3 ms·
Yes, but how exactly are the bulbs arranged? Wikipedia says 'Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main ca
by OscarCunningham 3y ago
Yes, but how exactly are the bulbs arranged? Wikipedia says 'Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy'. Which sequences of bulbs have little copies at the end of them? And how do the little copies attach?
- clintonc 3y agoThe combinatorics of how the Mandelbrot set is put together is well-studied, and rather independent of MLC. The arrangement of the bulbs on the boundary of the "main cardiod" (which is where there is an attracting fixed point) is described here: https://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_and_period_bulbs https://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_a.... Generally, the patterns are given by something called Lavaur's Algorithm; see https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/lavaurs https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_com... for some explanation. Attachment points are always at the "root" of the Mandelbrot set, which is the cusp of the main cardioid. A consequence of MLC is that the combinatorial picture given by Lavaur's algorithm and related analyses is "complete" -- all dynamical information is available from the combinatorial models.
- OscarCunningham 3y agoThank you, that's very helpful!