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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