s_{i} = 2,-2, … (two step repeating)

Center:-1.059+0.792i; Zoom = 4

As mentioned yesterday, (non-polynomial) rational functions do not have points that “escape to infinity”. That forces some changes when making pictures of the capture set. You cannot set a large radius and say when the orbit leaves that radius that it has escaped. In fact, the orbit is almost guaranteed to return. Perhaps another way to look at it is that since infinity, ∞, is just another point, nothing ever escapes.

To be sure rational functions still have critical points, and fixed and periodic points which may be attracting, repelling or indifferent. All the rich dynamics of polynomials are still present.

These images are generated without an “escape test”, and also without an “inside color”. There is of course an iteration limit. All orbits are run up to the full iteration limit. It does not make sense to color by iteration count, or to paint all the points that reach the iteration limit the same color. The coloring is based on a scheme I described in the series Smooth Colors in 2019. This scheme works well even when the concept of escape and capture are absent.

There are still attracting fixed points and cycles. With this coloring, they show up as smooth areas with gradually changing color.