s_{i} = 0.5, 2.5, … (two step repeating)

Center: 0.0+0.0i; Zoom = 0.1

Much more could be said about the rational functions in the previous posts. Rational Functions deserve their own topic, so I will not go there now. Let’s switch to non-integer exponents, starting with half steps (square roots).

If you are familiar with complex numbers you know that the square root function has a discontinuity along the negative real axis. At every iteration step, for those orbits that are close to the negative real axis we put them into two groups. If you are above (or on) the axis you go here, if you are below then you go there. Of the points near -4, half will be sent close to 2i and the other half close to -2i. This creates discontinuities (tears) in our pictures.

Choosing the negative axis is a convention. The cut could be made along any line from 0 to ∞. (For that matter it could be any simple curve from 0 to ∞.) There is no way to avoid the discontinuity. It is always there, and always starts at 0. This is true of all non-integer exponents.

Today’s picture is only here to demonstrate the cracks and tears that will be present for the rest of the pictures in this series.