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

z²+c+s_i
s_i = -1, 1, -1, 1, …
Center:C2.04 (0.633096, 1.288065), Zoom = 16

z²+c+s_i
s_i = -1, 1, -1, 1, …
Center:C2.04 0.633096, 1.288065, Zoom = 16

z²+c+s_i
s_i = -1, 1, -1, 1, …
Center: C2.00: 0.053427, 0.781326, Zoom = 40

$ z^2+c+s_i$
s_i = -1, 1, -1, 1, …
Center: C2.00: 0.053427, 0.781326, Zoom = 16

  6 two cycles
C2.00:   0.053427,  0.781326
C2.01:   0.053427, -0.781326
C2.02:   0.633096,  1.288065
C2.03:   0.633096, -1.288065
C2.04:   0.813477,  1.815442
C2.05:   0.813477, -1.815442

Everything is reflected across the real axis. C2.01 is exactly the same, just flipped vertically. So I will ignore everything going on in the bottom half of the complex plane.

$ z^2+c+s_i$
s_i = -1, 1, -1, 1, …
Center: C1.00: 0.500000, 1.322876, Zoom = 0.8

z₀ = 0 is a critical point and will again be the default starting point. The other two critical points will be ignored without guilt. They just bring us back to the sequence starting 1,-1…, about which we already know more than we ever wanted to know.

Fixed points are found when f(0,c) = 0. so c = $ \frac{1\pm\sqrt{-7}}{2}$ = 0.5 ± 1.322876i.

  2 fixed points
C1.00:   0.500000,  1.322876
C1.01:   0.500000, -1.322876

I will start naming named points with a leading letter to avoid confusion with decimal numbers.

$ z^2+c+s_i$
s_i = -1, 1, -1, ,1, …
Center:0+0.i, Zoom = 0.25,
Variable plane (Julia), c₀ = 0

Here is a picture of the Julia set for the new sequence (starting with -1). It is disconnected, has interior, yet is not dust.

If you are reading these posts you probably know Julia sets. But just in case, here is the wiki definition: Julia set. The picture is the Filled Julia Set. The actual Julia set is the boundary around the white areas.

Stepping way back to the basics.(I should have explained this a long time ago, I just assumed it was obvious.) We have a complex function in two variables, f(z,c). I like this view, it has symmetry, calling out both variables avoids confusion. Equivalently, and traditionally, we are looking at a family of complex functions f_c(z), indexed by a complex value, c. z is the variable and c is the parameter. Fix two numbers c₀, z₀, and define a sequence of numbers by iteration, z_i+1=f_c0(z_i), called an orbit. The question is how does this orbit behave in the limit. The simplest dichotomy is the orbit goes to infinity or it does not. In all of the pictures the non-escaping points are colored white (traditionally they are black, I am just changing things up to be different). The escaping points are colored (approximately) by how quickly the orbit rushes off to infinity.

In the previous pictures in this series, z₀ = 0 (usually) and c₀ depends on the pixel. A linear map maps pixel coordinates to complex numbers. Such pictures could (and probably should) be called "Non-escaping parameter plane" pictures. For the function family z²+c, this is the Mandelbrot set.

In today's picture the c value is fixed, in this case c₀ = 0, and the pixels are mapped to the complex plane for the starting z₀ value. z is the variable so these are "Non-escaping variable plane" pictures.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:0+0i , Zoom = 0.25

Way back at the start Sequence Fractals Part I #1 we had these definitions:
f₀(z,c) = z²+c+1
f₁(z,c) = z²+c-1
f_c(z) = f(z,c) =f₁(f₀(z,c),c) = (z²+c+1)²+c-1.
Derivative: f_c'(z) = 4z(z²+c+1)

Critical points are where f_c'(z) = 0. One critical point is 0, which we have studied in some detail. The other two are the roots of (z²+c+1), $ \pm\sqrt{-c-1}$

The sign does not matter (it disappears with the first z²), both roots behave the same. So, for today only, let z₀ be one of these roots and the starting point for the fractal iteration. Also forget about the two step function for now, and look at the original sequence fractal definition, and write out the first few iterations (notice the alternating function subscript)
z₀,f₀(z₀,c), f₁(f₀(z₀,c),c), f₀(f₁(f₀(z₀,c),c),c), f₁(f₀(f₁(f₀(z₀,c),c),c), c)…

z₀ was defined to be a root of (z²+c+1)=0 which is exactly f₀(z,c). How convenient! We can rewrite the orbit as
z₀, 0, f₁(0,c), f₀(f₁(0,c),c), f₁(f₀(f₁(0,c),c),c)...

After the first step this is exactly the sequence fractal for s_i = -1, 1, -1, 1, … starting at 0. That sequence will be the subject of the next several posts.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:--3.051388+0.210786i , Zoom = 10000

For orientation, this center of this picture is up and to the right of Sequence Fractals Part I #12. Using the same notation, as Sequence Fractals Part I #16, $ z_3=z_1\ and\ z_2\neq z_0$ so for this c, the critical point orbit is preperiodic with pre-period 1 and period 2.

In general if z_n+k=z_k and n,k are the smallest values for which this is true, then the orbit is preperiodic with pre-period k, period n, and denoted M_k,n. Such points should be called Misiurewicz points. See https://en.wikipedia.org/wiki/Misiurewicz_point

The wiki article says that it is not a Misiurewicz point unless you are looking at a polynomial conjugate to z^d+c for d >=2. If you are not familiar with the term 'conjugate', it is a particular, precise mathematical concept for a type of "similar to". The key property is that there is only one critical point. Our polynomial has three critical points. (We have been ignoring the other two.) I have been careful to avoid calling something a Mandelbrot set that is not a Mandelbrot set. So I will also avoid calling these Misiurewicz points. However I will stick with the M_ notation.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.631717+0i , Zoom = 6000

This is a zoom into the far left side of #1, also see #9, #10, #11, for more views of the neighborhood. The center is another M_1,1 point. This point is similar to the point -2, a M_2,1 point in the Mandelbrot set. (The Mandelbrot set has no M_1,1 points.)

Self-similarity is in full force here. Every zoom looks the same. I offer the following "proof by recursive pictures": For the parameter space non-escape set, M, for our function, -3.631717… is an accumulation point of M, a boundary point of M, and not connected to any other point of M.

In the Mandelbrot set, -2 has the first two properties, but not the third.

If you liked the bit about derivatives in yesterday's post, for today's picture

 f'(z1)  = 24.1509+0i
|f'(z1)| = 24.1509

So there is self-similarity with a 24x zoom and no rotation.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:0.034695+0.488109i , Zoom = 40

Take a look back at Sequence Fractals Part I #7, the two-cycle 2.05. Today we are looking at a zoom of the left most subgroup of the group of islands in the top left.

No, the camera did not shake. The center is as intended, the bottom of the small group in the center. You may be sensing a pattern here.

Denote the critical point orbit by z_i. More formerly, z₀ = 0 our critical point, and z_i+1 = f(z_i,c).

For c at the center of today's image, $ z_2=z_1\ and\ z_1\neq z_0$. So z₀ is not a fixed point, but after one step it becomes a fixed point. z₀ is called a preperiodic point with pre-period 1 and period 1. Often denoted M_1,1.

The fixed point, z₁ = -0.34122 + 1.43075i. It is a repelling fixed point. The derivative is

 f'(z1)  = 4.11157-4.85809i
|f'(z1)| = 6.364437

When referring to fixed points and cycles, the derivative is sometimes called the eigenvalue or the multiplier, and is denoted λ. |f'| > 1 indicates a repelling fixed point. f' also gives a clue to the self-similarity behavior. In polar coordinates f'(z₁) = re^iϴ, r=6.3644, ϴ=--0.868 = -50 degrees. Take the image, blow it up by a factor of 6, and rotate it -50 degrees and you get the original image back image.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:0.104301+0.479303i , Zoom = 4000

Hosts for three three-cycles, 3.22, 3.24, 3,26 surround the two cycle host 2.04. (3.23, 3.25, 3.27, 2.05 reflections below the axis). See Sequence Fractals Part I #11 and Sequence Fractals Part I #6 for the definition of these points. 3.26 is today's picture. In Sequence Fractals Part I #7 3.22 is the biggest mini in the lower left. Find 3.24 and 3.26 are up and to the right of the main feature.