$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: 0; Zoom = 0.25

Zooming out to see the big picture. C1.00 and C1.01 points are the larger distorted brots above and below. C2.02 and C2.01 are the small white area to the left, above and below the central chaos. See Sequence Fractals Part II #18 for a list of named point values for this fractal. The nearby mini slightly to the right appears to host a M1,3 point.

Here are the first few iterates of -0.074+0.384i near the center of the right side mini, which gives evidence that an attracting M1,3 point is nearby.

0 :  0.0000000000,  0.0000000000
1 :  1.1186973098, -0.2226672276
2 :  0.7986249557,  0.2897826579
3 : -0.0192503327, -0.0160305150
4 :  1.1180435564, -0.2235550021
5 :  0.7967073694,  0.2888181202
6 : -0.0135847570, -0.0190021871
7 :  1.1185795766, -0.2236186754
8 :  0.7975798288,  0.2882324026
9 : -0.0136806798, -0.0158793598
10:  1.1184661988, -0.2234036595
11:  0.7975737434,  0.2887061709
12: -0.0148163399, -0.0167138452
13:  1.1184113007, -0.2234957519
14:  0.7973969097,  0.2885950187
15: -0.0142411724, -0.0169491521
16:  1.1184598547, -0.2234949011
17:  0.7974818960,  0.2885534577
18: -0.0142878044, -0.0166716323
19:  1.1184478907, -0.2234766718

My M point calculator refuses to find this point. This is probably a shortcoming in my program. A M1,3 point is a root of a 128 degree polynomial, the program is only finds half of them before giving up. There are things I could do to improve the program, and I would like that additional confirmation. I choose not to head down that rabbit hole right now. So let me get by with "appears to be".

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1.1.00 (for critical point 0); Zoom = 2
Start: z₀ = $ \pm \sqrt{0.716-c}$

Wondering about the other critical point? Here is a look at the same area, starting the iteration with one of the other critical points. Zoom is reduced by a factor of 1000, otherwise the picture would be solid white. The previous two pictures are fully contained in the period two bulb. If you go back to Sequence Fractals Part II #19, you can see small portions of the valley between period two and the cardioid in this picture on the right edges of the main shape in the earlier picture.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1,1.00; Zoom = 2000

Let's get a lot closer. Zoom is 50x the previous picture. Sequence Fractals Part II #20, center is back at M1,1.00. M1,1.01 on the other side looks similar. It is a little bigger and somewhat squished horizontally. The distortions can be seen in the previous picture.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: -1.6 + 0.0i; Zoom = 40

Zooming in 4x from yesterday, Sequence Fractals Part II #19, and picking a center point midway between M1,1.00 and M1,1,01. The two M points are at the tip of the nubs on the left and the right.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1,1.00; Zoom = 10

M1,1.00 at the same zoom as Sequence Fractals Part II #14.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.716
Center: M1,1.00; Zoom = 1

Another small, 0.001 increase for a to -0.716, and we have colliding M points. Boom an explosion of white.

Named Points:
 2 fixed points
C1.00:   0.216000,  1.087198
C1.01:   0.216000, -1.087198
  6 two cycles
C2.00:  -0.302421,  0.452026
C2.01:  -0.302421, -0.452026
C2.02:   0.412158,  1.039344
C2.03:   0.412158, -1.039344
C2.04:   0.538264,  1.634787
C2.05:   0.538264, -1.634787
  4 1/1 preperiodic points
M1,1.00:  -0.176018,  0.000000
M1,1.01:  -0.146881, -0.000000
M1,1.02:   0.593450,  1.627621
M1,1.03:   0.593450, -1.627621

Notice what happened to the M1,1 points. M1,1.00 and M1,1.01 are no longer a conjugate pair. Now both lie on the real axis. If you imagine a as time you can picture the M1,1 points moving as a increases. They have been moving nearly vertically down and up towards each other. Sometime between the last frame and this one, they collided and are now moving away from each other along the real axis.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center: M1,1.00; Zoom = 1

Normal view of M1.1.00 again. Zoom is 1/10 of Sequence Fractals Part II #14. Notice that we are getting dangerously close to the reflected (complex conjugate) M1,1.01 point directly below. (This is of course just a set up for tomorrow's picture.)

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center -0.167756+0.065847:; Zoom = 200
Start: z₀ = $ \pm \sqrt{0.717-c}$

Here is the view from the other critical point of the same area, same zoom and colors. The Mandelbrot surface is complete. Except for the chunks knocked loose in the previous picture Sequence Fractals Part II #15, the white areas are the same.

Something similar can be seen with the quadratic family z²+c. Within this family you can prove "If an attracting fixed point exists, then the critical point is attracted to it." This is again a consequence of the fact that there is only one critical point. And basically the reason we always want to start with the critical point. If you try to draw a Mandelbrot set starting with any other point, you will find pieces missing. Always less, never more. If you start far enough away, none of it remains.

I have not worked through the details, but I am pretty sure that what we see here is closely related.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center -0.167756+0.065847:; Zoom = 200

Here is a zoom into the area a little north of M1,1.00. We are trying to answer the question, "Does this look like the surface of the Mandelbrot set?". Any Star Wars fans? The center bulb reminds me of the death star 2.0. Is the Empire building a Mandelbrot death star? Or perhaps our M1,1 point is attacking and destroying the Mandelbrot set.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.717
Center: M1,1.00; Zoom = 10

Another 0.001 increase in the value of a. The zoom factor is 1/12 of the similar picture for a = -0.718. Sequence Fractals Part II #11. The self-similarity makes it hard to eyeball the zoom. At the same zoom as the other image, you cannot see the blue and orange swirlies.

Named Points:
  2 fixed points
C1.00:   0.217000,  1.088118
C1.01:   0.217000, -1.088118
  6 two cycles
C2.00:  -0.301061,  0.453422
C2.01:  -0.301061, -0.453422
C2.02:   0.412829,  1.040308
C2.03:   0.412829, -1.040308
C2.04:   0.539231,  1.635462
C2.05:   0.539231, -1.635462
  4 1/1 preperiodic points
M1,1.00:  -0.160367,  0.037868
M1,1.01:  -0.160367, -0.037866
M1,1.02:   0.594367,  1.628305
M1,1.03:   0.594367, -1.628305

Notice the transition between white and colors about a quarter of the way down from the top. Does it almost look like a section of the surface of the Mandelbrot set?

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.718
Center: M1,1.00; Zoom = 5
Start: 0
Julia / dynamic / variable view.

Going back to critical point 0 as the start point, and switching to Julia view. Compare to Sequence Fractals Part II #9. This one appears connected, but it is not. Under deep zoom there is a small gap on the center where the two shapes come together. This is expected as M1,1.00 is a repelling fixed point after one iteration. Note however there are large connected components.

Compare the Julia pictures and also compare the parameter space pictures M1,1.00 in Sequence Fractals Part II #8 (a=-0.179) and Sequence Fractals Part II #11 (a=-0.718). In the first picture of each set (a=-0.719) imagine M1,1.00 moving through a field of settled dust, in its wake it kicks up the dust produces the colorful swirlies. Yes, water would be a better analogy, but I need dust for the next part. Now imagine in the second picture (a=-0.718) there is a vacuum cleaner nearby. It sucks up most of the dust that gets disturbed, exposing the clean white table underneath.

I think the analogy comes pretty close to the actual dynamics. Points near a repelling cycle or pre-cycle will tend to dance around a lot before heading off to infinity. This is exactly the colorful spirals that make fractal pictures so interesting. If there is an attracting basin nearby, such as the attracting fixed point associated with some other critical point, some will get sucked in during the dance.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.718
Center: -0.159284, 0.055500; Zoom = 120
Start: z₀ = $ \pm \sqrt{0.718-c}$

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, $ \pm \sqrt{-a-c}$ (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 z²+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.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.718
Center: M1,1.00; Zoom = 120

M1,1.00 again with another small increase in the parameter a to -0.718.

Named Points:
  2 fixed points
C1.00:   0.218000,  1.089036
C1.01:   0.218000, -1.089036
  6 two cycles
C2.00:  -0.299701,  0.454817
C2.01:  -0.299701, -0.454817
C2.02:   0.413502,  1.041271
C2.03:   0.413502, -1.041271
C2.04:   0.540199,  1.636137
C2.05:   0.540199, -1.636137
  4 1/1 preperiodic points
M1,1.00:  -0.159284,  0.055500
M1,1.01:  -0.159284, -0.055500
M1,1.02:   0.595284,  1.628989
M1,1.03:   0.595284, -1.628989

Whoa! What happened? The Misiurewicz point itself still escapes, but many of the nearby points are captured (white). Up until now, the M points all have baby Mandelbrot sets nearby which capture orbits. But they are small, and usually invisible without high magnification. Here, the white space is huge and looks nothing like the Mandelbot set.

The informal answer is that we are seeing interference from the other critical points.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.719
Center: M1,1.00; Zoom = 5000
Julia / dynamic / variable view.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.719
Center: M1,1.00; Zoom = 5
Julia / dynamic / variable view.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.719
Center: M1,1.00; Zoom = 120

Back to looking at M1,1.00, now with a minuscule increase of 0.001 in the sequence parameter a to -0.719.

Named Points:
  2 fixed points
C1.00:   0.219000,  1.089954
C1.01:   0.219000, -1.089954
  6 two cycles
C2.00:  -0.298343, -0.456209
C2.01:  -0.298342,  0.456209
C2.02:   0.414176,  1.042233
C2.03:   0.414176, -1.042233
C2.04:   0.541166,  1.636811
C2.05:   0.541166, -1.636811
  4 1/1 preperiodic points
M1,1.00:  -0.158201,  0.068754
M1,1.01:  -0.158201, -0.068753
M1,1.02:   0.596201,  1.629673
M1,1.03:   0.596201, -1.629673

$ z^2+c+s_i$
s_i = 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.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.72
Center: M1,1.00; Zoom = 5
Julia / dynamic / variable view.

Julia set for M1,1.00. It is not connected. Is it totally disconnected? I am not sure, it might be. (More on totally disconnected Julia sets tomorrow.)

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.72
Center: M1,1.00; Zoom = 120

100x zoom of yesterday's picture, with slightly brighter colors. Observe that it still consists of disconnected pieces.

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.72
Center: M1,1.00; Zoom = 1.2

Here is a look at M1,1.00 after increasing a to -0.72.

Named Points:
  2 fixed points
C1.00:   0.220000,  1.090871
C1.01:   0.220000, -1.090871
  6 two cycles
C2.00:  -0.296985,  0.457598
C2.01:  -0.296985, -0.457598
C2.02:   0.414851,  1.043195
C2.03:   0.414851, -1.043195
C2.04:   0.542134,  1.637485
C2.05:   0.542134, -1.637485
  4 1/1 preperiodic points
M1,1.00:  -0.157118,  0.079838
M1,1.01:  -0.157118, -0.079838
M1,1.02:   0.597118,  1.630356
M1,1.03:   0.597118, -1.630356