Sequence Fractals Part I #16

$ 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.