The logistic map can be related to the real axis of the Mandelbrot set, looking at the different cycle lengths as you pass through all the various nodules along the real axis. But there are other nodules around the outside of the first region of the set, and they are imperfectly self similar. Is there a curve that starts in the first region, but instead of going along the real axis, it curves up or down into one of the nodules where the cycle length is 3, and then continues through that nodule into another, and another, the same way the real axis passes through nodules perfectly where they join each other, giving a different bifurcation diagram sort of like the logistic map, but with different behavior? I've been curious about this for a while, and if someone can point me in the direction of what function would define such a curve, or of a paper showing that no such curve exists, I would really appreciate it. I don't have the time or knowhow to find such a curve myself.
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- $\begingroup$ It is known that the Mandelbrot set is "almost path connected" See Theorem 5.6 in arxiv.org/abs/math/9902156 but usually such a curve is not smooth and difficult (or impossible) to write down. Also, there are small copies of the Mandelbrot set, and the dynamics in such a copy can be related to the corresponding parameter in terms of renormalization. $\endgroup$inyo– inyo2025-03-17 10:21:31 +00:00Commented Mar 17 at 10:21
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