s_{0}=1.0, s_{i+1}=(0.6+0.8i)*s_{i}

Center: -1.141+1.458; Zoom = 4

Shallow zoom into the trash heap in the upper left of yesterday’s Sequence Fractals Part V #18 picture.

Complex exponentiation is a multi-valued function. Yes, by definition a function must have be single valued. But sometimes in complex analysis that requirement is ignored. (One work around is the consider the function that returns sets of complex numbers. The set, i.e. the function value, may consist of more than one element, but the set itself is a singular, specific, entity.)

You are already familiar with this. 1 and -1 are both reasonable values for 1^{0.5}. If w is a rational number, say a/b in lowest terms, then there are b candidates for z^{w}. If w is irrational, or if w is complex and either the real or imaginary part is irrational, then
z^{w} has infinite values. Of course the computer does not deal very well with sets and infinity. So we just pick one value and ignore the rest.

In summary, we cannot find critical points, we cannot find a meaningful escape radius. On top of that each of thousands of iteration steps involves an arbitrarily pick of one number out of an infinity of choices for the of exponentiation calculation. I think one could say that even though the pictures are generated with numbers, they are mathematically worthless.