As it turns out, the idea of exponentiation of a negative base is not just a good analogy, it is in fact directly relevant.

Find a fixed (preferably real) point of iteration. If you subtract the fixed point from successive iterates, you'll find that an iterate is a multiple of the distance from the fixed point. Iterative multiplication is, of course, exponentiation. The slope of the function at that point is then the base of exponentiation for determining the iterates. Only when the slope is 1 can we fall back on parabolic iteration methods.

Therefore, if the slope is negative, then you're going to be dealing with exponentiation of a negative base. This explains why every other value is high or low, because the iterates of multiplication of a negative number alternate positive and negative.

Fractional iteration of a negative number will necessarily be complex. Therefore, we should expect a complex spiral around the fixed point of the tetrational bases between e^-e and 1. We start at the fixed point at positive infinity, create a complex spiral of period 2 and infinitesimal radius, and then take iterative logarithms to get back to the origin. Voila!

For bases less than e^-e, the fixed point is actually repelling, much as the upper asymptote for bases between 1 and eta is repelling. This complicates the matter but does not make it intractable. We can likely start at the fixed point and exponentiate our way to some complex ring asymptote, then use logarithms just outside this ring to recover the tetration. However, this last idea is at best a guess until I can investigate it.

Finally, complex slopes should be similarly solvable, allowing us to solve iterative exponentiation of complex bases, so long as we can find a suitable (preferably real) fixed point.

The main remaining question, of course, is what to do with bases greater than eta. We still lack real fixed points. For bases of the form , with k non-zero, we might actually be able to find complex fixed points. But for the "primary branch", no real fixed points exist. Other methods exist which seem to work, but their inner workings are far more esoteric than the simplicity of hyperbolic iteration from a fixed point.

Find a fixed (preferably real) point of iteration. If you subtract the fixed point from successive iterates, you'll find that an iterate is a multiple of the distance from the fixed point. Iterative multiplication is, of course, exponentiation. The slope of the function at that point is then the base of exponentiation for determining the iterates. Only when the slope is 1 can we fall back on parabolic iteration methods.

Therefore, if the slope is negative, then you're going to be dealing with exponentiation of a negative base. This explains why every other value is high or low, because the iterates of multiplication of a negative number alternate positive and negative.

Fractional iteration of a negative number will necessarily be complex. Therefore, we should expect a complex spiral around the fixed point of the tetrational bases between e^-e and 1. We start at the fixed point at positive infinity, create a complex spiral of period 2 and infinitesimal radius, and then take iterative logarithms to get back to the origin. Voila!

For bases less than e^-e, the fixed point is actually repelling, much as the upper asymptote for bases between 1 and eta is repelling. This complicates the matter but does not make it intractable. We can likely start at the fixed point and exponentiate our way to some complex ring asymptote, then use logarithms just outside this ring to recover the tetration. However, this last idea is at best a guess until I can investigate it.

Finally, complex slopes should be similarly solvable, allowing us to solve iterative exponentiation of complex bases, so long as we can find a suitable (preferably real) fixed point.

The main remaining question, of course, is what to do with bases greater than eta. We still lack real fixed points. For bases of the form , with k non-zero, we might actually be able to find complex fixed points. But for the "primary branch", no real fixed points exist. Other methods exist which seem to work, but their inner workings are far more esoteric than the simplicity of hyperbolic iteration from a fixed point.

~ Jay Daniel Fox