s_{i} = a,-a,…; a = -0.718

Center: -0.159284, 0.055500; Zoom = 120

Start: z_{0} =

There is much I have been sweeping under the rug, it is time to come clean. However, I want to keep it somewhat informal, it is a difficult balance to maintain.

We are looking at the two step function which is actually a quartic (degree 4) polynomial. I did not formally define the two step function for Part II of this series. You can go back to the first post of Part I, Sequence Fractals Part I #1 and change “1” to “a” at appropriate places if you need it. (Or just ask in the comments.)

We are looking at parameter space pictures. Each pixel represents a different function, f_{c}. The pixel color is a piece of information about that function. It is easy to just take the parameter value, plug it into a program loop that cranks out a color and just call the whole process “paint by numbers”, and then ignore the rest.

All functions, except for a few extremely boring functions, have fixed points, cycles, and preperiodic (M_{m,n}) points. Lots of them!
Each function f_{c}, that is each pixel, has lots of C and M points. The short lists I have been providing are the parameter value c where the critical point of the function f_{c} is also a C or M value for that function. Adding the critical point requirement makes the C and M list much shorter.

But that is not all. Each f_{c} is a degree 4 polynomial, each one has a degree 3 derivative and so three critical points The critical points are 0, (today, a=-0.718). 0 is a critical point for all functions f_{c}, the other two depend on the function parameter c, and move around. There is no single “the” critical point. When I say “the critical point”, you should say “which one?”. The picture and all the C and M points depend on which critical point is selected. Up until now, 0 has been the chosen critical point.
As a side note, each function in the ubiquitous quadratic family of z^{2}+c, truly has a single critical point and it is always 0. So none of these complications show up in discussions for the Mandelbrot set.

Whew, there is still more, but that is enough for today. The starting point for today’s picture is one of the other critical points. It does not matter which one, they both behave the same. The center is M1,1.00 for the critical point 0. Repeating for emphasis, the starting point is determined by one critical point, the center by a different critical point.
We see that the M_{1,1} point, picture center, for 0 lies inside a hyperbolic component (cool math speak for interior) of the other critical point.