Sequence Fractals Part II #4

$z^2+c+s_i$
si = 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
```
The real axis is slightly below the center of the picture. Also seen here, C2.00 near the top and M1,1.01, the reflection across the real axis.

Sequence Fractals Part II #3

$z^2+c+s_i$
si = 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

Sequence Fractals Part II #2

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

Sequence Fractals Part II #1

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

Sequence Fractals Part I #34

z2+c+si
si = -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.

Sequence Fractals Part I #33

z2+c+si
si = -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: fc(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.

Sequence Fractals Part I #32

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

And here is the first (left most) M2,1 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
``````

Sequence Fractals Part I #31

z2+c+si
si = -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.)

Sequence Fractals Part I #30

z2+c+si
si = -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 M1,1 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 Mm,n, and λ is the derivative of the n-cycle, λ = (fcn)'(0), then for arbitrary small ε there is a radius r such that |x- λ(fcn)(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(fcn)(M) near c.

Sequence Fractals Part I #29

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

C3.02 with 50x zoom, 2x color

I am not going to show all three-cycle hosts. They all look similar. They are found in an interesting pattern. Three are connected to C1.00 in the usual Mandelbrot way, top and bottom bulbs, and largest mini off the period 2 bulb. Each of lower cycle hosts, C1.00, C2.00, and C2.04 are surrounded by four satellite three cycles.

`````` 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
``````