Feigenbaum's bifurcation diagram can be found in the Mandelbrot set. Click on the thread to see another striking animation showing this in conjunction with the corresponding Julia sets.https://twitter.com/bernatree/status/930192390010306562 …
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The De Rham curves/Takagi function in a polar plot of the Cauliflower (c=0.25) might be integrable. In the appropriate hyperbolic metric all of the (forget complex) Fatou components have the same area. The total area would not be unattainable, and it would be invariant...
by the action of a discrete group. The total Euclidean area would be some kind of modular form? I will send you email.
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