s_{i} = 2,-2, … (two step repeating)

Center:-0.40+-0.07i; Zoom = 0.5

Now the sequence alternates positive and negative exponents.

Up until now with the sequence fractals, each step of the iteration is a polynomial function. Different polynomials to be sure, but always a polynomial. Now with the addition of negative (integer) exponents we enter the realm of rational functions.

I have worked with rational functions in the past. That was prior to the most recent full site reboot, and I have not restored those older images yet. In general, the dynamics of rational functions are well-studied. Rational functions are studied on the Reimann sphere which is the complex numbers with the addition point, ∞, infinity. This makes “division by zero” possible. Also “escape to infinity” is just convergence to a point (∞). Lines are circles, and other cool stuff.

On a Reimann sphere, ∞ is an attracting fixed point when iterating a polynomial. But for a non-polynomial rational function, ∞ is just another point. An orbit can easily land on ∞ and then move on to other points. As an example, consider that the negative exponent (a power of the reciprocal) of a very large number is a very small number.