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.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:0.146654+152514.i , Zoom = 256

There are four three cycles surrounding c=0, see them at the four corners surrounding the main shape in Sequence Fractals Part I #2. They are 3.15, 3.16, 3.28, 3.29 in the Sequence Fractals Part I #11 list. This is a zoom of 3.28 in the NE corner. 3.29 is a reflection, 3.15, 3.16 are similar.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-2.216607+-2.216607i , Zoom = 2048

Three cycle host 3.12 (3.13). Located above the two cycle near -2.2. Top feature in Sequence Fractals Part I #5.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.053155+0.209961i , Zoom = 10000

Here is 3.06 (also reflected 3.07) in yesterday's three-cycle list. It is the little dust speck above the c=-3 brot in the world map Sequence Fractals Part I #1.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.630601+0.i , Zoom = 400000

Grab the electron microscope and crank up the zoom into the center of Sequence Fractals Part I #10. There is the mini brot that hosts a three-cycle.

Here is a list of the thirty c values that host three-cycles.

3.00:  -3.630601,  0.000000
3.01:  -3.613512,  0.000000
3.02:  -3.595016,  0.000000
3.03:  -3.568618,  0.000000
3.04:  -3.272072,  0.000000
3.05:  -3.136341,  0.000000
3.06:  -3.053155,  0.209961
3.07:  -3.053155, -0.209961
3.08:  -3.012519,  0.060576
3.09:  -3.012519, -0.060576
3.10:  -2.768406,  0.000000
3.11:  -2.308309,  0.000000
3.12:  -2.216607,  0.033713
3.13:  -2.216607, -0.033713
3.14:  -2.147552,  0.000000
3.15:  -0.324744,  0.181271
3.16:  -0.324744, -0.181271
3.17:  -0.320181,  0.000000
3.18:  -0.024650,  0.495512
3.19:  -0.024650, -0.495512
3.20:  -0.006748,  0.135925
3.21:  -0.006748, -0.135925
3.22:  -0.004470,  0.435048
3.23:  -0.004470, -0.435048
3.24:   0.072243,  0.489918
3.25:   0.072243, -0.489918
3.26:   0.104301,  0.479303
3.27:   0.104301, -0.479303
3.28:   0.146654,  0.152514
3.29:   0.146654, -0.152514

Several are on the real axis, they can be found by looking at the other axis-specks in Sequence Fractals Part I #1, they all look similar to today's picture. 3.22 and 3.23 are found in Sequence Fractals Part I #8. In the Mandelbrot set, the largest real-axis mini, and the top and bottom bulbs host three-cycles. 3.05, 3.08, 3.09, are attached to c=-3, and 3.17 3.20. 3.21 to c=0 in the same way.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.630601+0.i , Zoom = 1600

Now move to the left side of yesterday's picture Sequence Fractals Part I #9, and increase the zoom by 50x. The image is boring, but it is interesting how flat things look even at this magnification.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.604653+0.i , Zoom = 32

Go back to the picture that kicked off this series, Sequence Fractals Part I #1, and notice the small speck on the far left. Today we zoom into that area. At the center is the two-cycle visited in Sequence Fractals Part I #4. At the far left of this image is a three-cycle, and the subject of tomorrow's post.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-0.02465+0.495512i , Zoom = 1000

Now zoom into the feature in the upper left of yesterday's image Sequence Fractals Part I #7. This c value hosts a three-cycle. $ f(f(f(z_0,c),c),c)=z_0\ and\ f(z_0,c) \ne z_0$ . The left side of the first equation in the definition is a degree 32 polynomial. Take away the two fixed points and we expect to find 30 c values that host three cycles.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:0.044082+0.464238i , Zoom = 10

Here is a zoom of the top center of Sequence Fractals Part I #2, the little wing above the c=0 shape. This is also home to two-cycle 2.04 in the list in yesterday's post. 2.05 is a mirror image reflected across the real axis.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-2.21745+0i , Zoom = 800

Same location as yesterday Sequence Fractals Part I #5, with 10x zoom and 50% more color entropy. This is another two-cycle.

A two cycle means $ f(f(z_0,c),c)=z_0 \ and \ f_c(z_0) \ne z_0$. The second iteration comes back to the start point. The second part of the definition excludes fixed points. f is degree 4 polynomial in z and degree 2 in c. With the fixed start point z₀=0 the double iteration in the definition is a degree 8 polynomial in c. It has 8 roots, with 6 left over after removing the fixed points (0 and -3).

Here are the (c-parameter values) of the six two-cycles

2.00:  -3.604653,  0.000000
2.01:  -3.079229,  0.000000
2.02:  -2.217445,  0.000000
2.03:  -0.186836,  0.000000
2.04:   0.044082,  0.464238
2.05:   0.044082, -0.464238

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-2.21745+0i , Zoom = 8

If you look back at Sequence Fractals Part I #1, you will see there is something in the middle, near c=-2, that was cut off from the zooms to the right Sequence Fractals Part I #2 and left Sequence Fractals Part I #3. Here is a zoom of that thing in the middle. I am not sure whether this is a part of the feature as c=-3 or c=0, or if it is something by itself. I am not sure how to define "part of". I think I will call this a part of c = -3. Notice the satellites are nicely placed at 90 and 45 degrees.

$ z^2+c+s_i$
s_i = 1, -1, 1, -1, …
Center:-3.604665+0i , Zoom = 1024

Here is a zoom into the dark spot on the left side of Sequence Fractals Part I #3.

Recall from Sequence Fractals Part I #1 the two-step iteration formula, $ f(z,c) = (z^2+c+1)^2+c-1$. Call the center of this image c_this = -3.604665+0i. Then f(f(0,c_this),c_this) = 0. I may (will definitely) get sloppy at times and say that c_this is a 2-cycle. But these are parameter space (c) pictures. z is the variable that is iterated and it is the z orbit has properties like convergence, divergence and being a two cycle. Not the parameter c. I should say that the z₀=0 orbit with the parameter value c = c_this, is a two cycle. Or maybe that c hosts a two-cycle.

But that is too much of a mouthful, so I will from time to time use the technically incorrect expression. And as long as I am being technical, this is a two-cycle of the two-step function. So it is a four-cycle of the original sequence fractal definition.

And as long as I am dwelling in the minutiae… I have been trying to use the same palette for this set of pictures. The previous palette version of this picture was too dark in the area around the mini. I doubled the color entropy (colors change twice as rapidly) to expose more detail.