero) The operation $y_{k+1} = t^{y_k}-1$ means to move from the inside to the outside, showing the expected divergence. The small seagreen dots are computed using the interpolation of the log-polar coordinates of the $y_k$ based on 6 values $y_{-120} \ldots y_{-115}$ with a 5-order polynomial in 1/50 steps for one unit. (using simply the standard polinterpolate procedure in Pari/GP). The thin dotted line is the additional cubic spline interpolation provided by Excel.
One would now guess, that besides some extremal values, that curve of the orbit/of the flow would continue to expand - although with some exotic/erratic shape.
The following picture gives the orbit of the $z_k$ and its interpolated flow just by setting $z_k = (y_k +1) \cdot t$:

Of course, the analysis must be continued to really answer the original question.