How can I identify when the iterative sequence $z \to z^z$ is not going to head for infinity for some starting value $z$?
For example, $z=1$ is fixed, since $1^1 = 1$. If $z$ is real and between 0 and 1, the iteration heads for 1, where it stays. When starting $z$ is real and greater than 1, then the iteration heads for infinity.
When $z$ starts out negative, however, complex numbers quickly come into play. $-1+0i$, for instance, quickly zeros in on $1+0i$; however, $-0.5+0i$ quickly exceeds computing parameters. Restating $z$ as $x + yi$ and showing the iterations to the point of exceeding computing capacity are as follows:
$$-0.5 \to -1.414213562i \to 691.6428072-369.0287525i \to NAN.$$
Guidelines for complex exponentiation here:http://mathworld.wolfram.com/ComplexExponentiation.html
This reminds me of the iteration behind the Mandelbrot set: $z\to z^2+c$. In the case of the Mandelbrot set, however, it's easy to identify when a $z$ is reached that will head for infinity (when the absolute value of $z$ exceeds 2). When plotting the Mandelbrot set on the complex plane, the program merely needs to catch when 2 is exceeded, and excludes the point being evaluated from the set.
I would really like to see a plot of the set of points $x+iy$ on the complex plane where the iteration $x+yi \to (x+yi)^{x+yi}$ does not head for infinity. I suspect all such points zero-in on the value $1+0i$ but I am not sure they do. There may be other 'equilibrium' values, or there may be values where a repeated cycle forms (as is the case for points in the Mandelbrot set).
I have done a plot for this, however, in this case I was simply assuming that the iteration was headed for infinity whenever $x$ or $y$ exceeded 10 - I'm not sure this is the case! Another problem is that the plot sometimes exclude points when they come close to $0+0i$ (calculation problem - I could fix this but the other problem of excessively high values remains).
Here's the plot image:
Obviously, the set is fractal in nature, so I would really like to see it calculated correctly.
Most of the points excluded in the plot are excluded because they yield an iteration where $x$ or $y$ exceeds 10. I am pretty sure, however, that there are some very high numbers that do not head for infinity - having the imaginary unit in the exponent means that a very high positive input can become a very high negative input in the next iteration, such that the high negative real number in the exponent creates a real number in the iteration following that one that is very close to zero, thus zeroing-in on $1 + 0i$ in iterations following that.
I even suspect that numbers that exceed my computers computing capacity (something like $10^{360}$ or something) may eventually come back to $1+0i$ if my computer could just compute further iterations. Might this in fact be the case for all starting values that contain a non-zero imaginary component? Could this be proven one way or the other?
First, I don't think it really makes sense consider whether the orbit escapes to $\infty$ or not because $z^z$ is very small when $z$ is a negative number with large absolute value. Note that $z^z$ shares this property with $e^z$ and I'd expect the global structure of the iterates of these to functions to have a lot in common.
On a local level, we might expect the iteration of $f(z)=z^z$ to have a lot in common with its quadratic approximation near its fixed point, since: $$f(z) = 1+(z-1)+(z-1)^2+O\left((z-1)^3\right).$$ Thus, we might examine $$g(z) = 1+(z-1)+(z-1)^2.$$ Indeed, both these functions have fixed points at $z_0=1$ with multiplier (value of the derivative) equal to one. In the case of $g(z)$, it's well known that the the disk of radius $1/2$ centered at the point $1/2$ shrinks to $1$ under iteration of $g$. See, for example, Example 6.5.3 of Alan Beardon's Iteration of Rational Functions, which examines a very similar function. This is also illustrated in the figures below:
Thus, we might examine the behavior of the iteration of $f$ on this same disk. The result is:
Now, the point of all this is the following:
Thus, we have a strategy for generating a picture: Iterate until we either enter that disk, overflow, or reach a maximum bailout. The result looks like so:
Note that this does have a bit in common with images of some exponential Julia sets, like this one, as we might expect. The image also seems to suggest an answer to your question concerning the orbits of points off of the positive real axis - namely, the picture indicates that most such points to eventually converge to 1 but not all points. We can in fact prove this by finding points off of the positive real axis that map to some real number larger than 1. For example, $$z=2.469470714651633682 + 3.01399440678144i$$ maps (approximately) to 2, as you can verify numerically.