Sequence Fractals Part II #7
$ z^2+c+s_i$
si = a,-a,…; a = -0.72
Center: C1.00; Zoom = 5
Julia / dynamic / variable view.
Intermezzo. Julia set for C1.00. I have not posted the parameter plane view of C1.00, it looks pretty much like C1.00 when a = -0.75 Sequence Fractals Part II #2
You will often see the word "dichotomy" in reference to Julia sets. For the quadratic family of functions z2+c, Julia sets are either connected or totally disconnected. Totally disconnected, also called Cantor sets or Fatou dust, means that every connected component consists of a single point. The Mandelbrot set is often defined in this context as the set of c where the Julia set of z2+c is connected. It can be shown that the Julia set is connected when the critical point orbit is bounded, and that the Julia set is totally disconnected when the critical point orbit is unbounded. That equivalence justifies the algorithm used to generate these pictures.
It is not always pointed out that the dichotomy is not true in general. It only holds for quadratic polynomials and a few other carefully selected families. For example it is true for z3+c. But this is a small slice of the parameter space for cubic polynomials, not the full picture. The full parameter space (affine-conjugate equivalency classes, don't ask) for cubic polynomials is z3+bz+c where it is not true. The dichotomy also does not hold for quartics as today's picture demonstrates. It is disconnected with large connected components.
The actual theorem (by Fatou and Julia, discovered over 100 years ago, without a personal computer) states that the Julia set is connected if all critical point orbits are bounded, and totally disconnected if all critical point orbits are unbounded. Quadratic polynomials have a single critical point, so one=all. But most functions have multiple critical points, so there exists the additional possibility of some bounded and some unbounded.
The theorem can be found in math text books, inaccessible to most folks. I could not find a Wikipedia reference to this theorem. If you google (or duckduck) "Julia set dichotomy theorem" you will find many references that state and demonstrate the incomplete theorem for quadratic functions, with no reference to the general case. Here is a page (more like a footnote of a footnote) from the class notes of a Yale math class that describes the general case https://users.math.yale.edu/.../MandelCritPts.html.