The Mathematics of Beauty > Timeline > Fatou Set
What looks like ramifying branches in this picture, in reality consists of a mosaic of smaller black regions. E.g. the origin is not in the Julia Set, as it seems, but in the Fatou Set, because there is a pole at it. Its component of the Fatou Set starts on the negative real axis at approx. -c. But the resolution of the picture is not high enough to see this: The distance between -c and 0 is 1/4 pixel. The point -c1/3 is not in the Julia set too, because there the function has a zero. It is the center of the slighly bigger knot a bit left of the origin. Right of it at approx. -sqrt(c) there is a repelling fixed point. So in this small intervall there are points of the Julia as of the Fatou Set. Indeed this intervall is one of the inverse images of the whole negative real axis: Even if it seems white on the picture, in detail is looks like the negative real axis. |
f(z)=(z3+c)/z with c=0.001, shown on [-1.5;1.5]×[-1.5;1.5].