Sequence Fractals Part I #30

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.