Asymptotic self-similarity of the Mandelbrot set
Get acquainted with the phenomena of asymptotic scaling and similarity in an interactive session with the Java applet Juliette below. Click the gray column to give it keyboard focus (do not click the images). Use the following commands or the buttons below:
- The yellow cross in the left window is indicating the current parameter c=a, which is a Misiurewicz point of preperiod 3 and period 1. The critical value z=a is mapped to the fixed point α_{a} in 3 iterations of f_{a}(z). You may follow the orbit in the right window by hitting the key f a few times.
- Hit t repeatedly to rescale the parameter plane around c=a with the complex scaling factor ρ=f_{a}'(α_{a})=2α_{a}. The image is enlarged by |ρ| and rotated by the argument of ρ, which is approximately 120^{o}. The rescaled images show that ρ^{n}(M-a) is converging to an asymptotic model set: M is asymptotically self-similar at c=a.
- The Julia set K_{a} in the right window is asymptotically self-similar as well, and both sets are asymptotically similar to each other: (K_{a}-a)≈λ(M-a). Hit p to see the subset of K_{a} corresponding to the subset of M. Hit a to restart.
- There is a sequence of small Mandelbrot sets spiraling towards a. The centers c_{n} are labeled such that the period is n+7. Hit m to see period 7, and hit t repeatedly to increase the period. After n steps, the distance to a is reduced by the factor ρ^{n}, and the image is rescaled with ρ^{2n}. The small Mandelbrot sets are converging to a copy of M, and the decorations are getting longer and thinner.
- There are asymptotic models for ρ^{γn}(M-c_{n}) on the levels γ=1, γ=3/2, γ=7/4, γ=15/8 ... Hit 0, 1, 2, or 3 to choose the level, and hit t repeatedly to increase the order n. Hit p to see corresponding subsets of K_{c}.
- To choose any Misiurewicz point a, set the current point close to it, hit x, and enter the preperiod and period. The commands a, t, and p refer to the new Misiurewicz point afterwards. To enable the levels 0, 1, 2, 3, and m, you must set the current parameter inside the hyperbolic component around the center c_{0} before hitting x. E.g., hit i and paste “-0.04421 0.98658” for a component of period 5. Then hit x and enter “1,2” to find a Misiurewicz point of preperiod 1 and period 2, which is a=i.
Download Juliette. See the
context menu for an overview of commands and keyboard shortcuts.
Or read this introduction.