$ 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

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.75
Center: M1,1.00; Zoom = 1.5

The first Misiurewicz point. This is the little clump near the bottom of yesterday's picture. The picture looks off center because M1,1.00, the center of the picture is at the bottom of the cluster.

C2.00 is the center of the biggest visible 'brot near the top. Compare the c=-1 case C2.00 Sequence Fractals Part I #22, M1,1.00 Sequence Fractals Part I #30

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.75
Center: C1.00; Zoom = 0.25

Here is a look at one of the fixed points. Compare with Sequence Fractals Part I #21 for the a=-1 case.

Some named points
  2 fixed points
C1.00:   0.250000,  1.118034
C1.01:   0.250000, -1.118034
  6 two cycles
C2.00:  -0.256743,  0.498200
C2.01:  -0.256743, -0.498200
C2.02:   0.435577,  1.071747
C2.03:   0.435577, -1.071747
C2.04:   0.571165,  1.657567
C2.05:   0.571165, -1.657567
  4 1/1 preperiodic points
M1,1.00:  -0.124684,  0.236497
M1,1.01:  -0.124684, -0.236497
M1,1.02:   0.624684,  1.650711
M1,1.03:   0.624684, -1.650711

Cn.xx are cycles (periodic) of period n. C1 are fixed points. Mm,n.xx are Misiurewicz (preperiodic) points with pre-period m and period n. Actually this is parameter space (pixel=c) and periodicity happens in dynamic space (pixel=z). So to be technically correct I should say the critical point, 0, is (pre)periodic for the parameter Cn or Mm,n. On the other hand, the anti-technical definition is interesting stuff happens here. The points in the same group are numbered left-to-right (increasing real part) then top to bottom (decreasing imaginary part).

$ z^2+c+s_i$
s_i = a,-a,…; a = -0.75
Center: 0,0; Zoom = 0.25

Today starts Part II. Mainly because Part I was getting too long. A slight generalization is made to the previous formulas. We will be looking at the two step sequence a, -a, a, -a, … for various values of a. This generalization includes the two most recent sets, a=-1, and before that, a=1, as well as the Mandelbrot set, a=0.

One would expect the pictures to vary continuously as the parameter a changes. Yet in the three examples a=1,0,-1 the pictures are totally different. Ok, smaller steps are needed. Here a = -0.75. The picture is similar to the a = -1 case, compare with Sequence Fractals Part I #19. The features are a little bit bigger, closer together, and a slight distortion can be seen in the Mandelbrot sets.

z²+c+s_i
s_i = -1, 1, -1, 1, …
Center: 0.00555195607569483, 0.796788357586998, Zoom = 8388608000

Summary so far: Comparing to z²+c, the escape set picture for this formula looks surprising normal. There are two main, separated, cardioids. They are surrounded by many little satellites. We have found more and "thinner" Misiurewicz points. However, despite the sparseness, every neighborhood of every M point contains infinitely many minis. (Conjecture supported by many pictures.)

When I first started looking at sequence fractals, my approach was quite random and the early results were nothing like these pictures. I had to dial back to the simplest cases to find familiar territory. We have been exploring in almost familiar territory, in the suburbs just outside of Mandelbrot city. Next let's take a cautious step into the jungle.

z²+c+s_i
s_i = -1, 1, -1, 1, …
Center: 0.82813995, 1.39705173, Zoom = 5120000

This is just a random zoom. It has nothing to do with today's topic. I just feel obligated to always attach a new picture to each post.

Recall that we are actually looking at pictures generated by two alternating functions, see Sequence Fractals Part I #19
f₀(z,c) = z²+c-1
f₁(z,c) = z²+c+1
Two step function: f_c(z) = f(z,c) =f₁(f₀(z,c),c)

Let's go back to the original definition today. The fixed points and the n-cycles for the two-step function are actually 2-cycles and 2n-cycles in the original setup. A natural question is "Are there any fixed points?".

The answer is no. A fixed point would require f₀(0,c) = 0 and f₁(0,c) = 0. Observe that f₀(0,c) = f₁(0,c)-2, . So the simultaneous equalities cannot be solved.

It is essentially impossible for any odd cycle to exist. We need to solve two equations with one unknown. An old cycle gives rise to two polynomials
g₀ = f₀f₁…f₀
g₁ = f₁f₀…f₁
Please forgive the abuse of notation, this is function composition not multiplication (don't make me write all of the parenthesis). An odd cycle requires
g₀(0,c) = 0
g₁(0,c) = 0
Except in very rare, carefully designed situations, two equations and one unknown has no solutions.

Side note: I tried to add another degree of freedom so that the system of equations could be solved. I was able to find a sequence which had an odd cycle, but the picture did not look interesting. So that search in on the back burner now.

z²+c+s_i
s_i = -1, 1, -1, 1, …
M2,1.00: -0.028482, 0.832244, Zoom = 5000

And here is the first (left most) M_2,1 Misiurewicz point.

    26 2/1 preperiodic points
M2,1.00:   -0.028,   0.832  f'=   5.429,   8.069, |f'|=  9.725
M2,1.01:   -0.028,  -0.832  f'=   5.429,  -8.069, |f'|=  9.725
M2,1.02:   -0.004,   0.767  f'=   3.550,  -4.296, |f'|=  5.573
M2,1.03:   -0.004,  -0.767  f'=   3.550,   4.296, |f'|=  5.573
M2,1.04:    0.108,   0.771  f'=  -1.304,   2.085, |f'|=  2.459
M2,1.05:    0.108,  -0.771  f'=  -1.304,  -2.085, |f'|=  2.459
M2,1.06:    0.143,   0.691  f'=   3.432,  -5.565, |f'|=  6.539
M2,1.07:    0.143,  -0.691  f'=   3.432,   5.565, |f'|=  6.539
M2,1.08:    0.252,   1.245  f'=   3.815,  -8.645, |f'|=  9.449
M2,1.09:    0.252,  -1.245  f'=   3.815,   8.645, |f'|=  9.449
M2,1.10:    0.464,   1.552  f'=   7.181,  10.114, |f'|= 12.404
M2,1.11:    0.464,  -1.552  f'=   7.181, -10.114, |f'|= 12.404
M2,1.12:    0.727,   1.039  f'=   7.840,   8.341, |f'|= 11.447
M2,1.13:    0.727,  -1.039  f'=   7.840,  -8.341, |f'|= 11.447
M2,1.14:    0.756,   1.270  f'=   4.497,   0.392, |f'|=  4.514
M2,1.15:    0.756,  -1.270  f'=   4.497,  -0.392, |f'|=  4.514
M2,1.16:    0.772,   1.834  f'=   2.074, -12.196, |f'|= 12.371
M2,1.17:    0.772,  -1.834  f'=   2.074,  12.196, |f'|= 12.371
M2,1.18:    0.804,   1.833  f'=   4.556,   3.965, |f'|=  6.040
M2,1.19:    0.804,  -1.833  f'=   4.556,  -3.965, |f'|=  6.040
M2,1.20:    0.823,   1.408  f'=   1.284, -10.258, |f'|= 10.338
M2,1.21:    0.823,  -1.408  f'=   1.284,  10.258, |f'|= 10.338
M2,1.22:    0.826,   1.806  f'=  -2.921,  -2.545, |f'|=  3.874
M2,1.23:    0.826,  -1.806  f'=  -2.921,   2.545, |f'|=  3.874
M2,1.24:    0.857,   1.810  f'=   8.568,  10.873, |f'|= 13.843
M2,1.25:    0.857,  -1.810  f'=   8.568, -10.873, |f'|= 13.843

z²+c+s_i
s_i = -1, 1, -1, 1, …
M1,1.00: 0.142733, 0.691465, Zoom = 0.5
Julia, variable plane, view.

Julia view of M1,1.00. Tan Lei's paper similarityMJ.pdf also shows how the Julia set is self-similar, this time without the λ factor. It also describes the similarity between the Julia view and the parameter space non escaping set. The discussion and examples in the paper are about the Mandelbrot set. The theorems are stated more generally for rational functions, and so applies to our situation. (The two-step function is a degree 4 polynomial.)

z²+c+s_i
s_i = -1, 1, -1, 1, …
M1,1.00: 0.142733, 0.691465, Zoom = 50

  4 1/1 preperiodic points, with derivatives
M1,1.00:    0.143,   0.691  f'=   3.432,  -5.565, |f'|=  6.539
M1,1.01:    0.143,  -0.691  f'=   3.432,   5.565, |f'|=  6.539
M1,1.02:    0.857,   1.810  f'=   8.568,  10.873, |f'|= 13.843
M1,1.03:    0.857,  -1.810  f'=   8.568, -10.873, |f'|= 13.843

Here is the first M_1,1 Misiurewicz point.

Back in Sequence Fractals Part I #16 I remarked how the derivative near a Misiurewicz point describes the self-similarity. This is well-known in the fractal circles, however I could find only one sentence on my go-to reference, Wikipedia. Here is a link to the original proof by Tan Lei. similarityMJ.pdf

If c is a M_m,n, and λ is the derivative of the n-cycle, λ = (f_cⁿ)'(0), then for arbitrary small ε there is a radius r such that |x- λ(f_cⁿ)(x)| < ε whenever |x-c| < r.

Or more simply, if M is the set of non-escaping parameter space points (the picture), then M ≈ λ(f_cⁿ)(M) near c.

z²+c+s_i
s_i = -1, 1, -1, 1, …
C3.02: 0.000323, 0.777540, Zoom = 5000

 15 +i half plane 3 cycles
C1.00:   0.500000,  1.322876  - connected   Sequence Fractals Part I #21
 C3.12:   0.475692,  1.211907 
 C3.14:   0.543537,  1.408400
 C3.16:   0.729946,  1.274941
C1.00:   0.500000,  1.322876  - satellites  Sequence Fractals Part I #21
 C3.08:   0.256240,  1.261825
 C3.10:   0.456809,  1.544145 
 C3.18:   0.731182,  1.053796 
 C3.26:   0.828125,  1.397028 
C2.00:   0.053427,  0.781326  - satellites  Sequence Fractals Part I #22
 C3.00:  -0.028857,  0.822581 
 C3.02:   0.000323,  0.777540
 C3.04:   0.123345,  0.786543 
 C3.06:   0.123768,  0.684849
C2.04:   0.813477,  1.815442  - satellites  Sequence Fractals Part I #24
 C3.20:   0.774380,  1.836743
 C3.22:   0.802629,  1.829248 
 C3.24:   0.827269,  1.802481 
 C3.28:   0.855612,  1.812967

z²+c+s_i
s_i = -1, 1, -1, 1, …
C3.02: 0.000323, 0.777540, Zoom = 40

z²+c+s_i
s_i = -1, 1, -1, 1, …
Center:C3.00 -0.028857, 0.822581, Zoom = 5000

z²+c+s_i
s_i = -1, 1, -1, 1, …
Center:C3.00 -0.028857, 0.822581, Zoom = 100

  30 three cycles
C3.00:  -0.028857,  0.822581
C3.01:  -0.028857, -0.822581
C3.02:   0.000323,  0.777540
C3.03:   0.000323, -0.777540
C3.04:   0.123345,  0.786543
C3.05:   0.123345, -0.786543
C3.06:   0.123768,  0.684849
C3.07:   0.123768, -0.684849
C3.08:   0.256240,  1.261825
C3.09:   0.256240, -1.261825
C3.10:   0.456809,  1.544145
C3.11:   0.456809, -1.544145
C3.12:   0.475692,  1.211907
C3.13:   0.475692, -1.211907
C3.14:   0.543537,  1.408400
C3.15:   0.543537, -1.408400
C3.16:   0.729946,  1.274941
C3.17:   0.729946, -1.274941
C3.18:   0.731182,  1.053796
C3.19:   0.731182, -1.053796
C3.20:   0.774380,  1.836743
C3.21:   0.774380, -1.836743
C3.22:   0.802629,  1.829248
C3.23:   0.802629, -1.829248
C3.24:   0.827269,  1.802481
C3.25:   0.827269, -1.802481
C3.26:   0.828125,  1.397028
C3.27:   0.828125, -1.397028
C3.28:   0.855612,  1.812967
C3.29:   0.855612, -1.812967